Microsoft Word - Calculus 2 Formula Cheat Sheet

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

− u tan 1   + c a a

1

∫ a

2

∫ 

u du = sin −1   + c a a2 − u2

+ u2 1

du =

 y