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Calculus 1 Reference

Derivatives

Function

Derivative

xn nxn-1
ax axln a
ex ex
ln x 1/x
sin x cos x
cos x - sin x
tan x sec2 x
cot x - csc2 x
sec x sec x tan x
csc x - csc x cot x
y y'

Integrals

Function Integral
xn xn+1/(n+1) + c
ax ax/ln a  + c
ex ex + c
1/x ln |x| + c
sin x - cos x + c
cos x sin x + c

Geometry

Area of a circle r2
Circumference of a circle 2r
Surface area of a sphere I think 4∏r2
Volume of a sphere (4/3)∏r3
Volume of a cylinder r2h
Volume of a cone (1/3)∏r2h

Trigonometry

0 ∏/6 ∏/4 ∏/3 ∏/2 2∏
sin 0 1/2 1/√2 √3/2 1 0 0
cos 1 √3/2 1/√2 1/2 0 - 1 1
tan 0 1/√3 1 √3 infinity 0 0

sin2 x + cos2 x = 1 sin 2x = 2sin x cos x
sec2 x = tan2 x + 1 sin2 x = (1/2)(1 - cos 2x)
csc2 x = cot2 x + 1 cos2 x = (1/2)(1 + cos 2x)
cos (-x) = cos (x) sin (-x) = -sin (x)

Law of Cosines:

For a triangle with angles A, B and C, and sides opposite their respective angles of length a, b, and c, you can use the following formula for any angle:
a2 = b2 + c2 - [2 (b) (c) cos A]

To convert from logs to natural logs:

Logb a = (ln a)/(ln b)

 

Calculus 2 Reference

More Derivatives

Function Derivative
arctan x 1/(1 + x2)
arcsin x 1/√(1 - x2)

More Integrals

Function Integral
sec x ln |sec x + tan x| + c
csc x - ln |csc x + cot x| + c
1/(1 + x2) arctan x + c
1/√(1 - x2) arcsin x + c
sink x cos x Notice how each one of these is next to its derivative, before it's solved. sink+1 x/(k+1) + c
cosk x sin x - cosk+1 x/(k+1) + c
tank x sec2 x tank+1 x/(k+1) + c
seck x (sec x tan x) seck+1 x/(k+1) + c

In Integrals, the following trig substitutions are allowed:

√(a2 - x2) = a cos when x = a sin
√(x2 - a2) = a tan when x = a sec
√(a2 + x2) = a sec when x = a tan

Areas and Volumes with Integrals

Cartesian (Standard) Coordinates
Area between curves ∫ f(x) - g(x) dx
Length of a curve ∫ √[1 + f'(x)2] dx
Volume around x axis (disc/donut) ∏ * ∫ f(x)2 - g(x)2 dx
Volume around y axis (shells) 2∏ * ∫ x[f(x) - g(x)] dx
Polar Coordinates
Area inside curves (1/2) * ∫ r2 d
Length of a curve ∫ √[r2 + (r')2] d

Polar Coordinates

sin  = y/r tan = y/x
cos = x/r r2 = x2 + y2 

Simpson's Rule

(n is always even)

[(b-a)/3n][f(x0) + 4f(x1) + 2f(x2) + 4f(x3) + ... + f(xn)]

 
Copyright 2004 Bruce
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