1.
2
x
Let f Let f ( x) x) = cos( x ) and g and g ( x) x) = e , for –1.5 ≤ x ≤ 0.5. Find the area of the region enclosed by the graphs of f and g . (Total 6 marks)
2.
The following diagram shows a waterwheel with a bucket. The wheel rotates at a constant rate in an anticlockwise (counterclockwise) (counterclockwise) direction.
diagram not to scale
The diameter of the wheel is 8 metres. The centre of the wheel, A, A, is 2 metres above the water level. After t seconds, seconds, the height of the bucket above the water level is given by h = a sin bt 2.
(a)
!how that a = ". (2)
The wheel turns at a rate of one rotation every #$ seconds.
(b)
!how that b = 15 . (2)
%n the first rotation, there are two values of t when when the bucket is descending at a rate of –1 $.& m s . (c) (c)
'ind 'ind the these valu value es of t . (6)
IB Questionbank Maths SL
1
(d)
eter etermi mine ne whet whether her the the bucke buckett is under underwa water ter at the the secon second d value value of t . (4) (Total 14 marks)
d y
3.
A gradient gradient function is given by d x when x = *.
= 10e 2 x − 5
. hen x hen x = $, y = = 8. 'ind the value of y
(Total 8 marks)
4.
+et f +et f ( x x ) = x = x cos x cos x , for $ ≤ x ≤ !. (a)
'ind f "( x ). ). "( x (3)
IB Questionbank Maths SL
2
(b)
n the grid below, sketch the gra-h of y = f "( x ).
(4) (Total 7 marks)
IB Questionbank Maths SL
3
5.
x
+et f ( x ) = e sin 2 x *$, for $ ≤ x ≤ #. $art of the graph of f is given below.
There is an x interce-t at the -oint A, a local ma/imum -oint at 0, where x = p and a local minimum -oint at 1, where x = q.
(a)
rite down the x coordinate of A. (1)
(b)
'ind the value of (i)
p
(ii)
q. (2)
q
(c)
'ind
∫ f ( x)d x . 3/-lain why this is not the area of the shaded region. p
(3) (Total 6 marks)
IB Questionbank Maths SL
4
6.
2
4onsider f ( x ) = x ln(" – x ), for –2 % x 5 2. The gra-h of f is given below.
(a)
+et 6 and 7 be -oints on the curve of f where the tangent to the gra-h of f is -arallel to the x a/is. (i)
'ind the x coordinate of 6 and of 7.
(ii)
4onsider f ( x ) = k . rite down all values of k for which there are e/actly two solutions. (5)
#
2
+et g ( x ) = x ln(" – x ), for –2 % x 5 2.
(b)
!how that g "( x ) =
− 2 x # + & x 2 ln(# − x 2 ) 2 # − x . (4)
(c)
!ketch the gra-h of g ". (2)
(d)
4onsider g "( x ) = w . rite down all values of w for which there are e/actly two solutions. (3) (Total 14 marks)
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5
7.
–1 The velocity v m s of an obect after t seconds is given by v (t ) = 15 t − &t , for $ ≤ t ≤ 25.
(a)
n the grid below, sketch the gra-h of v , clearly indicating the ma/imum -oint.
(3)
+et be the distance travelled in the first nine seconds. (b)
(i)
rite down an e/-ression for .
(ii)
9ence, write down the value of . (4) (Total 7 marks)
8.
#
2
+et f "( x ) = –2# x : x # x *. (a)
There are two -oints of infle/ion on the gra-h of f . rite down the x coordinates of these -oints. (3)
(b)
+et g ( x ) = f '( x ). 3/-lain why the gra-h of g has no -oints of infle/ion. (2) (Total 5 marks)
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!
9.
2
+et f ( x ) = x ln(" – x ), for –2 % x 5 2. The gra-h of f is shown below.
The gra-h of f crosses the x a/is at x = a, x = $ and x = b.
