All Calculus Sl

1.

Find the equation of the normal to the curve with equation 3

y = x + 1 at the point (1, 2).

Working:

Answer : ......................................................................... (Total 4 marks)

2.

The graph represents the function

f : x

p cos x, p ∈

.

y 3

π 2

–

x

3

1

Find (a )

the value of p ;

(b) (b)

the the are areaa of of the the shad shaded ed regi region on..

Working:

Answers: (a) .......... ............... ........... ............ ............ ............ ........... .......... ........... ............ ........... ..... (b) ............ .................. ............ ............ ............ ............ ........... .......... ........... ............ ........ .. (Total 4 marks)

2

3.

Differentiate with respect to x

3 − 4x

(a) (b)

e

sin x sin x

Working:

Answers: (a) ............ .................. ............ ............ ............ ............ ........... .......... ........... ............ ........ .. (b) .......... ............... ........... ............ ............ ............ ........... .......... ........... ............ ........... ..... (Total 4 marks)

4.

The function f is given by

2 x + 1 f ( x x) = x − 3 , x ∈ (a )

(i)

, x ≠ 3.

Show that y = 2 is an asymptote of the graph of y y = f ( x x). (2)

(ii) (ii)

Find Find the the vert vertic ical al asy asymp mptot totee of of the the gra graph ph.. (1)

(iii) (iii)

Write Write down down the coordi coordinat nates es of the the point point P at which the asymptotes intersect. (1)

(b)

Find Find the the point pointss of inters intersect ection ion of of the the graph graph and the axes. axes. (4)

(c) (c)

Hen Hence sket sketch ch the the grap graph h of y y = f ( x x), showing the asymptotes by dotted lines. (4)

(d)

Show that f ′ ′ ( x x) =

−7 ( x − 3) 2

and hence find the equation of the tangent at 3

the point S where x = 4. (6)

(e) (e)

The The tange angent nt at the the poin pointt T on the graph is parallel to the tangent at S . Find the coordinates of T . (5)

(f)

Show that P is the midpoint of [ ST ]. ]. (l) (Total 24 marks)

5.

The function f is such that f ″ ( x x) = 2 x – 2. When the graph of f f is drawn, it has a minimum point at (3, –7). (a )

2

Show that f ′ ′ ( x x) = x – 2 x – 3 and hence find f ( x x). (6)

(b)

Find f (0), f (–1) and f ′ ′ (–1). (3)

(c) (c)

Hen Hence sket sketch ch the the grap graph h of f f , labelling it with the information obtained in part (b). (4)

(Note: It is not necessary to find the coordinates of the points where the graph cuts the x-axis.) (Total 13 marks)

x

6.

2 The diagram shows part of the graph of y = e .

y

y =

x 2

e

P l n

(a) (a)

2

x

Find Find the the coo coord rdin inat ates es of the the poi point nt P , where the graph meets the y-axis. (2)

The shaded region between the graph and the x-axis, bounded by x = 0 and x = ln 2, is rotated through 360° about the x-axis. 4

(b)

Write Write down down an integ integral ral whic which h repres represent entss the volu volume me of the the solid solid obta obtaine ined. d. (4)

(c) (c)

Sho Show tha that this this volu volume me is π . (5) (Total 11 marks)

7.

2

The parabola shown has equation y = 9 x.

y

2

=

9

x

y P

M

Q

(a )

x

Verify that the the point int P (4, 6) is on the parabola. (2)

The line ( PQ PQ) is the normal to the parabola at the point P , and cuts the x-axis at Q. (b)

(i)

Find the equation of ( PQ) in the form ax + by + c = 0. (5)

(ii) (ii)

Find Find the the co coordin rdinaates tes of of Q. (2)

5

S is the point (c )

  94 , 0   .  

Verify that SP = SQ. (4)

(d)

The line ( PM ) is parallel to the x-axis. From part (c), explain why ˆ (QP ) bisects the angle SPM. (3) (Total 16 marks)

8.

2

The diagram shows part of the graph of y = 12 x (1 – x x).

y

0 (a)

x

Write Write down down an integr integral al which which repr represe esents nts the area area of of the the shade shaded d regio region. n.

6

(b) (b)

Find Find the the are areaa of of the the shad shaded ed regi region on..

Working:


7

9.

Differentiate with respect to x: 2

2

(a )

( x x + l) .

(b)

1n(3 x – 1).

Working:


10.

1 x The diagram shows part of the graph of y = . The area of the shaded region is 2 units. y

0

1

a

x

8

Find the exact value of a.

Working:

Answer: ...................................................................... (Total 4 marks)

øR 11. 11.

The diagram shows the graph of the function f given by

f ( x x) = A sin

  π x    2  + B,

for 0 ≤ x ≤ 5, where A and B are constants, and x is measured in radians.

y

( 1

, 3

)

( 5

,

2 ( 0

,

1

)

x 0

1

2

3 ( 3

4 ,

–

1

5

)

The graph includes the points (1, 3) and (5, 3), which are maximum points of the graph. (a) (a)

Writ Wr itee down down the the valu values es of f f (1) and f (5). (2)

(b) (b)

Sho Show tha that the the perio eriod d of f f is 4. (2)

9

The point (3, –1) is a minimum point of the graph. (c )

Show that A = 2, and find the value of B B. (5)

(d)

Show that f ′ ′ ( x x) = π cos

  π x    2  . (4)

The line y = k – π x is a tangent line to the graph for 0 ≤ x ≤ 5. (e )

Find (i) (i)

the the poi point nt wher wheree this this tang tangen entt mee meets ts the the cur curve ve;;

(ii)

the value of k . (6)

(f)

Solve the equation f ( x x) = 2 for 0 ≤ x ≤ 5. (5) (Total 24 marks)

10

(a) (a) Find Find the the equa equatio tion n of the the tang tangen entt line line to to the the curv curvee y = ln x at the point (e, 1), and verify that the origin is on this line.

12.

(4)

(b)

d Show that d x ( x x ln x – x x) = ln x. (2)

(c)

The diagra diagram m shows shows the region region enclos enclosed ed by by the the curve curve y = ln x, the tangent line in part (a), and the line y = 0. y

1

( e

1

0

2

,

1

)

x

3

Use the result of part (b) to show that the area of t his region is

1 2

e – 1. (4) (Total 10 marks)

13.

2

A curve has equation y = x( x x – 4) . (a )

For th this curve fi find (i)

the x-intercepts;

(ii) (ii)

the the coo coord rdina inate tess of of the the maxim maximum um poin point; t;

(iii) (iii)

the coordi coordinat nates es of of the the point point of infl inflexi exion. on. (9)

(b)

Use your your answe answers rs to to part part (a) (a) to to sketc sketch h a grap graph h of the curve curve for for 0 ≤ x ≤ 4, clearly indicating the features you have found in part (a). (3)

11

(c )

(i) On your sketch indicate by shading the region whose area is given by the following integral: 4

∫ x( x − 4) dx. 2

0

(ii)

Explain, Explain, using using your your answer answer to part part (a), (a), why the value value of this integral integral is greater greater than than 0 but less than 40. (3) (Total 15 marks)

14.

2

Find the coordinates of the point on the graph of y = x – x at which the tangent is parallel to the line y = 5 x.

Working:


12

15.

  π 2   If f f ′ ( x x) = cos x, and f   = – 2, find f ( x x). Working:


13

16.

3

Let f ( x x) = x .

(a )

f (5 + h) − f (5) h Evaluate for h = 0.1.

(b)

f (5 + h) − f (5) h What number does approach as h approaches zero?

Working:


17.

The main runway at Concordville airport is 2 km long. An airplane, landing at Concordville, touches down at point T, and immediately starts to slow down. The point A is at the southern end of the runway. A marker is located at point P on the runway. P

A

T 2

k

B

m

Not to scale

14

As the airplane slows down, its distance, s, from A, is given by 2

s = c + 100t – 4t , where t is the time in seconds after touchdown, and c metres is the distance of T from A.

(a) (a)

The The airp airplan lanee tou touch ches es down down 800 800 m from from A, ( ie c = 800). (i)

Find the dist distanc ancee travel travelled led by by the airp airplan lanee in the the first first 5 secon seconds ds after after touc touchdo hdown. wn. (2)

(ii) (ii)

Write Write down down an expres expressio sion n for for the the veloci velocity ty of of the airpla airplane ne at at time time t seconds after touchdown, and hence find the velocity after 5 seconds. (3)

–1

The airplane passes the marker at P with a velocity of 36 m s . Find (iii) (iii)

how many many seco seconds nds afte afterr touchd touchdown own it it passes passes the the marker; marker; (2)

(iv) (iv)

the the dis dista tanc ncee fro from m P to A. (3)

(b)

Show Show that if the the airpla airplane ne touche touchess down down before before reach reaching ing the the point point P, it can can stop stop before before reaching the northern end, B, of the runway. (5) (Total 15 marks)

(a) Sketch th the gr graph of of y y = π sin x – x, –3 ≤ x ≤ 3, on millimetre square paper, using a scale of 2 cm per unit on each axis.

18.

Label and number both axes and indicate clearly the approximate positions of the x-intercepts and the local maximum and minimum points. (5)

(b) (b)

Find Find the the sol solut utio ion n of of the the equa equati tion on

π sin x – x x = 0,

x > 0. (1)

15

(c) (c)

Find Find the the inde indefi fini nite te inte integr gral al

∫ (π sin x − x)dx and hence, or otherwise, calculate the area of the region enclosed by the graph, the x-axis and the line x = 1. (4) (Total 10 marks)

19.

The diagram shows part of the graph of the curve with equation 2 x

y = e

cos x.

y

P a( ,

0

(a )

b

)

x

d y 2 x Show that d x = e (2 cos x – sin – sin x). (2)

d 2 y (b)

2

Find d x . (4)

16

There is an inflexion point at P ( a, b). (c) (c)

Use Use the the resu result ltss from from par parts ts (a) (a) and and (b) (b) to to prov provee that that::

(i)

3; tan a = 4 (3)

(ii) (ii)

2a

the the gra gradi dien entt of of the the curv curvee at at P is e . (5) (Total 14 marks)

20.

A curve with equation y =f ( x x) passes through the point (1, 1). Its gradient function is f ′ ′ ( x x) = –2 x + 3. Find the equation of the curve.

Working:


17

21.

3

Given that f ( x x) = (2 x + 5) find (a)

(b)

f ′ ′ ( x);

∫ f ( x)dx.

Working:


18

22.

1 The diagram shows the graph of the function y = 1 + x , 0 < x ≤ 4. Find the exact value of the area of the shaded region.

4 1 1= –+ x

y 3 2 1

1

3

1

0

1

3

2

4

Working:


23.

A rock-climber slips off a rock-face and falls vertically. At first he falls freely, but after 2 seconds a safety rope slows him down. The height h metres of the rock-climber after t seconds of the fall is given by: 2

0 ≤ t ≤ 2

h = 50 – 5 t , 2

h = 90 – 40 t + 5t ,

(a) (a)

2≤ t≤ 5

Find Find the the heig height ht of the the roc rockk-cl climb imber er when when t = 2. (1)

19

(b)

Sketch a graph of h against t for 0 ≤ t ≤ 5. (4)

(c )

dh Find dt for: (i)

0 ≤ t ≤ 2

(ii)

2 ≤ t ≤ 5 (2)

(d) (d)

Find Find the the velo veloci city ty of of the the rock rock-c -cli limb mber er when when t = 2. (2)

(e)

Find Find the the times times when when the the velo velocity city of the rock-c rock-climb limber er is zero zero.. (3)

(f) (f)

Find Find the the min minimu imum m heig height ht of of the the rock rock-c -cli limb mber er for for 0 ≤ t ≤ 5. (3) (Total 15 marks)

20

24.

