AP Calc AB/BC

when does a limit fail?

1) limit approaches 2 different values

2) limit approaches an asymptote or non real #

3) function oscillates

when is the function discontinuous?

limx→af(x) DNE - nonremovable

f(a) DNE - removable

limx→af(x) = f(a) non removable

when do derivatives fail?

1) when the slope is undefined or vertical

2) sharp point

3) when the function is discontinuous

derivative of sinx

cosx

derivative of cosx

-sinx

derivative of tanx

sec²x

derivative of secx

secxtanx

derivative of cscx

-cscxcotx

derivative of cotx

-csc²x

Rolle’s Theorem

given f is continuous on [a, b] and differentiable on (a,b), if f(a) = f(b), there has to be a horizontal tangent such that f’© = 0

MVT

given f is continuous on [a, b] and differentiable on (a,b), there exists a number c in (a,b) that f’(c ) = f(b) - f(a)/b-a

motion equation

s(t) = .5gt² + v₀t + s₀; where s is position

feet = -16

m = 4.9

how to find increasing decreasing

f’’(x) = 0 anf find inflection points

extrema theorem

if a function is continuous on [a,b] there will always be a max and min.

integration by parts

integral of udv = uv - integral of vdu

limx-c(f(x)g(x)

multiple the functions

growth rates from fastest to slowest

x^x, x!, a^x, x^p, xln(x), ln(x)

IVT

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

definition of a derivative

f’(x) = limh→0 = f(x+h) - f(x) / h

taylor series

f(x) = f(a) + f’(a)(x-a) + f’’(a)(x-a)²/2! …

maclaurin series e^x

1 + x + x²/2! + x³/3!… + x^N/N!

maclaurin series sinx

x - x³/3! + x⁵/5! - x⁷/7! + … + (-1)^Nx^2N+1/(2N+1)!

maclaurin series cosx

1 - x²/2! + x⁴/4! - x⁶/6! + … + (-1)^Nx^2N/(2N)!

maclaurin series 1/x

1 - (x-1) + (x-1)² + … + (-1)^N(x-1)^N

maclaurin series ln(x)

(x-1) - (x-1)²/2 + … + (-1)^N-1(x-1)^N/N

trig identity

sin²x + cos²x = 1

disc formula

3.14 integral of a, b r²dx

washer formula

3.14 integral of a, b (R² - r²) dx

cross section

integral of a, b Adx

first fundamental theorem

d/dx∫ag(x)f(t)dt = f(g(x))(g’(x))

second fundamental theorem

∫ab f(x) dx = F(b) - F(a).