(a)
'ind the value of a and of b. (3)
The gra-h of f has a ma/imum value when x = " . (b)
'ind the value of " . (2)
(c)
The region under the gra-h of f from x = $ to x = " is rotated #;$ abot the x a/is. 'ind the volume of the solid formed. (3)
(d)
+et # be the region enclosed by the curve, the x a/is and the line x = " , between x = a and x = " . 'ind the area of # . (4) (Total 12 marks)
IB Questionbank Maths SL
$
10.
2
+et f ( x ) = x ( x – 5) , for $ ≤ x ≤ !. *he follo+ing diagra sho+s the graph of f .
+et # be the region enclosed by the x a/is and the curve of f .
(a)
'ind the area of # . (3)
(b)
'ind the volume of the solid formed when # is rotated through #;$ abot the x a/is. (4)
IB Questionbank Maths SL
%
(c)
The diagram below shows a -art of the gra-h of a
The area of the shaded region is e
11.
+et f ( x ) = #sin x " cos x , for – 2 ≤ x ≤ 2. (a)
!ketch the gra-h of f . (3)
(b)
rite down (i)
the am-litude
(ii)
the -eriod
(iii)
the x interce-t that lies between
−
2 and $. (3)
(c)
9ence write f ( x ) in the form p sin (qx & ). (3)
IB Questionbank Maths SL
'
(d)
rite down one value of x such that f "( x ) = $. (2)
(e)
rite down the two values of k for which the e
(f)
+et g ( x ) = ln( x *), for $ ≤ x ≤ . There is a value of x , between $ and *, for which the gradient of f is e
12.
5 cos x # and g ( x ) = –0.5 x 2 & x – -, for 0 ≤ x ≤ . +et f ( x ) =
(a)
n the same diagram, sketch the gra-hs of f and g . (3)
(b)
4onsider the gra-h of f . rite down (i)
the x interce-t that lies between x = $ and x =#
(ii)
the -eriod
(iii)
the am-litude. (4)
(c)
4onsider the gra-h of g . rite down (i)
the two x interce-ts
(ii)
the e
(d)
+et # be the region enclosed by the gra-hs of f and g . 'ind the area of # . (5) (Total 15 marks)
IB Questionbank Maths SL
1(
13.
x
2
+et f ( x ) = e (* – x ). (a)
x
2
!how that f " ( x ) = e (* – 2 x – x ). (3)
6art of the gra-h of y = f ( x ), for – ! ≤ x ≤ 2, is shown below. The x coordinates of the local minimum and ma/imum -oints are & and s res-ectively.
(b)
rite down the equation of the horiontal asym-tote. (1)
(c)
rite down the value of & and of s. (4)
(d)
+et L be the normal to the curve of f at 6($, *). !how that L has e
(e)
+et # be the region enclosed by the curve y = f ( x ) and the line L. (i)
'ind an e/-ression for the area of # .
(ii)
4alculate the area of # . (5) (Total 17 marks)
IB Questionbank Maths SL
11
14.
+et f ( x ) = x cos ( x – sin x ), $ ≤ x ≤ &) (a)
!ketch the gra-h of f on the following set of a/es.
(3)
(b)
The gra-h of f intersects the x a/is when x = a, a / 0. rite do+n the ale of a. (1)
(c)
The gra-h of f is revolved #;$ abot the x a/is from x = $ to x = a. 'ind the volume of the solid formed. (4) (Total 8 marks)
15.
2 x
+et f ( x ) = e cos x , –1 ≤ x ≤ 2) (a)
2 x
!how that f "( x ) = e (2 cos x – sin x ). (3)
+et the line L be the normal to the curve of f at x = $. (b)
'ind the e
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12
The gra-h of f and the line L intersect at the -oint ($, *) and at a second -oint 6. (c)
(i)
'ind the x coordinate of 6.
(ii)
'ind the area of the region enclosed by the gra-h of f and the line L) (6) (Total 14 marks)
16.
x
The function f is defined as f ( x ) = e sin x , where x is in radians. 6art of the curve of f is shown below.