In this this quest question ion you you shoul should d note note that that radian radianss are used used thro through ughout out.. 2

(a) (i) Sketch the graph of y = x cos x, for 0 ≤ x ≤ 2 making clear the approximate positions of the positive x-intercept, the maximum point and the end points. (ii)

Write down the approximate coordinates of the positive x-intercept, the maximum point and the end-points. (7)

(b)

Find the exact value of the positive x-intercept for 0 ≤ x ≤ 2. (2)

Let R be the region in the first quadrant enclosed by the graph and the x-axis. (c )

(i) (ii) (ii)

Shade R on your diagram.

Write Write down down an integr integral al whic which h repr represe esents nts the area area of of R. R. (3)

(d)

Evaluate Evaluate the the integral integral in part part (c)(ii) (c)(ii),, either either by using using a graphic graphic display display calculator, calculator, or by by using the following information.

d 2 2 d x ( x x sin x + 2 x cos x – 2 sin x) = x cos x. (3) (Total 15 marks)

21

25.

In this this part part of the the quest question ion,, radian radianss are used used throu througho ghout. ut.

The function f is given by 2

f ( x x) = (sin x) cos x. The following diagram shows part of the graph of y = f ( x x).

y

A

C

B

x

O

The point A is a maximum point, the point B lies on the x-axis, and the point C is a point of inflexion.

(a )

Give the period of f f . (1)

(b) (b)

From From cons consid ider erat atio ion n of of the the gra graph ph of of y y = f ( x x), find to an accuracy of one significant figure the range of f . (1)

22

(c )

Find f ′ ′ ( x x).

(i)

(ii) (ii)

Henc Hencee sho show w that that at the the poi point nt A, cos cos x =

(iii (iii))

Find Find the the exac exactt max maxim imum um valu value. e.

1 3.

(9)

(d) (d)

Fin Find the the exac exactt va value lue of of th the x-coordinate at the point B. (1)

(e )

(i) (ii) (ii)

Find

∫ f ( x) d x.

Find the area area of of the the shaded shaded region region in the diagram diagram.. (4)

(f)

3

Given that f ″ ″ ( x x) = 9(cos x) – 7 cos x, find the x-coordinate at the point C. (4) (Total 20 marks)

26.

2

Let f ′ ′ ( x x) = 1 – x x . Given that f (3) = 0, find f ( x x).

Working:

Answer: .................................................................... (Total 4 marks)

23

27.

2

Given the function f ( x x) = x – 3bx + (c + 2), determine the values of b and c such that f (1) = 0 and f ′ ′ (3) = 0.

Working:

Answer: .................................................................... (Total 4 marks)

x-coordinate at the point B. (1)

(e)

∫ f ( x)

(i)

Find

(a )

Draw th the gr graph of y y = π + x cos x, 0 ≤ x ≤ 5, on millimetre square graph paper, using a scale of 2 cm per unit. Make clear

28.

Note: Radians are used throughout this question.

(i)

the in integer va values of of x x and y on each axis;

(ii) (ii)

the the appr approx oxim imat atee posi positi tion onss of the the x-intercepts and the turning points. (5)

(b)

Without the use of a calculator , show that π is a solution of the equation π + x cos x = 0. (3)

(c) (c)

Find Find ano anoth ther er sol solut utio ion n of the the equ equat atio ion n π + x cos x = 0 for 0 ≤ x ≤ 5, giving your answer to six significant figures. (2)

24

(d)

Let R be the region enclosed by the graph and the axes for 0 ≤ x ≤ π . Shade R on your diagram, and write down an integral which represents the area of R . (2)

(e) (e)

Evalu Evaluat atee the the integ integra rall in part part (d) (d) to to an accur accurac acy y of six significant figures. (If you consider d ( x sin x + cos x ) = x cos x .) it necessary, you can make use of the result d x (3) (Total 15 marks)

29.

A ball is thrown vertically upwards into the air. The height, h metres, of the ball above the ground after t seconds is given by 2

h = 2 + 20 t – 5t , t ≥ 0 (a )

Find the initial height above the ground of the ball (that is, its height at the instant when it is released). (2)

(b)

Show Show that that the the heigh heightt of the ball ball after after one second second is 17 17 metre metres. s. (2)

(c) (c)

At a late laterr time time the the ball all is is again at a height of 17 metres. (i) (i)

Write rite down own an equa equati tion on tha that t must satisfy when the ball is at a height of 17 metres.

(ii)

Solve lve th the eq equatio ation n algebraically. (4)

25

dh (d)

Find dt .

(i) (ii)

Find the initial velocity of the ball (that is, its velocity at the instant when it is released).

(iii)

Find when the ball reaches its maximum height.

(iv) (iv)

Find Find the the maxi maximu mum m heig height ht of of the the ball. ball. (7) (Total 15 marks)

30.


2 x f ( x) =1 – 1 + x 2

(i) To display the graph of y = f ( x x) for –10 ≤ x ≤ 10, a suitable interval for y y, a ≤ y ≤ b must be chosen. Suggest appropriate values for a and b .

(a )

(ii) (ii)

Give Give the the equ equat atio ion n of the the asy asymp mpto tote te of of the the grap graph. h. (3)

f ′ ( x) = (b)

Show that

2 x 2 – 2 (1 + x 2 ) 2

. (4)

(c)

Use your answer to part (b) to find the coordinates of the maximum point of the graph. (3)

26

(d)

(i)

Either by inspection or by using an appropriate substitution, find

∫ f ( x) dx (ii)

Hence find the exact area of the region enclosed by the graph of f , the x-axis and the y-axis. (8) (Total 18 marks)

1 31.

The point P ( 2

, 0 ) lies on the graph of the curve of y = sin (2 x –1).

Find the gradient of the tangent to the curve at P.

Working:

Answer: ....................................................................... (Total 4 marks)

27

32.

Find

(a)

∫ sin (3 x + 7)dx;

(b)

∫ e

– 4 x

dx

.

Working:

Answers: (a) ............ .................. ........... .......... ........... ............ ............ ............ ............ ............ ........ .. (b) ........... ................. ............ ............ ............ ............ ............ ............ ............ ........... ....... .. (Total 4 marks)

33.

f ( x) =


1n 2 x

f ′ ( x) = (a )

(i)

Show that

x

,

x

1 – 1n 2 x

x 2

> 0. .

Hence (ii) (ii)

prov rove th that the the gra grap ph of of f f can have only one local maximum or minimum point;

(iii) (iii)

find the the coord coordinat inates es of the maxi maximum mum poin pointt on the the graph graph of f f . (6)

28

f ′′ ( x ) (b) (b)

By show showin ing g that that the the sec secon ond d der deriv ivat ativ ivee otherwise, find the coordinates of the point of inflexion on the graph of f .

=

2 1n 2 x – 3 x 3

or

(6)

(c )

The region S is enclosed by the graph of f f , the x-axis, and the vertical line through the maximum point of f f , as shown in the diagram below. y y = h 0 (i)

x

Would Would the the trap trapezi ezium um rule rule overes overestima timate te or or under underest estima imate te the the area area of of S ? Justify your answer by drawing a diagram or otherwise. (3)

(ii)

Find

∫ f ( x) dx , by using the substitution u = ln 2 x, or otherwise. (4)

(iii)

Using

∫ f ( x) dx , find the area of S . (4)

(d)

The Newton Newton–Ra –Raphs phson on meth method od is is to be used used to solv solvee the the equat equation ion f ( x x) = 0. (i)

Show Show that that it is not not poss possibl iblee to find a solut solution ion using using a starti starting ng valu valuee of x1 = 1. (3)

(ii)

Starting wi with x1 = 0.4, calculate successive approximations x2 , x3, ... for the root of the equation until the absolute error is less than 0.01. Give all answers correct to five decimal places. (4) (Total 30 marks)

29

34.

Consider the function f ( x x) = k sin x + 3 x, where k is a constant. (a )

Find f ′ ( x x). π

(b)

When x = 3 , the gradient of the curve of f f ( x x) is 8. Find the value of k.

Working:


30

35.

The diagram below shows part of the graph of the function – x 3 + 2 x 2 + 15 x .

f : x y 4 3 3 2 2 1 1

Q

5

A –

0 5 0 5 0 5 0

3

–

2

P

–

–1 – – –

B 5 1 1 2

1

2

3

4

5

x

0 5 0

The graph intercepts the x-axis at A(–3, 0), B(5, 0) and the origin, O. There is a minimum point at P and a maximum point at Q.

(a) (a)

: The The fun funct ctio ion n may may also also be writt written en in the the for form m f x where a < b. Write down the value of (i)

a;

(ii)

b.

– x( x – a) ( x – b),



(2)

(b)

Find (i)

f ′ ( x x);

(ii)

the exact values of x x at which f '( x x) = 0;

(iii (iii))

the the val value ue of the the fun funct ctio ion n at at Q. (7)

31

(c )

Find the equation of the tangent to the graph of f at O.

(i) (ii) (ii)

This This tang tangen entt cut cutss the the grap graph h of of f f at another point. Give the x-coordinate of this point. (4)

(d) (d)

Dete Determ rmin inee the the area area of the the sha shade ded d reg regio ion. n. (2) (Total 15 marks)

36.

A ball is dropped vertically from a great height. Its velocity v is given by –0.2t –0.2t

v = 50 – 50e

, t ≥ 0

where v is in metres per second and t is in seconds. (a )

Find th the value of of v when (i)

t = 0;

(ii)

t = 10. (2)

(b)

(i) (ii)

Find an expression for the acceleration, a, as a function of t .

What is is the the value lue of of a when t = 0? (3)

(c )

(i)

As t becomes large, what value does v approach?

(ii)

As t becomes large, what value does a approach?

(iii)

Explain Explain the the relation relationship ship between between the answers answers to parts (i) and (ii). (3)

32

(d)

Let y metres be the distance fallen after t seconds. –0.2t –0.2t

(i)

Show that y = 50t + 250e

+ k , where k is a constant.

(ii)

Given that y = 0 when t = 0, find the value of k .

(iii)

Find the the time time required required to fall fall 250 m, giving giving your answer answer correct correct to to four significant figures. (7) (Total 15 marks)

37.

Radian measure is used, where appropriate, throughout the question.

y = Consider the function

3 x – 2 2 x – 5

.

The graph of this function has a vertical and a horizontal asymptote. (a) (a)

Writ Wr itee dow down n the the equa equati tion on of (i)

the ve vertical as asymptote;

(ii) (ii)

the the hori horizo zont ntal al asym asympt ptot ote. e. (2)

d x (b)

Find

dy

, simplifying the answer as much as possible. (3)

(c)

How many many points points of infl inflexi exion on does does the the graph graph of this this func functio tion n have? have? (1) (Total 6 marks)

33

1 38.

The derivative of the function f is given by f ′ ( x x) = 1 + x – 0.5 sin x, for x x ≠ –1. The graph of f passes through the point (0, 2). Find an expression for f ( x f passes x).

Working:

Answer : ...................................................................... (Total 6 marks)

34

39.

Figure 1 shows the graphs of the functions f 1, f 2, f 3, f 4. Figure 2 includes the graphs of the derivatives of the functions shown in Figure 1, eg the derivative of f f 1 is shown in diagram (d). Figure 1

Figure 2

y

y ( a

f 1

) O

O

x

y

y ( b

f 2

O

) O

x

y

x

y ( c

f 3

O

x

)

x

O

y

x

y ( d

f 4

)

O

x

x

O

y ( e

) O

x

35

Complete the table below by matching each function with its derivative. Function

Derivative diagram

f 1

(d)

f 2 f 3 f 4

Working:

(Total 6 marks)

40.

–kx

Consider functions of the form y = e 1

(a )

Show that

∫ 0

e – kx dx

1 – k k = k (1 – e ).

