AP Calculus AB Formulas

What is the limit?

y’ = m = lim h→ 0 [f(a+h) - f(a)]/ h

What is the limit 2?

y’ = m = lim x → a [f(x) - f(a)]/x-a

y’ = m = lim h → 0 [f(a+h) - f(a-h)]/2h

f’© = f(b) - f(a)/ b-a

What is the equation for l’Hopital?

L’H → lim x→ 0 f(x)/g(x) = lim x→ a f’(x)/g’(x)

What are critical points?

f’(x) = 0, f’(x) = DNE

How to find increasing/decreasing?

f’(x) > 0 → inc, f’(x) < 0 → dec

Concavity?

f”(x) > 0 → conc. up (min), f”(x) < 0 → conc. down (max)

Local Extrema?

f’(x) = 0 and f”(x) > 0 → min

f’(x) = 0 and f”(x) < 0 → max

Point of inflection?

a. f”(x) = 0

b. f”(x) = DNE

c. where f’(x) changes sign (means… changes concavity)

Linearization?

L(x) = f(a) + f’(a)(x-a)

Newton’s method?

Xn + t = Xn - f(Xn)/f’(Xn)

Approximating binomials

(t + X)^k = x + Kx

y’ of ln(u)

1/u

y’ of loga u

1/ulna

y’ of e^x

e^x

y’ of lnx

1/x

y’ of a^x

a^x lna

y’ of x^n

nX^(n-1)

y’ of cu

c(du/dx)

y’ of sinx

cosx

y’ of cosx

-sinx

y’ of tanx

sec²x

y’ of cscx

-cscxcotx

y’ of secx

secxtanx

y’ of cotx

-csc²x

y’ of inverse sin

1/sqrt(1-u²)

y’ of inverse cos

-1/sqrt(1-u²)

y’ of inverse tan

1/(1+u²)

y’ of inverse csc

-1/u sqrt(u²-1)

y’ of inverse sec

1/u sqrt(u²-1)

y’ of inverse cot

-1/1+u²