Calc BC Everything to Memorize

ill be adding to the cards as the year goes on :)

limits existence theorem

function continuous when f(x) = lim (x→c^-) f(x) = lim (x→c⁺) f(x)

squeeze theorem

h(x) ≤ g(x) ≤ f(x), lim (x→a) h(x) = c = lim (x→a) f(x), lim g(x) (x→a) = c

intermediate value theorem

function continuous on [a,b], f(a) ≤ f(c) ≤ f(b)

derivative intermediate value theorem

function differentiable on [a,b], f’(a) ≤ f’ ≤ f’(b)

composition of limits (lim (x→a) g(f(x)))

lim (x→a) f(x) = c, lim (x→c) g(x)

lim (θ→0) sin cθ / cθ

1

limit definition of a derivative (x→a)

f’(a) = lim (x→a) (f(x) - f(a)) / (x - a)

limit definition of a derivative (h→0)

f’(x) = lim (h→0) (f(x+h) - f(x)) / h

limit definition of a derivative (h→0) at a point

f’(a) = lim (h→0) (f(a+h) - f(a)) / h

power rule

d / dx (xⁿ) = nx^n-1

product rule y = uv

y’ = uv’ + vu’

quotient rule y = u / v

(vu’ - uv’) / v²

chain rule y = f(g(x))

y’ = f’(g(x))(g’(x))

outside/inside rule y = (f(x))ⁿ

y’ = n(f(x))^n-1f’(x)

derivative of an inverse function

(f^-1)’(x) = 1 / f’(f^-1(x))

derivative of an exponential function y = a^x

y’ = (ln a)(a^x)(dx)

derivative of a natural log y = ln x

y’ = dx / x

derivative of a log y = log_ax

y’ = dx / x(ln a)

d / dx cos x

-sin x

d / dx sin x

cos x

d / dx sec x

sec x tan x

d /dx csc x

-csc x cot x

d/ dx tan x

sec²x

d / dx cot x

-csc²x

d / dx cos^-1x

-dx (1 - (x)²)^-1/2

d / dx sin^-1x

dx (1 - (x)²)^-1/2

d / dx sec^-1x

dx (|x|((x)²-1)^1/2)^-1

d / dx csc^-1x

-dx (|x|((x)²-1)^1/2)^-1

d / dx tan^-1x

dx / (1 + (x)²)

d / dx cot^-1x

-dx / (1 + (x)²)

extreme value theorem

function continuous on [a,b], function has at least 1 max or min on [a,b] - if not a closed interval, there are no absolute maxes or mins

mean value theorem

function continuous on [a,b] and differentiable on (a,b), (f(b) - f(a)) / (b - a) = f’(c)

Rolle’s theorem

function continuous on [a,b] and differentiable on (a,b), f(a) = c = f(b), (c - c) / (b - a) = 0