(3)

(b)

Let k = 0.5 –0.5 x

, for –1 ≤ x ≤ 3, indicating the coordinates of the

(i)

Sketch the graph of y y = e y-intercept.

(ii) (ii)

Shad Shadee the the regi region on enc enclos losed ed by by this this gra graph ph,, the the x-axis, y-axis and the line x = 1.

(iii (iii))

Find Find the the area area of this this regi region on.. (5)

d y (c )

(i)

–kx Find d x in terms of k , where y = e . 36

–kx

The point P(1, 0.8) lies on the graph of the function y = e (ii)

Find th the va value of of k in this case.

(iii) (iii)

Find the gradie gradient nt of of the tangen tangentt to the the curve curve at at P.

.

(5) (Total 13 marks)

2 41.

3 Let the function f be defined by f ( x x) = 1 + x , x ≠ –1.

(a )

(i)

Write down the equation of the vertical asymptote of the graph of f f .

(ii) (ii)

Write Write down down the the equati equation on of the the horiz horizont ontal al asymp asymptote tote of of the grap graph h of f f .

(iii (iii))

Sket Sketch ch the the gra graph ph of f f in the domain –3 ≤ x ≤ 3. (4)

– 6 x (b)

(i)

2

3 2 Using the fact that f ′ ( x x) = (1 + x ) , show that the second derivative

(

)

12 x 2 x 3 – 1

f ″ ″ ( x x) =

(ii)

(1 + x 3 ) 3

.

Find the x-coordinates of the points of inflexion of the graph of f . (6)

37

(c) (c)

The The tab table le belo below w giv gives es some some valu values es of f f ( x x) and 2 f ( x x).

x

(i)

f ( x x)

2 f ( x x)

1

1

2

1.4

0.534188

1.068376

1.8

0.292740

0.585480

2.2

0.171703

0.343407

2.6

0.107666

0.215332

3

0.071429

0.142857

Use the trap trapeziu ezium m rule rule with with five five subsub-int interv ervals als to to approx approxima imate te the the integr integral al 3

∫ f ( x) dx. 1

3

(ii)

f ( x ) d x ∫ Given that = 0.637599, use a diagram to explain why your answer is 1

greater than this. (5) (Total 15 marks)

38

42.

Let f ( x x) = (a)

(b)

x 3 . Find

f ′ ′ ( x);

∫ f ( x)dx.

Working:


43.

3

2

The graph of y y = x – 10 x +12 x + 23 has a maximum point between x = –1 and x = 3. Find the coordinates of this maximum point.

Working:


44.

The diagram shows part of the curve y = sin x. The shaded region is bounded by the curve and 3π . 4 the lines y = 0 and x = 39

y

3π 4

3π

2

Given that sin 4 = 2

3π

π

x

2

and cos 4 = – 2 , calculate the exact area of the shaded region.

Working:


40

45.

x

The diagram below shows a sketch of the graph of the function y = sin (e ) where –1 ≤ x ≤ 2, and x is in radians. The graph cuts the y-axis at A, and the x-axis at C and D. It has a maximum point at B.

y B A

–

(a) (a)

1

0

1

C

D

2 x

Fin Find the the coor coord dinat inates es of A. A. (2)

41

(b) (b)

The The coor coordin dinat ates es of C may may be wri writt tten en as (ln (ln k , 0). Find the exact value of k . (2)

(c )

(i)

Write down the y-coordinate of B.

d y (ii)

Find d x .

π (iii (iii))

Henc Hence, e, show show that that at B, x = ln 2 . (6)

(d)

(i) (ii) (ii)

Write down the integral which represents the shaded area.

Eva Evaluat luatee this this int integ egra ral. l. (5)

(e )

(i)

Copy the above diagram into your answer booklet. (There is no need 3

to copy the shading.) On your diagram, sketch the graph of y = x . (ii) (ii)

The The two two grap graphs hs inte inters rsec ectt at the the point point P. P. Find Find the the x-coordinate of P. (3) (Total 18 marks)

46.

In this question, s represents displacement in metres, and t represents time in seconds.

d s (a )

–1 The velocity v m s of a moving body may be written as v = dt = 30 – at , where a is a constant. Given that s = 0 when t = 0, find an expression for s s in terms of a and t.

(5)

Trains approaching a station start to slow down when they pass a signal which is 200 m from the station. (b) (b)

The The veloc elocit ity y of Trai Train n 1 t seconds after passing the signal is given by v = 30 – 5 t. (i) (i)

Writ Wr itee dow down n its its velo veloci city ty as it pass passes es the the sign signal al..

(ii) (ii)

Show Show that that it will will stop stop befo before re reachin reaching g the the station station.. (5)

42

(c)

Train Train 2 slow slowss down down so so that that it stop stopss at the the stati station. on. Its velo velocity city is give given n by d s

v = dt = 30 – at , where a is a constant. (i)

Find, in terms of a, the time taken to stop.

(ii) (ii)

Use your your solut solution ionss to part partss (a) (a) and (c)(i) (c)(i) to to find find the value value of of a. (5) (Total 15 marks)

47.

2 The diagram below shows the shaded region R enclosed by the graph of y y = 2 x 1 + x , the x-axis, and the vertical line x = k .

y y =

2

2

x 1

R

k

x

d y (a )

Find d x . (3)

(b) (b)

2

Usin Using g the the subst ubstit itut utio ion n u = 1 + x or otherwise, show that

∫

2 x 1 + x 2

2

3

2 2

d x = 3 (1 + x )

+ c. (3)

(c )

Given that the the area of R R equals 1, find the value of k . (3) (Total 9 marks)

43

1

48.

5

Consider the function h ( x x) = x . (i) (i)

Find Find the the equa equati tion on of the the tang tangen entt to to the the grap graph h of of h at the point where x = a, (a ≠ 0). Write the equation in the form y = mx + c.

(ii) (ii)

Show Show that that this this tang tangen entt inte inters rsec ects ts the the x-axis at the point (–4 a, 0). (Total 5 marks)

x

49.

3 2 Let f ( x x) = e + 5 cos x. Find f ′ ′ ( x x).

Working:

Answer : .................................................................. (Total 6 marks)

44

3

50.

Given that 3

(a)

∫ g ( x)dx = 10, deduce the value of 1

1

∫ 2 g ( x)d x; 1

3

(b)

∫ ( g ( x) + 4)dx. 1

Working:


51.

–1

An aircraft lands on a runway. Its velocity v m s

at time t seconds after landing is given by the

–0.5t –0.5t

equation v = 50 + 50e (a) (a)

, where 0 ≤ t ≤ 4.

Find Find the the vel veloc ocit ity y of of the the airc aircra raft ft (i)

when it lands;

(ii)

when t = 4. (4)

(b)

Write down an integra integrall which which repres represents ents the distan distance ce travelle travelled d in the first four seconds. seconds. (3)

45

(c)

Calcul Calculate ate the distanc distancee trave travelle lled d in the first first four four seco seconds nds.. (2)

After four seconds, the aircraft slows down (decelerates) at a constant rate and comes to rest when t = 11. (d)

Sketch a graph of velocity against time for 0 ≤ t ≤ 11. Clearly label the axes and mark on the graph the point where t = 4. (5)

(e)

Find Find the const constant ant rate rate at whic which h the aircr aircraft aft is slow slowing ing down down (dec (decele elerat rating ing)) between between t = 4 and t = 11. (2)

(f) (f)

Calcu Calcula late te the the dist distan ance ce trav travel elle led d by the the aircr aircraf aftt betwe between en t = 4 and t = 11. (2) (Total 18 marks)

52.

Consider the function f ( x x) = cos x + sin x.

π (a )

(i) (ii)

Show that f (– 4 ) = 0.

Find in terms of π , the smallest positive value of x x which satisfies f ( x x) = 0. (3)

46

x

The diagram shows the graph of y y = e (cos x + sin x), – 2 ≤ x ≤ 3. The graph has a maximum turning point at C( a, b) and a point of inflexion at D. y 6

C a ( ,

b

)

4 D 2

–

2

–

1

1

2

3

x

d y (b)

Find d x . (3)

(c )

Find the exact value of a and of b. (4)

π

(d)

Show that at D, y =

2e 4 . (5)

(e) (e)

Find Find the the are areaa of of the the shad shaded ed regi region on.. (2) (Total 17 marks)

47

d y 53.

3 It is given that d x = x +2 x – 1 and that y = 13 when x = 2.

Find y in terms of x x.

Working:

Answer : .................................................................. (Total 6 marks)

48

(a)

54.

(b)

Find ∫ (1 + 3 sin ( x + 2))d x.

The diagra diagram m show showss part part of the graph graph of the functio function n f ( x x) = 1 + 3 sin ( x + 2). a

∫ f ( x)d x . 0

The area of the shaded region is given by y

4

2

–

4

–

2

0

–

2

4

x

2

Find the value of a.

Working:

Answers: (a) .......... ............... ........... ............ ............ ............ ........... .......... ........... ............ ........... ..... (b) .......... ............... ........... ............ ............ ............ ........... .......... ........... ............ ........... ..... (Total 6 marks)

49

55.

The diagram shows the graph of y y = f ( x x).

y

0

x

50

On the grid below sketch the graph of y = f ′ ′ ( x x).

y

x

0

(Total 6 marks)

56.

–2 x

Consider the function f ( x x) = 1 + e (a )

(i) (ii) (ii)

.

Find f ′ ( x x).

Expl Explain ain brie briefly fly how how thi thiss sho shows ws that that f ( x x) is a decreasing function for all values of x (ie that f ( x x) always decreases in value as x increases). (2)

1 Let P be the point on the graph of f where x = – 2 . (b) (b)

Find Find an expr expres essi sion on in term termss of of e for for (i)

the y-coordinat inatee of P;

(ii) (ii)

the the grad gradie ient nt of of the the tang tangen entt to the the cur curve ve at at P. (2)

51

(c)

Find Find the equa equatio tion n of the the tangen tangentt to the the curve curve at P, P, giving giving your your answ answer er in the the form form y = ax + b. (3)

(d)

(i)

Sketch the curve of f for –1 ≤ x ≤ 2.

1 (ii)

Draw th the ta tangent at at x = – 2 .

(iii) (iii)

Shade Shade the the area area enclo enclosed sed by by the curv curve, e, the the tangen tangentt and the the y-axis.

(iv)

Find ind th this area. (7) (Total 14 marks)

57.

Note: Radians Note: Radians are used throughout used throughout this question.

A mass is suspended from the ceiling on a spring. It is pulled down to point P and t hen released. It oscillates up and down.

d

i a g r a s c a l e

P

Its distance, s cm, from the ceiling, is modelled by the function s = 48 + 10 cos 2 πt where t is the time in seconds from release. (a )

(i) (ii) (ii)

What is the distance of the point P from the ceiling?

How How lon long g is is it it unt until il the the mas masss is is nex nextt at P? (5)

52

d s (b)

(i) (ii) (ii)

Find dt .

Wher Wheree is the the mass mass whe when n the the velo veloci city ty is is zero zero?? (7)

A second mass is suspended on another spring. Its distance r cm from the ceiling is modelled by the function r = 60 + 15 cos 4 π t . The two masses are released at the same instant. (c )

Find th the value of of t when they are first at the same distance below the ceili ng. (2)

(d)

In the the first first three three seco seconds nds,, how many many time timess are the the two two masses masses at at the same same heig height? ht? (2) (Total 16 marks)

2 x 2 – 13 x + 20 58.

Consider the function f given by f ( x x) = A part of the graph of f f is given below.

( x – 1) 2

,

x ≠ 1.

y

0

x

The graph has a vertical asymptote and a horizontal asymptote, as shown. (a )

Write down the equation of the vertical asymptote. (1)

53

(b)

f (100) = 1.91

f (–100) = 2.09 f (1000) = 1.99

(i)

Evaluate f (–1000).

