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L'Hopitals rule

use to find indeterminate limits, find derivative of numerator and denominator separately then evaluate limit

indeterminate forms

All these are indeterminate (which means that if any of these are a limit of something then that limit is undefined):

6th degree Taylor Polynomial

polynomial with finite number of terms, largest exponent is 6, find all derivatives up to the 6th derivative

Taylor Series

If a function is differentiable infinitely many times in some interval around x=a, then the Taylor Series centered at a for f is:

Nth Term Test

If Limit as K approaches infinity, then the Series of K Diverges.

Alternating Series Test

If An is positive, the series ∑(-1)An converges if & only if..

converges absolutely

alternating series converges and general term converges with another test

converges conditionally

alternating series converges and general term diverges with another test

Ratio Test

limit as n→∞ of an+1/an

find interval of convergence

use ratio test, set > 1 and solve absolute value equations, check endpoints

find radius of convergence

use ratio test, set > 1 and solve absolute value equations, radius = center - endpoint

Integral Test

take integral and evaluate, if it goes to a number it is convergent, if there is ∞in it, its divergent

limit comparison test

Let A and B be series with positive terms such that p = limit A/B (to infinity) and 0 < p < infinity, then both series converge or both diverge

Geometric Series Test

general term = a₁r^n, converges if -1 < r < 1

P-Series Test

general term = 1/n^p, converges if p > 1

x(t) and y(t)

dy/dx = dy/dt / dx/dt

second derivative of parametrically defined curve

find first derivative, dy/dx = dy/dt / dx/dt, then find derivative of first derivative, then divide by dx/dt

length of parametric curve

∫ √ (dx/dt)² + (dy/dt)² over interval from a to b

√(dx/dt)² + (dy/dt)² not an integral!

given velocity vectors dx/dt and dy/dt, find speed

given velocity vectors dx/dt and dy/dt, find total distance travelled

∫ √ (dx/dt)² + (dy/dt)² over interval from a to b

area inside polar curve

1/2 ∫ r² over interval from a to b, find a & b by setting r = 0, solve for theta

area inside one polar curve and outside another polar curve

1/2 ∫ R² - r² over interval from a to b, find a & b by setting equations equal, solve for theta.

1

sin² θ + cos² θ =

sec² θ

1 + tan² θ =

csc² θ

1 + cot² θ =

sin(-θ) =

-sin θ

cos(-θ) =

cos θ

tan(-θ) =

-tan θ

sinAcosB + sinBcosA

sin(A + B) =

sinAcosB - sinBcosA

sin(A - B) =

cosAcosB - sinAsinB

cos(A + B) =

cosAcosB + sinAsinB

cos(A - B) =

2sinθcosθ

sin 2θ =

cos 2θ =

cos² θ - sin² θ

tan θ =

sin θ / cos θ

cot θ =

cos θ / sin θ

sec θ =

1 / cos θ

csc θ =

1 / sin θ

cos² θ =

(1 / 2)(1 + cos 2θ)

sin² θ =

(1 / 2)(1 - cos 2θ)

d/dx (x^n) =

nx^(n - 1)

d/dx (fg) =

fg' + gf'

d/dx (f / g) =

(gf' - fg') / g^2

f'(g(x))g'(x)

Chain Rule

cos x

d/dx (sin x) =

-sin x

d/dx (cos x) =

sec² x

d/dx (tan x) =

-csc² x

d/dx (cot x) =

secxtanx

d/dx (sec x) =

-cscxcotx

d/dx (csc x) =

e^x

ƒ'(x) of e^x

d/dx (a^x) =

a^x * ln a

1 / x

d/dx (ln x) =

d/dx (Arcsin x) =

1 / √(1 - x²)

d/dx (Arctan x) =

1 / (1 + x²)

d/dx (Arcsec x) =

1 / (|x| * √(x² - 1))

d/dx (Arccos x) =

-1 / √(1 - x²)

d/dx (Arccot x) =

1 / (1 + x²)

d/dx (Arccsc x) =

-1 / (|x| * √(x² - 1))

0

d/dx [c] =

cf'(x)

d/dx [cf(x)] =

∫a dx =

ax + C

∫x^n dx =

(x^n+1) / (n + 1) + C,

ln |x| + C

∫1 / x dx =

∫e^x dx =

e^x + C

∫a^x dx =

a^x / ln a + C

∫ln x dx =

xln x - x + C

∫sin x dx =

-cos x + C

∫cos x dx =

sin x + C

∫tan x dx =

ln |sec x| + C

∫cot x dx =

ln |sin x| + C

∫sec x dx =

ln |sec x + tan x| + C

∫csc x dx =

-ln |csc x + cot x| + C

∫sec² x dx =

tan x + C

∫secxtanxdx =

sec x + C

∫csc² x dx =

-cot x + C

∫cscxcotx dx =

-csc x + C

∫tan² x dx =

tan x - x + C

∫1 / (a² + x²) dx =

(1 / a)(Arctan (x / a)) + C

Arcsin (x / a) + C

∫1 / (√(a² - x²)) dx =

∫1 / (x * √(x² - a²)) =

(1 / a)(Arcsec (|x| / a)) + C

lim(x→a) f(x) = L

The limit lim(x→a) exists if and only if...

A function y = f(x) is even if...

f(-x) = f(x) for every X in the function's domain.

A function y = f(x) is odd if...

f(-x) = -f(x) for every X in the function's domain.

f(x + p) = f(x) for every value of X.

A function f(x) is periodic with period p(p > 0) if...

Intermediate-Value Theorem:

A function y = f(x) that is continuous on a closed interval [a,b] takes on every value between f(a) and f(b).

2π / |B|

or y = Acos(Bx + C) is...

|a|

or y = Acos(Bx + C) is...

π

y = tan x is...

If f is continuous on [a,b] and f(a) and f(b) differ in sign, then...

the equation f(x) = 0 has at least one solution in the open interval (a,b).

lim(x→±∞) f(x) / g(x) = 0 if...

the degree of f(x) < the degree of g(x).

lim(x→±∞) f(x) / g(x) = ∞ if...

the degree of f(x) > the degree of g(x).

lim(x→±∞) f(x) / g(x) = [c] if...

the degree of f(x) = the degree of g(x).

lim(x→∞) f(x) = b

A line y = b is a horizontal asymptote of the graph y = f(x) if either...

lim(x→a⁺) f(x) = ±∞

A line x = a is a vertical asymptote of the graph y = f(x) if either...

((f(x₁) - f(x₀)) / (x₁ - x₀))

If (x₀,y₀) and (x₁,y₁) are points on the graph of y = f(x), then the average rate of change of y with respect to x over the interval [x₀,x₁] is...

f'(x₀).

If (x₀,y₀) is a point on the graph of y = f(x), then the instantaneous rate of change of y with respect to x at x₀ is...