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DnD Fifth Edition Cheat Sheet/DM Screen Sheets. contains some valuable rules that can be found quickly on first glance.
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LOGIKA 1. IF 2. IFERROR 3. OR 4. AND 5. NOT TEKS 25. LOWER 26. MID 27. PROPER 28. REPLACE 29. REPT 30. RIGHT 31. SUBSTITUTE 32. TEXT 33. UPPER 34. VALUE LOOKUP & REFERENSI 6. CHOOSE 7. HLOOKUP 8....
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Calculus II Final Exam Cheat Sheet L’Hospital’s Rule: When taking a limit, if you get an indeterminate form i.e.
±∞ 0 , ,etc you take ±∞ 0
the derivative of the top and bottom and evaluate the limit again… Integration by Parts Trig Substitution If the integral contains the following root udv = uv − vdu use the given substitution and formula to b convert into an integral involving trig and = uv |ba − vdu a functions. choose u and dv from integral and compute du by a differentiating u and computing v by integrating dv a 2 − bx 2 ⇒ x = sin θ b Trig Stuff a sin 2 x = 2 sin x cos x bx 2 − a 2 ⇒ x = secθ sin 2 x + cos 2 x = 1 b 1 1 + tan 2 x = sec2 x cos 2 x = (1 + cos 2 x) a 2 a 2 + bx 2 ⇒ x = tan θ 2 2 1 + cot = csc x b 1 sin 2 x = (1 − cos 2 x) 2
∫
∫
∫
Product and Quotients of Trig Functions
∫
∫
For sin n x cos m xdx we have the following:
For tan n secm xdx we have the
1. n odd. Strip 1 sine out and convert conv ert rest to 2 cosines using sin = 1 − cos2 x , then use the substitution u = cos x . 2. m odd. Strip 1 cosine out and convert rest to sines using cos 2 = 1 − sin 2 x , then use the substitution u = sin x . 3. n and m both odd. Use either 1 or 2 4. n and m both even. Use double angle and/or half angle formulas to reduce the integral into a form that can be integrated.
1.
2.
3. 4.
following: n odd. Strip 1 tangent and 1 secant out and convert the rest to secants using tan 2 x = sec 2 x − 1 , then use the substitution u = sec x m even. Strip 2 secants out and covert rest to tangents using sec2 = 1 + tan 2 x , then use the substitution u = tan x . n odd and m even. Use either 1 or 2. n even and m odd. Each integral will be dealt with differently.
Centroid _
x=
1
b
x[ f ( x) − g ( x)]dx A ∫
_
y=
a
x 2 = 4 py
2
Parabola focus : (0, p)
= 4 px
Directrix y = − p focus : ( p, 0) directrix : x = − p
1
b
1
{[ f ( x))]] A ∫ 2
− [ g ( x))]]2 } dx
a
x 2
y2
+ =1 a 2 b2 Vertices and foci are always on major axis c 2 = a 2 − b2 Make a box with sides determined by the square root of the denominators. Ellipse
2
x 2
y2
− = 1 or a 2 b2 y 2 x 2 − =1 a 2 b2 Foci and vertices are always on axis determined by positive squared term. Draw box and make diagonal asymptotes. c 2 = a 2 + b2 Hyperbola
∞
∑
Taylor Series
f '( a )( x − a) n n!
n =0
an +1
Ratio Test lim n →∞
Differential Equations P (t ) = P0e k ⋅t Exponential growth
Separable:
converges if < 1
an
dy dx
= g (x ) f ( y ) cross multiply
dy + P ( x) y = Q( x) use I.F.F Absolute Convergence: If the absolute value of the Linear: dx series converges the series is said to be absolutely convergent. Arc Length Cartesian b
L=
∫ a
2
b
dy 1 + dx if dx
= f ( x), a ≤ x ≤ b or L = ∫ a
2
dx 1 + dy if x = f ( y ), a ≤ x ≤ b dy
Parametric b
L=
∫ a
Polar
2
2
dx dy dt + dt dt
2
b
L=
∫ a
dr r + dθ d θ 2
Surface Area Cartesian & Parametric
Area of Polar (not surface area)
b
b
∫
S = 2π r ⋅ L
A=
a
1
∫ 2 r dθ 2
a
Midpoint Rule b
___
___
___
1
2
n
Cartesian to Polar: x = r cosθ
∫ f ( x)dx ≈ ∆x[ f ( x ) + f ( x ) + ... + f ( x )] a
y = r sin θ Polar to Cartesian: r 2 = x2 + y2
Trapezoid Rule b
∫
f ( x)dx ≈
∆ x 2
a
[ f ( x0 ) + 2 f ( x1 ) + 2 f ( x2 ) + ... + 2 f ( xn −1 ) + f ( xn )]
Simpson’s Rule b
∫ f ( x)dx ≈
∆ x
a
3
[ f ( x0 ) + 4 f ( x1 ) + 2 f ( x2 ) + 4 f ( x3 ) +... + 2 f ( xn −2 ) + 4 f ( xn−1 ) + f ( xn )]
tan θ =
Common Integrals
∫ kdx = kx + c ∫ ∫ x
x n dx = −1
1 n +1
x n +1
dx = ln | x | + c
1
1
∫ ax + b dx = a ln | ax + b | +c ∫ ln udu = u ln(u) − u + c ∫ e du = e + c n
u
∫ cos udu = sin u + c ∫ sin udu = − cos u + c ∫ sec udu = tan u + c ∫ sec u tan udu = sec u + c ∫ csc u cot udu = − csc u + c ∫ csc udu = − cot u + c 2
2
∫ tan udu = ln | sec u | + c ∫ sec udu = ln | sec u + tan u | +c 1