(ii)

Write down the equation of the horizontal asymptote. (2)

9 x – 27 3

(c )

( – 1) , Show that f ′ ( x x) = x

x ≠ 1. (3)

72 – 18 x 4 The second derivative is given by f ″ ( x x) = ( x – 1) ,

(d)

x ≠ 1.

Using va values of f f ′ ( x x) and f ″ ( x x) explain why a minimum must occur at x = 3. (2)

(e) (e)

Ther Theree is is a point point of infl inflex exio ion n on on the the grap graph h of of f f . Write down the coordinates of this point. (2) (Total 10 marks)

54

59.

3

2

Let f ( x x) = x – 2 x – 1. (a )

Find f ′ ′ ( x x).

(b) (b)

Find Find the the gra gradi dien entt of of the the curv curvee of of f f ( x x) at the point (2, –1).

Working:

Answers: (a) (a) ………… ……………… ………… ………… ………… ………… ………… …… (b) (b) ………… ……………… ………… ………… ………… ………… ………… …… (Total 6 marks)

60.

–1

A car starts by moving from a fixed point A. Its velocity, v m s

after t seconds is given by

– t t

v = 4t + 5 – 5e . Let d be the displacement from A when t = 4. (a) (a)

Write Wr ite down down an integ integra rall whi which ch repr repres esen ents ts d .

(b)

Calcu lculate the the va value lue of of d .

Working:

Answers: (a) (a) ………… ……………… ………… ………… ………… ………… ………… …….. .. (b) (b) ………… ……………… ………… ………… ………… ………… ………… …….. .. (Total 6 marks)

55

1 (a) Conside ider th the fu function ion f ( x x) = 2 + x − 1 . The diagram below is a sketch of part of the graph of y y = f ( x x).

61.

y 5 4 3 2 1 –

5 –

4 –

3 –

10

2 –

1

–

1

–

2

–

3

–

4

–

5

2

3

4

5

x

Copy and complete the sketch of f f ( x x). (2)

(b)

(i) (ii) (ii)

Write down the x-intercepts and y-intercepts of f f ( x x).

Writ Wr itee down down the the equ equat ation ionss of the the asy asymp mpto totes tes of f f ( x x). (4)

(c )

(i) (ii) (ii)

Find f ′ ( x x).

There There are are no no maxi maximum mum or minimu minimum m poin points ts on on the the grap graph h of of f f ( x x). Use your expression for f f ′ ( x x) to explain why. (3)

The region enclosed by the graph of f f ( x x), the x-axis and the lines x = 2 and x = 4, is labelled A, as shown below.

y 5 4 3 2

A

1 –

5 –

4 –

3 –

2 –

10

1

–

1

–

2

–

3

–

4

–

5

2

3

4

5

x

56

(d)

(i)

Find

∫ f ( x) d x.

(ii) (ii)

Write Write down down an expr express ession ion that that repre represen sents ts the the area area labelle labelled d A.

(iii (iii))

Find Find the the area area of A A. (7) (Total 16 marks)

62.

3

2 Let f ( x x) = 6 x . Find f ′ ( x).

Working:

Answer : …………………………………………........ (Total 6 marks)

57

63.

The displacement s metres of a car, t seconds after leaving a fixed point A, is given by 2

s = 10t – 0.5t . (a) (a)

Calcu alcula late te the the vel velo ocity city when when t = 0.

(b)

Calcu lculate the the va value lue of of t when the velocity is zero.

(c)

Calcul Calculate ate the the disp displace lacemen mentt of the the car car from from A when when the the velo velocit city y is zero zero..

Working:

Answers: (a) (a) ………… ……………… ………… ………… ………… ………… ………… …….. .. (b) (b) ………… ……………… ………… ………… ………… ………… ………… …….. .. (c) (c) ………… ……………… ………… ………… ………… ………… ………… …….. .. (Total 6 marks)

58

64.

The population p of bacteria at time t is given by p = 100e

0.05t 0.05t

.

Calculate (a )

the value of p when t = 0;

(b) (b)

the the rate rate of of incre increas asee of of the the popu populat lation ion when when t = 10.

Working:

Answers: (a) (a) ………… ……………… ………… ………… ………… ………… ………… …….. .. (b) (b) ………… ……………… ………… ………… ………… ………… ………… …….. .. (Total 6 marks)

1 65.

–2 x

The derivative of the function f is given by f ′ ( x x) = e

+ 1 − x , x < 1.

The graph of y y = f ( x x) passes through the point (0, 4). Find an expression for f ( x x).

Working:

Answer : …………………………………………........ (Total 6 marks)

59

3

66.

Let f be a function such that (a )

∫ f ( x) dx = 8 . 0

Deduce th the va value of 3

∫ 2 f ( x) d x; 0

(i)

3

∫ ( f ( x) + 2) d x. 0

(ii) d

(b)

f ( x − 2)dx = 8 ∫ If , write down the value of c and of d . c

Working:

Answers: (a) (i) ......... ............... ............ ............ ............ ........... .......... ........... ............ ........ (ii) ....................................................... ....................................................... (b) c = ......................., d = ....................... (Total 6 marks)

60

67.

Part of the graph of the periodic function f is shown below. The domain of f f is 0 ≤ x ≤ 15 and the period is 3.

f (

x ) 4 3 2 1 0 0

(a )

1

2

3

4

5

6

7

8

9

1

0

x

Find (i)

f (2);

(ii)

f ′ (6.5);

(iii) f ′ (14). (b) (b)

How How many many solu soluti tion onss are are the there re to to the the equa equatio tion n f ( x x) = 1 over the given domain?

Working:

Answers: (a) (a) (i) (i) ………… ……………… ………… ………… ………… ………… ……… … (ii) ……………………………………… (iii) ……………………………………… (b) …………………………………………… (Total 6 marks)

61

68.

2

The function f ( x x) is defined as f ( x x) = –( x – h) + k . The diagram below shows part of the graph of f f ( x x). The maximum point on the curve is P (3, 2). y 4

P

( 3

,

2

)

2 –

(a )

1

1 –

2

–

4

–

6

–

8

–

1

0

–

1

2

2

3

4

5

6

x

Write down th the va value lue of of (i)

h;

(ii)

k . (2)

(b)

2

Show that f ( x x) can be written as f ( x x) = – x + 6 x – 7. (1)

(c )

Find f ′ ( x x). (2)

The point Q lies on the curve and has coordinates (4, 1). A straight line L, through Q, is perpendicular to the tangent at Q. (d)

(i)

Calculate the gradient of L .

(ii)

Find ind th the eq equatio tion of of L L.

(iii)

The line L intersects the curve again at R. Find the x-coordinate of R. (8) (Total 13 marks)

62

Let f ( x x) = 1 + 3 cos (2 x) for 0 ≤ x ≤ π , and x is in radians.

69.

(a )

Find f ′ ′ ( x x).

(i) (ii)

Find ind the va values fo for x x for which f ′ ′ ( x x) = 0, giving your answers in terms of π . (6)

π The function g ( x x) is defined as g ( x x) = f (2 x) – 1, 0 ≤ x ≤ 2 . (b)

(i)

The graph of f may be transformed to the graph of g g by a stretch in the

x-direction with scale factor this other transformation. (ii) (ii)

1 2

followed by another transformation. Describe fully

Find Find the the solu soluti tion on to the the equa equati tion on g ( x x) = f ( x x) (4) (Total 10 marks)

70.

Let h ( x x) = ( x x – 2) sin ( x – 1) for –5 ≤ x ≤ 5. The curve of h ( x x) is shown below. There is a minimum point at R and a maximum point at S. The curve intersects the x-axis at the points (a, 0) (1, 0) (2, 0) and ( b, 0).

y 4 3 2

(a –

5 –

4 –

, 3 –

0

)

2 –

1

S

1

(b 1

–

1

–

2

–

3

–

4

–

5

–

6

–

7

R

2

3

4

, 5

x

63

(a )

Find ind th the ex exact va value lue of of (i)

a;

(ii)

b. (2)

The regions between the curve and the x-axis are shaded for a ≤ x ≤ 2 as shown. (b)

(i) Write down an expression which represents the total area of the shaded regions. (ii) (ii)

Calc Calcul ulat atee thi thiss tota totall area area.. (5)

(c )

(i) (ii) (ii)

The y-coordinate of R is –0.240. Find the y-coordinate of S.

Henc Hencee or oth other erwi wise se,, find find the the ran range ge of of valu values es of of k for which the equation ( x x – 2) sin ( x – 1) = k has four distinct solutions. (4) (Total 11 marks)

1 71.

2 Let f ( x x) = 1 + x .

(a)

Write Write down down the the equat equation ion of the the horizo horizonta ntall asympt asymptote ote of of the the graph graph of of f f . (1)

(b)

Find f ′ ( x x). (3)

6 x 2 (c) (c)

−2

2 3 The The sec secon ond d der deriv ivat ativ ivee is is giv given en by f ″ ( x x) = (1 + x ) .

Let A be the point on the curve of f where the gradient of the tangent is a maximum. Find the x-coordinate of A. (4)

64

1 (d)

1

Let R be the region under the graph of f f , between x = – 2 and x = 2 , as shaded in the diagram below

y 2

1

R

–

1

–

1 2

–

x

1

1 2

1

Write down the definite integral which represents the area of R. (2) (Total 10 marks)

Let y = g ( x x) be a function of x x for 1 ≤ x ≤ 7. The graph of g g has an inflexion point at P, and a minimum point at M.

72.

Partial sketches of the curves of g g ′ and g ″ are shown below.

g (

x)

’

g (

6

6

5

5

4

4

3

3

2

2

1

1

0

1

–

1

–

)x

’ ’

2

3

4

5

6

7

8

x

0

1

–

1

2

–

2

–

3

–

3

–

4

–

4

–

5

–

5

–

6

–

6

y =

’

g (

)x

2

3

y =

5

4

’

’

g

6

(

7

8

x

x)

65

Use the above information to answer the following. (a )

Write down the x-coordinate of P, and justify your answer. (2)

(b)

Write down the x-coordinate of M, and justify your answer. (2)

(c )

Given that g (4) = 0, sketch the graph of g g . On the sketch, mark the points P and M. (4) (Total 8 marks)

73.

The function f is given by f ( x x) = 2sin (5 x – 3). (a )

Find f " f " ( x x).

(b)

Write down

∫ f ( x)dx .

.......................................................................... ................................................................................................................ ........................................................... ..................... .......................................................................... ................................................................................................................ ........................................................... ..................... .......................................................................... ................................................................................................................ ........................................................... ..................... .......................................................................... ................................................................................................................ ........................................................... ..................... .......................................................................... ................................................................................................................ ........................................................... ..................... .......................................................................... ................................................................................................................ ........................................................... ..................... .......................................................................... ................................................................................................................ ........................................................... ..................... .......................................................................... ................................................................................................................ ........................................................... ..................... .......................................................................... ................................................................................................................ ........................................................... ..................... .......................................................................... ................................................................................................................ ........................................................... ..................... .......................................................................... ................................................................................................................ ........................................................... ..................... .......................................................................... ................................................................................................................ ........................................................... ..................... (Total 6 marks)

74.

–1

The velocity v m s

of a moving body at time t seconds is given by v = 50 – 10 t .

66

–2

(a) (a)

Fin Find its its acc accele elerati ration on in m s .