There is a -oint of infle/ion at A, and a local ma/imum -oint at >. The curve of f intersects the x a/is at the -oint 4. (a)
rite down the x coordinate of the -oint 4. (1)
(b)
(i)
'ind f ′ ( x ).
(ii)
rite down the value of f ′ ( x ) at the -oint >. (4)
(c)
x
!how that f ″( x ) = 2e cos x . (2)
(d)
(i)
rite down the value of f ″( x ) at A, the -oint of infle/ion.
(ii)
9ence, calculate the coordinates of A. (4)
IB Questionbank Maths SL
13
(e)
+et ? be the region enclosed by the curve and the x a/is, between the origin and 4. (i)
rite down an e/-ression for the area of # .
(ii)
'ind the area of # . (4) (Total 15 marks)
1
17.
The function f ( x ) is defined as f ( x ) = # (a)
2 x − 5
5
, x ≠ 2 .
!ketch the curve of f for 5 ≤ x ≤ &, showing the asym-totes. (3)
(b)
@sing your sketch, write down (i)
the e
(ii)
the value of the x interce-t
(iii)
the value of the yinterce-t. (4)
(c)
The region enclosed by the curve of f , the x a/is, and the lines x = # and x = a, is revolved through #;$ ° about the x a/is. +et * be the volume of the solid formed.
(i)
'ind
+ ! + 1 2 2 x − 5 ( 2 x − 5) d x .
∫
2- + & ln & & , find the value of a.
(ii)
9ence, given that * =
(10) (Total 17 marks)
IB Questionbank Maths SL
14
p −
18.
+et f ( x ) =
& x x 2
− q2
, where p, q∈
.
6art of the gra-h of f , including the asym-totes, is shown below.
(a)
The e
p
(ii)
q. (2)
(b)
+et # be the region bounded by the gra-h of f , the x a/is, and the y a/is. (i)
'ind the negative x interce-t of f .
(ii)
9ence find the volume obtained when # is revolved through #;$ ° about the x a/is. (7)
& ( x 2 + 1)
(c)
( x 2 −1) 2 .
(i)
!how that f " ( x ) =
(ii)
9ence, show that there are no ma/imum or minimum -oints on the gra-h of f . (8)
IB Questionbank Maths SL
15
(d)
+et g ( x ) = f " ( x ). +et + be the area of the region enclosed by the gra-h of g and the x a/is, between x = $ and x = a, where a $. Biven that + = 2, find the value of a. (7) (Total 24 marks)
19.
(2 x –1)
4onsider the function f ( x ) e (a)
5 ( ) 2 x 1 − , x ≠
1 2.
!ketch the curve of f for 2 ≤ x ≤ 2, including any asym-totes. (3)
(b)
(i)
rite down the e
(ii)
rite down which one of the following e/-ressions does not re-resent an area between the curve of f and the x a/is. 2
∫ 1
f ( x )d x
2
∫ 0
(iii)
f ( x )d x
Custify your answer. (3)
(c)
The region between the curve and the x a/is between x = * and x = *.& is rotated through #;$° about the x a/is. +et * be the volume formed. (i)
rite down an e/-ression to re-resent * .
(ii)
9ence write down the value of * . (4)
(d)
'ind f " ( x ). (4)
IB Questionbank Maths SL
1!
(e)
(i)
rite down the value of x at the minimum -oint on the curve of f .
(ii)
The e
20.
x
The function f is defined as f ( x ) = (2 x *) e , $ ≤ x ≤ #. The -oint 6($, *) lies on the gra-h of f ( x ), and there is a ma/imum -oint at 7. (a)
!ketch the gra-h of y = f ( x ), labelling the -oints 6 and 7. (3)
(b)
x
(i)
!how that f " ( x ) = (* 2 x ) e .
(ii)
'ind the eact coordinates of 7. (7)
(c)
The e
, has two solutions. rite down the range of (2)
(d)
x
Biven that f ″( x ) = e (& 3 2 x ), show that the curve of f has only one -oint of infle/ion. (2)
(e)
+et ? be the -oint on the curve of f with x coordinate #. 'ind the area of the region enclosed by the curve and the line (6?). (7) (Total 21 marks)
IB Questionbank Maths SL
1$
21.