(b) (b)

The The ini inittial ial di displa splace ceme ment nt s is 40 metres. Find an expression for s s in terms of t . .......................................................................... ................................................................................................................ ........................................................... ..................... .......................................................................... ................................................................................................................ ........................................................... ..................... .......................................................................... ................................................................................................................ ........................................................... ..................... .......................................................................... ................................................................................................................ ........................................................... ..................... .......................................................................... ................................................................................................................ ........................................................... ..................... .......................................................................... ................................................................................................................ ........................................................... ..................... .......................................................................... ................................................................................................................ ........................................................... ..................... .......................................................................... ................................................................................................................ ........................................................... ..................... .......................................................................... ................................................................................................................ ........................................................... ..................... .......................................................................... ................................................................................................................ ........................................................... ..................... .......................................................................... ................................................................................................................ ........................................................... ..................... .......................................................................... ................................................................................................................ ........................................................... ..................... (Total 6 marks)

75.

The function f is defined by f : x (a )

– 0.5 x 2 + 2 x + 2.5.

Write down (i)

f ′ ( x x);

(ii)

f ′ (0). (2)

(b)

Let N be the normal to the curve at the point where t he graph intercepts the y-axis. Show that the equation of N N may be written as y = –0.5 x + 2.5. (3)

67

Let g : x (c )

(i) (ii)

– 0.5 x + 2.5



Find the solutions of f ( x x) = g ( x x).

Hence find the coord coordinates inates of the the other other point of intersec intersection tion of the the normal normal and the curve. (6)

(d)

Let R be the region enclosed between the curve and N . (i) (i)

Writ Wr itee dow down n an an exp expre ress ssio ion n for for the the are areaa of of R R.

(ii) (ii)

Henc Hencee writ writee dow down n the the area area of R. (5) (Total 16 marks)

76.

2 x

The diagram below shows the graphs of f f ( x x) = 1 + e , g ( x x) = 10 x + 2, 0 ≤ x ≤ 1.5. y f 1

6

1

2

g

p 8

4

0

(a )

. 5

1

1

x . 5

(i) Write down an expression for the vertical distance p between the graphs of f f and g . (ii)

Given that p has a maximum value for 0 ≤ x ≤ 1.5, find the value of x x at which this occurs. (6)

68

The graph of y y = f ( x x) only is shown in the diagram below. When x = a, y = 5. y

1

6

1

2

8 5 4

0

(b)

(i) (ii)

. a5

1

1

x . 5

–1

Find f ( x x).

Hence show that a = ln 2. (5)

(c)

The regi region on shad shaded ed in the diag diagram ram is is rotat rotated ed throu through gh 360° 360° abou aboutt the x-axis. Write down an expression for the volume obtained. (3) (Total 14 marks)

69

x – 2

h : x 77.

( x – 1) 2

Consider the function

, x ≠ 1.

A sketch of part of the graph of h is given below. A

y

P

x

N

o

t

t o

s c a

l e

B The line (AB) is a vertical asymptote. The point P i s a point of inflexion.

(a )

Write down the equation of the vertical asymptote. (1)

(b)

Find h′( x x), writing your answer in the form

a – x ( x – 1) n where a and n are constants to be determined. (4)

70

h ′ ( x ) = (c )

Given that

2 x – 8 ( x – 1) 4 , calculate the coordinates of P. (3) (Total 8 marks)

78.

5

Let f ( x x) = (3 x + 4) . Find (a)

f ′ ( x);

(b)

∫ f ( x x)d x.

Working:


71

79.

The curve y = f ( x x) passes through the point (2, 6). d y 2 Given that d x = 3 x – 5, find y in terms of x x.

Working:

Answer : ....…………………………………….......... (Total 6 marks)

80.

The table below shows some values of two functions, f and g , and of their derivatives f ′ and g ′.

x

1

2

3

4

f ( x x)

5

4

–1

3

g ( x x)

1

–2

2

–5

f ′ ( x x)

5

6

0

7

g ′ ( x x)

–6

–4

–3

4

72

Calculate the following.

(a)

d dx ( f f ( x x) + g ( x x)), when x = 4; 3

(b)

∫ ( g' ( x) + 6) dx . 1

Working:


81.

The equation of a curve may be written in the form y = a( x x – p p)( x – q). The curve intersects the x-axis at A(–2, 0) and B(4, 0). The curve of y = f ( x x) is shown in the diagram below.

y 4 2 A –

(a )

(i) (ii) (ii)

4

–

B 0

2

2

–

2

–

4

–

6

4

6 x

Write down the value of p and of q.

Given Given that that the point point (6, 8) is is on the curve, curve, find find the the value value of a. 73

(iii) (iii)

2

Write Write the the equat equation ion of the the curv curvee in the form form y = ax + bx + c. (5)

(b)

(i) (ii) (ii)

d y Find d x .

A tangen tangentt is drawn drawn to to the curv curvee at a point point P. P. The grad gradien ientt of this this tange tangent nt is 7. 7. Find the coordinates of P. (4)

(c )

The line L passes through B(4, 0), and is perpendicular to the tangent to the curve at point B. (i)

Find the equation of L L.

(ii)

Find the x-coordinate of the point where L intersects the curve again. (6) (Total 15 marks)

82.

3 x 2 Let f ( x x) = 5 x − 1 .

(a )

Write down the equation of the vertical asymptote of y y = f ( x x). (1)

(b)

ax 2 + bx 2 Find f ′ ( x x). Give your answer in the form (5 x − 1) where a and b ∈

. (4) (Total 5 marks)

74

83.

The function g ( x x) is defined for –3 ≤ x ≤ 3. The behaviour of g g ′ ( x x) and g ″ ( x x) is given in the tables below.

x

–3 < x < –2

–2

–2 < x < 1

1

1 < x < 3

g ′ ( x x)

negative

0

positive

0

negative

x

1 –3 < x < – 2

1 – 2

1 – 2 < x < 3

g ″ ( x x)

positive

0

negative

Use the information above to answer the following. In each case, justify your answer.

(a )

Write down th the va value lue of of x x for which g has a maximum. (2)

(b) (b)

On whic which h int inter erva vals ls is the the valu valuee of of g g decreasing? (2)

(c )

Write down th the va value lue of of x x for which the graph of g g has a point of inflexion. (2)

(d)

Given that g (–3) = 1, sketch the graph of g g . On the sketch, clearly indicate the position of the maximum point, the minimum point, and the point of inflexion. (3) (Total 9 marks)

75

k

84.

∫ x − 2 d x = ln 7, find the value of k . 3

Given

W

1

o

r k i n

g

:

A

n

s w

e

r s :

. . . . . . . . . . . . . . . . . . . . . . . . (Total 6 marks)

85.

3

2

Let f ( x x) = (2 x + 7) and g ( x x) cos (4 x). Find (a)

f ′ ( x x);

(b)

g ′ ( x x).

W

o

r k i n

g

:

A

n

s w

e

r s :

( a

)

. . . . . . . . . . . . . . . . . . .

( b

)

. . . . . . . . . . . . . . . . . . . (Total 6 marks)

76

86.

The following diagram shows a rectangular area ABCD enclosed on three sides by 60 m of fencing, and on the fourth by a wall AB.

Find the width of the rectangle that gives its maximum area.

W

o

r k i n

g

:

A

n

s w

e

r s :

. . . . . . . . . . . . . . . . . . . . . . . . (Total 6 marks)

87.

A particle moves with a velocity v m s (a )

−1

2

given by v = 25 − 4 t where t ≥ 0.

The di displacement, s metres, is 10 when t is 3. Find an expression for s s in terms of t . (6)

(b)

Find t when s reaches its maximum value. (3)

(c) (c)

The The part particl iclee has has a posi positi tive ve dis displ plac acem emen entt for for m ≤ t ≤ n. Find the value of m and the value of n. (3) (Total 12 marks)

88.

The graph of y y = sin 2 x from 0≤ x ≤ π is shown below.

77

The area of the shaded region is 0.85. Find the value of k . (Total 6 marks)

78

(a)

89.

5 x

Let f ( x x) = e . Write down f ′ ( x x).

(b)

Let g ( x x) = sin 2 x. Write down g ′ ( x x).

(c )

Let h ( x x) = e

5 x

sin 2 x. Find h′ ( x x).

........................................................................... ................................................................................................................. ................................................................... ............................. ........................................................................... ................................................................................................................. ................................................................... ............................. ........................................................................... ................................................................................................................. ................................................................... ............................. ........................................................................... ................................................................................................................. ................................................................... ............................. ........................................................................... ................................................................................................................. ................................................................... ............................. ........................................................................... ................................................................................................................. ................................................................... ............................. ........................................................................... ................................................................................................................. ................................................................... ............................. ........................................................................... ................................................................................................................. ................................................................... ............................. ........................................................................... ................................................................................................................. ................................................................... ............................. ........................................................................... ................................................................................................................. ................................................................... ............................. ........................................................................... ................................................................................................................. ................................................................... ............................. ........................................................................... ................................................................................................................. ................................................................... ............................. ........................................................................... ................................................................................................................. ................................................................... ............................. (Total 6 marks)

79

90.

The following diagram shows part of the curve of a function ƒ. The points A, B, C, D and E lie on the curve, where B is a minimum point and D is a maximum point.

(a) (a)

Comp Comple lete te the the fol follo lowin wing g tabl table, e, not notin ing g wheth whether er ƒ′( ƒ′( x x) is positive, negative or zero at the given points. A

B

E

f ′ ( x x)

(b) (b)

Comp Comple lete te the the foll follow owin ing g table table,, notin noting g wheth whether er ƒ′′( ƒ′′( x x) is positive, negative or zero at the given points. A

C

E

ƒ′′ ( x x) (Total 6 marks)

80

91.

−1

3

The velocity, v m s , of a moving object at time t seconds is given by v = 4t − 2t . When t = 2, the displacement, s, of the object is 8 metres. Find an expression for s s in terms of t . ........................................................................... ................................................................................................................. ................................................................... ............................. ........................................................................... ................................................................................................................. ................................................................... ............................. ........................................................................... ................................................................................................................. ................................................................... ............................. ........................................................................... ................................................................................................................. ................................................................... ............................. ........................................................................... ................................................................................................................. ................................................................... ............................. ........................................................................... ................................................................................................................. ................................................................... ............................. ........................................................................... ................................................................................................................. ................................................................... ............................. ........................................................................... ................................................................................................................. ................................................................... ............................. ........................................................................... ................................................................................................................. ................................................................... ............................. ........................................................................... ................................................................................................................. ................................................................... ............................. ........................................................................... ................................................................................................................. ................................................................... ............................. ........................................................................... ................................................................................................................. ................................................................... ............................. ........................................................................... ................................................................................................................. ................................................................... ............................. (Total 6 marks)

81

92.

The graph of a function g is given in the diagram below.

The gradient of the curve has its maximum value at point B and its minimum value at point D. The tangent is horizontal at points C and E. (a)

Comple Complete te the the tabl tablee below below,, by stating stating whethe whetherr the the first first deriva derivative tive g ′ is positive or negative, and whether the second derivative g ′′′′ is positive or negative. Interval

g ′

g ′′′′

a < x < b e < x < ƒ

(b)

Comple Complete te the tabl tablee below below by noting noting the the point pointss on the grap graph h descri described bed by by the follo followin wing g conditions. Conditions

Point

g ′ ( x x) = 0, g ′′′′ ( x x) < 0 g ′ ( x x) < 0, g ′′′′ ( x) = 0 (Total 6 marks)

82

93.

2

A part of the graph of y y = 2 x – x x is given in the diagram below.

The shaded region is revolved through 360 ° about the x-axis. (a)

Write Write down down an expr express ession ion for this this volum volumee of revolu revolution tion..

(b) (b)

Calcu alcula late te this this volu volume me..