+et h ( x ) = ( x – 2) sin ( x – 1) for –5 ≤ x ≤ &. The curve of h ( x ) is shown below. There is a minimum -oint at ? and a ma/imum -oint at !. The curve intersects the x a/is at the -oints (a, 0) (1, 0) (2, 0) and ( b, $). y # & 2
(a , 0) – 5
–#
–& –2
4
1 –1
(b , 0 ) 1
– 1
2
&
#
5
x
– 2 – & – # – 5 – ! – 6
(a)
'ind the e/act value of (i)
a
(ii)
b. (2)
The regions between the curve and the x a/is are shaded for a ≤ x ≤ 2 as shown. (b)
(i)
rite down an e/-ression which re-resents the total area of the shaded regions.
(ii)
4alculate this total area. (5)
(c)
(i)
The y coordinate of ? is –0.2#0. Find the y coordinate of !.
(ii)
9ence or otherwise, find the range of values of k for which the e
IB Questionbank Maths SL
1%
22.
4onsider the function f ( x ) = cos x sin x)
(a)
(i)
!how that f ( – # ) = $.
(ii)
'ind in terms of -, the smallest "ositi#e value of x which satisfies f ( x ) = $. (3)
x
The diagram shows the gra-h of y = e (cos x sin x ), – 2 ≤ x ≤ #. The gra-h has a ma/imum turning -oint at 4(a, b) and a -oint of infle/ion at . y !
8 (a, b)
# 7 2
– 2
–1
1
2
&
x
d y
(b)
'ind d x . (3)
(c)
'ind the eact value of a and of b. (4)
(d)
!how that at , y =
2e # . (5)
(e)
'ind the area of the shaded region.
IB Questionbank Maths SL
1'
(2) (Total 17 marks)
2
23.
& +et the function f be defined by f ( x ) = 1 + x , x ≠ –1.
(a)
(i)
rite down the e
(ii)
rite down the e
(iii)
!ketch the gra-h of f in the domain –& ≤ x ≤ #. (4)
– ! x 2
(b)
(i)
& 2 @sing the fact that f ′( x ) = (1 + x ) , show that the second derivative
12 x( 2 x & – 1)
f ″ ( x ) =
(ii)
(1 + x & ) &
.
'ind the x coordinates of the -oints of infle/ion of the gra-h of f ) (6)
(c)
The table below gives some values of f ( x ) and 2f ( x ).
(i)
x
f ( x )
2f ( x )
*
*
2
*."
$.&#"*88
*.$;8#D;
*.8
$.2:2D"$
$.&8&"8$
2.2
$.*D*D$#
$.#"#"$D
2.;
$.*$D;;;
$.2*&##2
#
$.$D*"2:
$.*"28&D
@se the tra-eium rule with five subintervals to a--ro/imate the integral &
∫ f ( x) dx. 1
&
(ii)
∫
f ( x ) d x Biven that 1 = $.;#D&::, use a diagram to e/-lain why your answer is greater than this. (5) (Total 15 marks)
IB Questionbank Maths SL
2(
24.
x
The diagram below shows a sketch of the gra-h of the function y = sin (e ) where –1 ≤ x ≤ 2, and x is in radians. The gra-h cuts the y,a/is at A, and the x,a/is at 4 and . %t has a ma/imum -oint at >. y : 9
– 1
(a)
0
1
8
7
2
x
'ind the coordinates of A. (2)
(b)
The coordinates of 4 may be written as (ln k , $). 'ind the eact value of k . (2)
(c)
(i)
rite down the y coordinate of >. d y
(ii)
'ind d x .
(iii)
9ence, show that at >, x = ln 2 . (6)
(d)
(i)
rite down the integral which re-resents the shaded area.