........................................................................... ................................................................................................................. ................................................................... ............................. ........................................................................... ................................................................................................................. ................................................................... ............................. ........................................................................... ................................................................................................................. ................................................................... ............................. ........................................................................... ................................................................................................................. ................................................................... ............................. ........................................................................... ................................................................................................................. ................................................................... ............................. ........................................................................... ................................................................................................................. ................................................................... ............................. ........................................................................... ................................................................................................................. ................................................................... ............................. ........................................................................... ................................................................................................................. ................................................................... ............................. ........................................................................... ................................................................................................................. ................................................................... ............................. ........................................................................... ................................................................................................................. ................................................................... ............................. ........................................................................... ................................................................................................................. ................................................................... ............................. ........................................................................... ................................................................................................................. ................................................................... ............................. ........................................................................... ................................................................................................................. ................................................................... ............................. (Total 6 marks)

94.

Consider the function ƒ : x

2

3 x – 5 x + k . 83

(a )

Write down ƒ′ ( x x).

The equation of the tangent to the graph of ƒ at x = p is y = 7 x – 9. Find the value of (b)

p;

(c)

k .

........................................................................... ................................................................................................................. ................................................................... ............................. ........................................................................... ................................................................................................................. ................................................................... ............................. ........................................................................... ................................................................................................................. ................................................................... ............................. ........................................................................... ................................................................................................................. ................................................................... ............................. ........................................................................... ................................................................................................................. ................................................................... ............................. ........................................................................... ................................................................................................................. ................................................................... ............................. ........................................................................... ................................................................................................................. ................................................................... ............................. ........................................................................... ................................................................................................................. ................................................................... ............................. ........................................................................... ................................................................................................................. ................................................................... ............................. ........................................................................... ................................................................................................................. ................................................................... ............................. ........................................................................... ................................................................................................................. ................................................................... ............................. ........................................................................... ................................................................................................................. ................................................................... ............................. ........................................................................... ................................................................................................................. ................................................................... ............................. (Total 6 marks)

84

95.

2 – x

The diagram below shows the graph of ƒ ( x) = x e at A and C and there is a maximum at B.

for 0 ≤ x ≤ 6. There are points of inflexion

(a)

Using Using the produc productt rule rule for differ different entiat iation ion,, find find ƒ′ ( x).

(b)

Find the exact value of the y-coordinate y-coordinate of B.

(c) (c)

The The sec secon ond d der deriv ivat ativ ivee of of ƒ is ƒ′′ ƒ′′ ( x) = ( x – 4 x + 2) e . Use this result to find the exact value of the x-coordinate of C.

2

–x

........................................................................... ................................................................................................................. ................................................................... ............................. ........................................................................... ................................................................................................................. ................................................................... ............................. ........................................................................... ................................................................................................................. ................................................................... ............................. ........................................................................... ................................................................................................................. ................................................................... ............................. ........................................................................... ................................................................................................................. ................................................................... ............................. ........................................................................... ................................................................................................................. ................................................................... ............................. ........................................................................... ................................................................................................................. ................................................................... ............................. ........................................................................... ................................................................................................................. ................................................................... ............................. ........................................................................... ................................................................................................................. ................................................................... ............................. ........................................................................... ................................................................................................................. ................................................................... ............................. ........................................................................... ................................................................................................................. ................................................................... ............................. ........................................................................... ................................................................................................................. ................................................................... ............................. ........................................................................... ................................................................................................................. ................................................................... ............................. (Total 6 marks)

85

96.

The displacement s metres at time t seconds is given by 2

s = 5 cos 3 t + t + 10, for t ≥ 0. (a) (a)

Writ Wr itee down down the the mini minimu mum m val value ue of s s.

(b) (b)

Fin Find the the acce accele lera rati tion on,, a, at time t .

(c )

Find th the value of of t when the maximum value of a first occurs.

........................................................................... ................................................................................................................. ................................................................... ............................. ........................................................................... ................................................................................................................. ................................................................... ............................. ........................................................................... ................................................................................................................. ................................................................... ............................. ........................................................................... ................................................................................................................. ................................................................... ............................. ........................................................................... ................................................................................................................. ................................................................... ............................. ........................................................................... ................................................................................................................. ................................................................... ............................. ........................................................................... ................................................................................................................. ................................................................... ............................. ........................................................................... ................................................................................................................. ................................................................... ............................. ........................................................................... ................................................................................................................. ................................................................... ............................. ........................................................................... ................................................................................................................. ................................................................... ............................. ........................................................................... ................................................................................................................. ................................................................... ............................. ........................................................................... ................................................................................................................. ................................................................... ............................. ........................................................................... ................................................................................................................. ................................................................... ............................. (Total 6 marks)

86

3 97.

2

Let f ( x x) = – 4 x + x + 4. (a )

(i)

Write down f ′ ( x x).

(ii) (ii)

Find Find the the equa equatio tion n of the the norm normal al to to the the curv curvee of f f at (2, 3).

(iii (iii))

This This nor norma mall inter interse sects cts the the curv curvee of f f at (2, 3) and at one other point P. Find the x-coordinate of P. (9)

Part of the graph of f f is given below.

(b)

Let R be the region under the curve of f f from x = −1 to x = 2. (i) (i)


(ii) (ii)

Calcu alcula late te this this are area.

(iii)

The re region R is revolved through 360 ° about the x-axis. Write down an expression for the volume of the solid formed. (6) k

(c )

Find

∫ f ( x) d x, giving your answer in terms of k . 1


98.

Consider the functions f and g where f ( x x) = 3 x – 5 and g ( x x) = x – 2. (a) (a)

−1

Fin Find the the inve inverrse func functi tion on,, f . (3)

87

(b)

–1

–1

Given that g ( x x) = x + 2, find ( g ◦ f ) ( x). (2)

x + 3 (c )

−1

Given al also th that ( f f

◦ g ) ( x)

–1 3 , solve ( f −1 ◦ g ) ( x x) = ( g ◦ f ) ( x x).

(2)

f ( x) Let h ( x x) =

g ( x)

, x

≠

2.

(i) Sketch the graph of h for −3 ≤ x ≤ 7 and −2 ≤ y ≤ 8, including any asymptotes.

(d)

(ii)

Write down the equations of the asymptotes. (5)

3 x − 5 (e )

The expression following.

(i)

Find

x − 3

1 may also be written as 3 +

x − 2

. Use this to answer the

∫ h ( x) d x. 5

(ii)

Hence, calculate the exact value of

∫ 3

h ( x x)d x. (5) 5

(f)

On your your sketch sketch,, shad shadee the the regi region on whose whose area area is repr represe esente nted d by

∫ 3

h ( x x)d x. (1) (Total 18 marks)

88

99.

The following diagram shows the graph of a function f .

Consider the following diagrams.

Complete the table below, noting which one of the diagrams above represents the graph of (a)

f ′( x x);

(b)

f ′′( x x). Graph (a)

f ′ ( x x)

(b)

f " f " ( x x)

Diagram

(Total 6 marks)

89

−1

2t−1

100. The velocity v in m s of a moving body at time t seconds is given by v = e the displacement of the body is 10 m. Find the displacement when t =1.

. When t = 0 5.

........................................................................... ................................................................................................................. ................................................................... ............................. ........................................................................... ................................................................................................................. ................................................................... ............................. ........................................................................... ................................................................................................................. ................................................................... ............................. ........................................................................... ................................................................................................................. ................................................................... ............................. ........................................................................... ................................................................................................................. ................................................................... ............................. ........................................................................... ................................................................................................................. ................................................................... ............................. ........................................................................... ................................................................................................................. ................................................................... ............................. ........................................................................... ................................................................................................................. ................................................................... ............................. ........................................................................... ................................................................................................................. ................................................................... ............................. ........................................................................... ................................................................................................................. ................................................................... ............................. ........................................................................... ................................................................................................................. ................................................................... ............................. ........................................................................... ................................................................................................................. ................................................................... ............................. ........................................................................... ................................................................................................................. ................................................................... ............................. (Total 6 marks)

90

101. The shaded region in the diagram below is bounded by f ( x x) = x , x = a, and the x-axis. The shaded region is revolved around the x-axis through 360 °. The volume of the solid formed is 0.845π .

Find the value of a. ........................................................................... ................................................................................................................. ................................................................... ............................. ........................................................................... ................................................................................................................. ................................................................... ............................. ........................................................................... ................................................................................................................. ................................................................... ............................. ........................................................................... ................................................................................................................. ................................................................... ............................. ........................................................................... ................................................................................................................. ................................................................... ............................. ........................................................................... ................................................................................................................. ................................................................... ............................. ........................................................................... ................................................................................................................. ................................................................... ............................. ........................................................................... ................................................................................................................. ................................................................... ............................. ........................................................................... ................................................................................................................. ................................................................... ............................. ........................................................................... ................................................................................................................. ................................................................... ............................. ........................................................................... ................................................................................................................. ................................................................... ............................. ........................................................................... ................................................................................................................. ................................................................... ............................. ........................................................................... ................................................................................................................. ................................................................... ............................. (Total 6 marks)

− x

102. The function f is defined as f ( x x) = (2 x +1) e , 0 ≤ x ≤ 3. The point P(0, 1) lies on the graph of f ( x x), and there is a maximum point at Q.

(a )

Sketch th the graph of y y = f ( x x), labelling the points P and Q. (3)

91

(b)

(i) (ii)

− x

Show that f ′ ( x) = (1− 2 x) e .

Find the exact coordinates of Q. (7)

(c )

The equation f ( x x) = k , where k ∈ k .

, has two solutions. Write down the range of values of (2)

(d)

− x

Given that f ″ ( x x) = e

(−3 + 2 x), show that the curve of f f has only one point of i nflexion. (2)

(e) (e)

Let Let R be the the poi point nt on the the cur curve ve of f f with x-coordinate 3. Find the area of the region enclosed by the curve and the line (PR). (7) (Total 21 marks)

92

−2t −2t

103. The velocity v of a particle at time t is given by v = e + 12t . The displacement of the particle at time t is s. Given that s = 2 when t = 0, express s in terms of t .

........................................................................... ................................................................................................................. ................................................................... ............................. ........................................................................... ................................................................................................................. ................................................................... ............................. ........................................................................... ................................................................................................................. ................................................................... ............................. ........................................................................... ................................................................................................................. ................................................................... ............................. ........................................................................... ................................................................................................................. ................................................................... ............................. ........................................................................... ................................................................................................................. ................................................................... ............................. ........................................................................... ................................................................................................................. ................................................................... ............................. ........................................................................... ................................................................................................................. ................................................................... ............................. ........................................................................... ................................................................................................................. ................................................................... ............................. ........................................................................... ................................................................................................................. ................................................................... ............................. ........................................................................... ................................................................................................................. ................................................................... ............................. ........................................................................... ................................................................................................................. ................................................................... ............................. ........................................................................... ................................................................................................................. ................................................................... ............................. (Total 6 marks)

93

104. The graph of the function y = f ( x x), 0 ≤ x ≤ 4, is shown below.

(a )

Write down th the va value lue of of (i)

f ′ (1);

(ii)

f ′ (3).

........................................................................... ................................................................................................................. ................................................................... ............................. ........................................................................... ................................................................................................................. ................................................................... ............................. ........................................................................... ................................................................................................................. ................................................................... ............................. ........................................................................... ................................................................................................................. ................................................................... ............................. ........................................................................... ................................................................................................................. ................................................................... .............................

94

(b) (b)

On the the diag diagra ram m belo below, w, draw draw the the grap graph h of of y y = 3 f (− x).

(c) (c)

On the the diag diagra ram m bel below ow,, draw draw the the grap graph h of of y y = f (2 x).

(Total 6 marks)

95

3

2

105. Let f ( x x) = x − 3 x − 24 x +1.

The tangents to the curve of f f at the points P and Q are parallel to the x-axis, where P is to the left of Q. (a) (a)

Calcu Calcula late te the coor coordi dina nate tess of of P and and of of Q. Q.

Let N 1 and N 2 be the normals to the curve at P and Q respectively.

(b) (b)

Write Wr ite down down the the coo coord rdin inat ates es of of the the point pointss wher wheree (i) (i)

the the tan tangent gent at P inter nterssects ects N 2;

(ii) (ii)

the the tan tange gent nt at Q int inter erse sect ctss N 1.

........................................................................... ................................................................................................................. ................................................................... ............................. ........................................................................... ................................................................................................................. ................................................................... ............................. ........................................................................... ................................................................................................................. ................................................................... ............................. ........................................................................... ................................................................................................................. ................................................................... ............................. ........................................................................... ................................................................................................................. ................................................................... ............................. ........................................................................... ................................................................................................................. ................................................................... ............................. ........................................................................... ................................................................................................................. ................................................................... ............................. ........................................................................... ................................................................................................................. ................................................................... ............................. ........................................................................... ................................................................................................................. ................................................................... ............................. ........................................................................... ................................................................................................................. ................................................................... ............................. ........................................................................... ................................................................................................................. ................................................................... ............................. ........................................................................... ................................................................................................................. ................................................................... ............................. ........................................................................... ................................................................................................................. ................................................................... ............................. (Total 6 marks)

96

3

106. It is given that

∫ 1

f ( x x)d x = 5. 3

(a )

Write down

∫ 1

2 f ( x x)d x. 3

(b)

Find the value of

∫ 1

2

(3 x + f ( x))d x.

........................................................................... ................................................................................................................. ................................................................... ............................. ........................................................................... ................................................................................................................. ................................................................... ............................. ........................................................................... ................................................................................................................. ................................................................... ............................. ........................................................................... ................................................................................................................. ................................................................... ............................. ........................................................................... ................................................................................................................. ................................................................... ............................. ........................................................................... ................................................................................................................. ................................................................... ............................. ........................................................................... ................................................................................................................. ................................................................... ............................. ........................................................................... ................................................................................................................. ................................................................... ............................. ........................................................................... ................................................................................................................. ................................................................... ............................. ........................................................................... ................................................................................................................. ................................................................... ............................. ........................................................................... ................................................................................................................. ................................................................... ............................. ........................................................................... ................................................................................................................. ................................................................... ............................. ........................................................................... ................................................................................................................. ................................................................... ............................. (Total 6 marks)

97

2

107. Let f ′ ( x x) = 12 x − 2.

Given that f (−1) =1, find f ( x x). ........................................................................... ................................................................................................................. ................................................................... ............................. ........................................................................... ................................................................................................................. ................................................................... ............................. ........................................................................... ................................................................................................................. ................................................................... ............................. ........................................................................... ................................................................................................................. ................................................................... ............................. ........................................................................... ................................................................................................................. ................................................................... ............................. ........................................................................... ................................................................................................................. ................................................................... ............................. ........................................................................... ................................................................................................................. ................................................................... ............................. ........................................................................... ................................................................................................................. ................................................................... ............................. ........................................................................... ................................................................................................................. ................................................................... ............................. ........................................................................... ................................................................................................................. ................................................................... ............................. ........................................................................... ................................................................................................................. ................................................................... ............................. ........................................................................... ................................................................................................................. ................................................................... ............................. ........................................................................... ................................................................................................................. ................................................................... ............................. (Total 6 marks)

98

108. The velocity, v, in m s the time in seconds.

−1

of a particle moving in a straight line is given by v = e

(a) (a)

Find Find the the acce accele lera ratio tion n of of the the part particl iclee at at t =1.

(b)

At what value of of t does the particle have a velocity of 22.3 m s ?

(c) (c)

Find Find the the dist distan ance ce tra trave vell lled ed in in the the firs firstt seco second nd..

3t −2 −2

, where t is

−1

........................................................................... ................................................................................................................. ................................................................... ............................. ........................................................................... ................................................................................................................. ................................................................... ............................. ........................................................................... ................................................................................................................. ................................................................... ............................. ........................................................................... ................................................................................................................. ................................................................... ............................. ........................................................................... ................................................................................................................. ................................................................... ............................. ........................................................................... ................................................................................................................. ................................................................... ............................. ........................................................................... ................................................................................................................. ................................................................... ............................. ........................................................................... ................................................................................................................. ................................................................... ............................. ........................................................................... ................................................................................................................. ................................................................... ............................. ........................................................................... ................................................................................................................. ................................................................... ............................. ........................................................................... ................................................................................................................. ................................................................... ............................. ........................................................................... ................................................................................................................. ................................................................... ............................. ........................................................................... ................................................................................................................. ................................................................... ............................. (Total 6 marks)

99

2

109. Let f ( x x) = 3 cos 2 x + sin x.

(a )

Show that f ′ ( x x) = −5 sin 2 x.

π (b)

In the in interval 4 ≤ x ≤

3π 4 , one normal to the graph of f f has equation x = k .

Find the value of k . ........................................................................... ................................................................................................................. ................................................................... ............................. ........................................................................... ................................................................................................................. ................................................................... ............................. ........................................................................... ................................................................................................................. ................................................................... ............................. ........................................................................... ................................................................................................................. ................................................................... ............................. ........................................................................... ................................................................................................................. ................................................................... ............................. ........................................................................... ................................................................................................................. ................................................................... ............................. ........................................................................... ................................................................................................................. ................................................................... ............................. ........................................................................... ................................................................................................................. ................................................................... ............................. ........................................................................... ................................................................................................................. ................................................................... ............................. ........................................................................... ................................................................................................................. ................................................................... ............................. ........................................................................... ................................................................................................................. ................................................................... ............................. ........................................................................... ................................................................................................................. ................................................................... ............................. ........................................................................... ................................................................................................................. ................................................................... ............................. (Total 6 marks)

100

110. The following diagram shows part of the graph of a quadratic function, with equation in the form y = ( x x − p)( x x − q), where p, q ∈ .

(a )

Write down (i)

the value of p and of q;

(ii) (ii)

the equatio equation n of of the the axis axis of of symm symmetr etry y of of the the curve. curve. (3)

(b) (b)

2

Find Find the the equ equat atio ion n of of the the func functio tion n in the the form form y = ( x x − h) + k , where h, k ∈

. (3)

d y (c )

Find d x . (2)

(d)

Let T be the tangent to the curve at the point (0, 5). Fi nd the equation of T . (2) (Total 10 marks)

101

x

111. The function f is defined as f ( x x) = e sin x, where x is in radians. Part of the curve of f f is shown below.

There is a point of inflexion at A, and a local maximum point at B. The curve of f intersects the x-axis at the point C. (a )

Write down the x-coordinate of the point C. (1)

(b)

(i) (ii) (ii)

Find f ′ ( x x).

Write rite down own th the val value ue of f f ′ ( x x) at the point B. (4)

(c )

x

Show that f ″ ( x x) = 2e cos x. (2)

(d)

(i) (ii) (ii)

Write down the value of f ″ ( x x) at A, the point of inflexion.

Henc Hence, e, calcu calcula late te the coor coordi dina nate tess of of A. A. (4)

(e) (e)

Let Let R be be the the reg regio ion n enc enclos losed ed by by the the curv curvee and and the the x-axis, between the origin and C. (i) (i)


(ii)

Find the area of R R. (4) (Total 15 marks)

102

1 112. The function f ( x x) is defined as f ( x x) = 3 +

(a )

2 x − 5

5 , x

≠

2.

Sketch th the curve of of f f for −5 ≤ x ≤ 5, showing the asymptotes. (3)

(b) (b)

Usin Using g you yourr ske sketc tch, h, writ writee down down (i) (i)

the the equ equat atio ion n of of eac each h asy asymp mpto tote te;;

(ii)

the value of th the x-intercept;

(iii (iii))

the the val value ue of the the y-in y-inte terce rcept pt.. (4)

(c) (c)

The The reg regio ion n enc enclo lose sed d by by the the curv curvee of of f f , the x-axis, and the lines x = 3 and x = a, is revolved through 360 ° about the x-axis. Let V be the volume of the solid formed.

(i)

Find

   9 + 6 + 1 2     2 x − 5 ( 2 x − 5)  d x.

∫

 28 + 3 ln 3    3  , find the value of a.

π (ii)

Hence, given that V =


103

p − 113. Let f ( x x) =

3 x

x 2

− q2

, where p, q∈

+

.

Part of the graph of f f , including the asymptotes, is shown below.

(a) (a)

The The equa equati tion onss of the the asy asympto mptote tess are are x =1, x = −1, y = 2. Write down the value of (i)

p;

(ii)

q. (2)

(b)

Let R be the region bounded by the graph of f f , the x-axis, and the y-axis. (i)

Find the negative x-intercept of f f .

(ii) (ii)

Henc Hencee find find the the vol volum umee obta obtaine ined d whe when n R is revolved through 360 ° about the x-axis. (7)

(

)

3 x 2 + 1 (c )

(i) (ii) (ii)

Show that f ′ ( x) =

( x 2 −1)

2

.

Hence, Hence, show show that that there there are are no maxi maximum mum or or minimu minimum m points points on on the grap graph h of f f . (8)

104

(d)

Let g ( x x) = f ′ ( x x). Let A be the area of the region enclosed by the graph of g and the xaxis, between x = 0 and x = a, where a > 0. Given that A = 2, find the value of a. (7) (Total 24 marks)

114. The following diagram shows part of the graph of y = cos x for 0 ≤ x ≤ 2π . Regions A and B are shaded.

(a) (a)

Write Wr ite down down an expr expres essi sion on for for the the area area of A. (1)

(b) (b)

Calcu alcula late te the the area area of A. (1)

105

(c) (c)

Find Find the the tota totall area area of the the sha shade ded d reg regio ions ns..

........................................................................... ................................................................................................................. ................................................................... ............................. ........................................................................... ................................................................................................................. ................................................................... ............................. ........................................................................... ................................................................................................................. ................................................................... ............................. ........................................................................... ................................................................................................................. ................................................................... ............................. ........................................................................... ................................................................................................................. ................................................................... ............................. ........................................................................... ................................................................................................................. ................................................................... ............................. ........................................................................... ................................................................................................................. ................................................................... ............................. ........................................................................... ................................................................................................................. ................................................................... ............................. ........................................................................... ................................................................................................................. ................................................................... ............................. ........................................................................... ................................................................................................................. ................................................................... ............................. ........................................................................... ................................................................................................................. ................................................................... ............................. ........................................................................... ................................................................................................................. ................................................................... ............................. (4) (Total 6 marks)

106

3

115. Consider the function f ( x x) = 4 x + 2 x. Find the equation of the normal to the curve of f at the point where x =1.

........................................................................... ................................................................................................................. ................................................................... ............................. ........................................................................... ................................................................................................................. ................................................................... ............................. ........................................................................... ................................................................................................................. ................................................................... ............................. ........................................................................... ................................................................................................................. ................................................................... ............................. ........................................................................... ................................................................................................................. ................................................................... ............................. ........................................................................... ................................................................................................................. ................................................................... ............................. ........................................................................... ................................................................................................................. ................................................................... ............................. ........................................................................... ................................................................................................................. ................................................................... ............................. ........................................................................... ................................................................................................................. ................................................................... ............................. ........................................................................... ................................................................................................................. ................................................................... ............................. ........................................................................... ................................................................................................................. ................................................................... ............................. ........................................................................... ................................................................................................................. ................................................................... ............................. (Total 6 marks)

107

116. Differentiate each of the following with respect to x.

(a)

y = sin 3 x (1)

(b)

y = x tan x (2)

ln x (c)

y = x (3)

........................................................................... ................................................................................................................. ................................................................... ............................. ........................................................................... ................................................................................................................. ................................................................... ............................. ........................................................................... ................................................................................................................. ................................................................... ............................. ........................................................................... ................................................................................................................. ................................................................... ............................. ........................................................................... ................................................................................................................. ................................................................... ............................. ........................................................................... ................................................................................................................. ................................................................... ............................. ........................................................................... ................................................................................................................. ................................................................... ............................. ........................................................................... ................................................................................................................. ................................................................... ............................. ........................................................................... ................................................................................................................. ................................................................... ............................. ........................................................................... ................................................................................................................. ................................................................... ............................. ........................................................................... ................................................................................................................. ................................................................... ............................. ........................................................................... ................................................................................................................. ................................................................... ............................. (Total 6 marks)

108

117. On the axes below, sketch a curve y = f ( x) which satisfies the following conditions.

x

f ( x x)

−2 ≤ x < 0 0

–1

0 < x < 1 1

2

1 < x ≤ 2

f ′ ( x x)

f ′′ ( x x)

negative

positive

0

positive

positive

positive

positive

0

positive

negative

(Total 6 marks)

118. Consider the function f ( x x) e

(a )

(2 x –1)

+

 5     − ( ) x 2 1   , x ≠

1 2.