(ii)
3valuate this integral. (5)
(e)
(i)
4o-y the above diagram into your answer booklet. (There is no need to # co-y the shading.) n your diagram, sketch the gra-h of y = x .
(ii)
The two gra-hs intersect at the -oint 6. 'ind the x coordinate of 6. (3) (Total 18 marks)
IB Questionbank Maths SL
21
25.
$oteE ?adians are used throughout this
raw the gra-h of y = - x cos x , $ ≤ x ≤ &, on millimetre s
the integer values of x and y on each a/is
(ii)
the a--ro/imate -ositions of the x interce-ts and the turning -oints. (5)
(b)
%it&out t&e use o! a calculator , show that - is a solution of the e
(c)
'ind another solution of the e
(d)
+et # be the region enclosed by the gra-h and the a/es for $ ≤ x ≤ -. !hade # on your diagram, and write down an integral which re-resents the area of # . (2)
(e)
3valuate the integral in -art (d) to an accuracy of si significant figures. (%f you consider it necessary, you can make use of the result d ( x sin x + cos x ) = x cos x .) d x (3) (Total 15 marks)
IB Questionbank Maths SL
22
26.
The function f is given by 2 x f ( x) =1 – 1 + x 2
(a)
(i)
To dis-lay the gra-h of y = f ( x ) for –10 ≤ x ≤ *$, a suitable interval for y , a ≤ y ≤ b must be chosen. !uggest a--ro-riate values for a and b .
(ii)
Bive the e
f ′ ( x)
(b)
!how that
=
2 x 2 – 2 (1 + x 2 ) 2
. (4)
(c)
'se our anser to "art (*) to find the coordinates of the ma/imum -oint of the gra-h. (3)
(d)
(i)
+it&er by ins-ection or by using an a--ro-riate substitution, find
∫ f ( x) dx (ii)
,ence find the e/act area of the region enclosed by the gra-h of f , the x a/is and the y a/is. (8) (Total 18 marks)
IB Questionbank Maths SL
23
27.
-n t&is question ou s&ould note t&at radians are used t&roug&out. (a)
2
(i)
!ketch the gra-h of y = x cos x- for $ ≤ x ≤ 2 making clear the a--ro/imate -ositions of the -ositive x interce-t, the ma/imum -oint and the end-oints.
(ii)
rite down the a""roimate coordinates of the -ositive x interce-t, the ma/imum -oint and the end-oints. (7)
(b)
'ind the eact #alue of the -ositive x interce-t for $ ≤ x ≤ 2) (2)
+et ? be the region in the first
(i)
!hade # on your diagram.
(ii)
rite down an integral which re-resents the area of #) (3)
(d)
3valuate the integral in -art (c)(ii), either by using a gra-hic dis-lay calculator, or by using the following information. d d x ( x 2 sin x 2 x cos x – 2 sin x ) = x 2 cos x . (3) (Total 15 marks)
IB Questionbank Maths SL
24
28.
-n t&is "art o! t&e question radians are used t&roug&out. The function f is given by 2
f ( x ) = (sin x ) cos x . The following diagram shows -art of the gra-h of y = f ( x ). y
9
8
:
x
;
The -oint A is a ma/imum -oint, the -oint > lies on the x a/is, and the -oint 4 is a -oint of infle/ion.
(a)
Bive the -eriod of f . (1)
(b)
'rom consideration of the gra-h of y . f ( x ), find to an accurac o! one signi!icant !igure the range of f . (1)
IB Questionbank Maths SL
25
(c)
(i)
'ind f ′ ( x ).
(ii)
9ence show that at the -oint A, cos x =
(iii)
'ind the e/act ma/imum value.
1 &.
(9)
(d)
'ind the e/act value of the x coordinate at the -oint >. (1)
(e)
∫ f ( x) d x .
(i)
'ind
(ii)
'ind the area of the shaded region in the diagram. (4)
(f)
#
Biven that f ″ ( x ) . :(cos x ) – 6 cos x , find the x coordinate at the -oint 4. (4) (Total 20 marks)
IB Questionbank Maths SL
2!