Sketch th the curve of of f f for −2 ≤ x ≤ 2, including any asymptotes. (3) 109

(b)

(i) (ii) (ii)

Write down the equation of the vertical asymptote of f .

Write Write down down which which one one of of the the follo followin wing g expre express ssion ionss does does not represent an area between the curve of f f and the x-axis. 2

∫ 1

f ( x x)d x

2

∫ 0

(iii (iii))

f ( x x)d x

Just Justif ify y you yourr ans answe wer. r. (3)

(c) (c)

The The reg regio ion n bet betwe ween en the the cur curve ve and and the the x-axis between x = 1 and x = 1.5 is rotated through 360° about the x-axis. Let V be the volume formed. (i) (i)

Writ Wr itee dow down n an an exp expre ress ssio ion n to to rep repre rese sent nt V .

(ii) (ii)

Henc Hencee wri write te down down the the val value ue of V . (4)

(d)

Find f ′ ( x x). (4)

(e )

(i) (ii)

Write down the value of x at the minimum point on the curve of f .

The equation f ( x x) = k has no solutions for p ≤ k < q. Write down the value of p p and of q. (3) (Total 17 marks)

110

119. A Ferris wheel with centre O and a radius of 15 metres is represented in the diagram below. π ˆ Initially seat A is at ground level. The next seat i s B, where AOB = 6 .

(a) (a)

Fin Find the the leng length th of the the ar arc AB. AB. (2)

(b) (b)

Find Find the the are areaa of of the the sect sector or AOB. AOB. (2)

111

2π (c) (c)

The The whee wheell turn turnss clo clock ckwi wise se thro throug ugh h an angl anglee of 3 . Find the height of A above the ground. (3)

The height, h metres, of seat C above the ground after t minutes, can be modelled by the function

h (t ) = 15 − 15 cos

 2t + π     4  .

π (d)

(i)

Find the height of seat C when t = 4 .

(ii) (ii)

Find Find the the init initia iall hei heigh ghtt of of seat seat C. C.

(iii)

Find the time at which which seat seat C first reaches reaches its its highes highestt point. point. (8)

(e )

Find h′ (t ). ). (2)

(f)

For 0 ≤ t ≤ π , (i)

sketch the graph of h′;

(ii) (ii)

find find the the time time at at which which the the heig height ht is is chang changing ing most most rapidly rapidly.. (5) (Total 22 marks)

112

1

(a)

120.

d x. ∫ 2 x + 3 Find (2)

1

(b)

d x ∫ 2 x + 3 Given that = ln

P , find the value of P P .

........................................................................... ................................................................................................................. ................................................................... ............................. ........................................................................... ................................................................................................................. ................................................................... ............................. ........................................................................... ................................................................................................................. ................................................................... ............................. ........................................................................... ................................................................................................................. ................................................................... ............................. ........................................................................... ................................................................................................................. ................................................................... ............................. ........................................................................... ................................................................................................................. ................................................................... ............................. ........................................................................... ................................................................................................................. ................................................................... ............................. ........................................................................... ................................................................................................................. ................................................................... ............................. ........................................................................... ................................................................................................................. ................................................................... ............................. ........................................................................... ................................................................................................................. ................................................................... ............................. ........................................................................... ................................................................................................................. ................................................................... ............................. ........................................................................... ................................................................................................................. ................................................................... ............................. (4) (Total 6 marks)

113

121. A particle moves along a straight line so that its velocity, v ms

−1

at time t seconds is given by v

3t

= 6e + 4. When t = 0, the displacement, s, of the particle is 7 metres. Find an expression for s s in terms of t . ........................................................................... ................................................................................................................. ................................................................... ............................. ........................................................................... ................................................................................................................. ................................................................... ............................. ........................................................................... ................................................................................................................. ................................................................... ............................. ........................................................................... ................................................................................................................. ................................................................... ............................. ........................................................................... ................................................................................................................. ................................................................... ............................. ........................................................................... ................................................................................................................. ................................................................... ............................. ........................................................................... ................................................................................................................. ................................................................... ............................. ........................................................................... ................................................................................................................. ................................................................... ............................. ........................................................................... ................................................................................................................. ................................................................... ............................. ........................................................................... ................................................................................................................. ................................................................... ............................. ........................................................................... ................................................................................................................. ................................................................... ............................. ........................................................................... ................................................................................................................. ................................................................... ............................. (Total 7 marks)

114

1 3 2 122. Consider f f ( x x) = 3 x + 2 x – 5 x. Part of the graph of f f is shown below. There is a maximum point at M, and a point of inflexion at N.

(a )

Find f ′ ( x x). (3)

(b)

Find the x-coordinate of M. (4)

(c )

Find the x-coordinate of N. (3)

(d)

The line L is the tangent to the curve of f f at (3, 12). Find the equation of L L in the form y = ax + b. (4) (Total 14 marks)

115

5

123. Let

∫ 3 f ( x) dx =12. 1

5

(a )

Show that

∫ f ( x) dx = − 4. 1

(2)

(b)

Find the value of

∫

5

1

( x + f ( x ) ) d x +

5

∫ ( x + f ( x) ) d x. 2

........................................................................... ................................................................................................................. ................................................................... ............................. ........................................................................... ................................................................................................................. ................................................................... ............................. ........................................................................... ................................................................................................................. ................................................................... ............................. ........................................................................... ................................................................................................................. ................................................................... ............................. ........................................................................... ................................................................................................................. ................................................................... ............................. ........................................................................... ................................................................................................................. ................................................................... ............................. ........................................................................... ................................................................................................................. ................................................................... ............................. ........................................................................... ................................................................................................................. ................................................................... ............................. ........................................................................... ................................................................................................................. ................................................................... ............................. ........................................................................... ................................................................................................................. ................................................................... ............................. ........................................................................... ................................................................................................................. ................................................................... ............................. ........................................................................... ................................................................................................................. ................................................................... ............................. (5) (Total 7 marks)

116

124. Let f : x

(a )

3

sin x. (i)

(ii)

Write down the range of the function f .

Consider f ( x x) =1, 0 ≤ x ≤ 2π . Write down the number of solutions to this equation. Justify your answer. (5)

(b)

p

q

Find f ′ ( x x), giving your answer in the form a sin x cos x where a, p, q ∈

. (2)

1

(c )

π

)2

3 sin x (cos x Let g ( x for 0 ≤ x ≤ 2 . Find the volume generated when the curve x) = of g g is revolved through 2 π about the x-axis. (7) (Total 14 marks)

117

125. The diagram below shows part of the graph of the gradient function, y = f ′ ( x).

y

p

(a) (a)

q

x

r

On the the gri grid d bel below ow,, ske sketc tch h a grap graph h of of y y = f ″ ( x x), clearly indicating the x-intercept.

y

p

q

r

x

(2)

(b) (b)

Comp Comple lete te the the tabl table, e, for for the the gra graph ph of y y = f ( x x).

x-coordinate (i)

Maximum point on f

(ii) (ii)

Inf Inflex lexion ion poi poin nt on on f (2)

(c) (c)

Just Justif ify y you yourr answ answer er to to par partt (b) (b) (ii (ii). ).

........................................................................... ................................................................................................................. ................................................................... ............................. ........................................................................... ................................................................................................................. ................................................................... ............................. ........................................................................... ................................................................................................................. ................................................................... ............................. ........................................................................... ................................................................................................................. ................................................................... ............................. ........................................................................... ................................................................................................................. ................................................................... ............................. (2) (Total 6 marks)

118

126. The following diagram shows the graphs of f f ( x x) = ln (3 x – 2) + 1 and g ( x x) = – 4 cos (0.5 x) + 2, for 1 ≤ x ≤ 10.

(a )

Let A be the area of the region enclosed by the curves of f f and g . (i)

Find ind an expression for A A.

(ii) (ii)

Calcu alcula late te the the val value ue of A. (6)

(b)

(i) (ii)

Find f ′ ( x).

Find g ′ ( x). (4)

(c )

There ar are tw two va values of of x x for which the gradient of f f is equal to the gradient of g g . Find both these values of x. (4) (Total 14 marks)

119

x –2

127. Let f ( x x) = 3 x – e

(a )

– 4, for –1 ≤ x ≤ 5.

Find the x-intercepts of the graph of f f .

........................................................................... ................................................................................................................. ................................................................... ............................. ........................................................................... ................................................................................................................. ................................................................... ............................. ........................................................................... ................................................................................................................. ................................................................... ............................. ........................................................................... ................................................................................................................. ................................................................... ............................. ........................................................................... ................................................................................................................. ................................................................... ............................. ........................................................................... ................................................................................................................. ................................................................... ............................. ........................................................................... ................................................................................................................. ................................................................... ............................. ........................................................................... ................................................................................................................. ................................................................... ............................. ........................................................................... ................................................................................................................. ................................................................... ............................. (3)

120

(b) (b)

On the the gri grid d bel below ow,, ske sketch tch the the gra graph ph of f f .

y 3 2 1 –

10 – 1

2 –

–

–

2

–

3

–

4

–

5

–

6

–

7

–

8

–

9 1

1

2

3

4

5

6

x

0 (3)

121

(c) (c)

Writ Wr itee dow down n the the gra gradi dien entt of the the gra graph ph of f f at x = 2.

........................................................................... ................................................................................................................. ................................................................... ............................. ........................................................................... ................................................................................................................. ................................................................... ............................. ........................................................................... ................................................................................................................. ................................................................... ............................. ........................................................................... ................................................................................................................. ................................................................... ............................. ........................................................................... ................................................................................................................. ................................................................... ............................. ........................................................................... ................................................................................................................. ................................................................... ............................. ........................................................................... ................................................................................................................. ................................................................... ............................. ........................................................................... ................................................................................................................. ................................................................... ............................. ........................................................................... ................................................................................................................. ................................................................... ............................. (1) (Total 7 marks)

122

x

2

128. Let f ( x x) = e (1 – x x ).

(a )

x

2

Show that f ′ ( x x) = e (1 – 2 x – x x ). (3)

Part of the graph of y = f ( x x), for – 6 ≤ x ≤ 2, is shown below. The x-coordinates of the local minimum and maximum points are r and s respectively.

(b)

Write down the equation of the horizontal asymptote. (1)

(c )

Write down th the va value lue of of r and of s s. (4)

(d)

Let L be the normal to the curve of f f at P(0, 1). Show that L has equation x + y = 1. (4)

(e )

Let R be the region enclosed by the curve y = f ( x x) and the line L. (i) (i)

Find Find an expr expres essi sion on for for the the area area of R R.

(ii) (ii)

Calcu alcula late te the the area rea of R R. (5) (Total 17 marks)

123

All Calculus Sl

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