ap calc bc stuff to remember

trig derivatives, exponential derivatives, log derivatives

d/dx sin x

cos x

d/dx cos x

-sin x

d/dx tan x

sec²x

d/dx cot x

-csc²x

d/dx sec x

sec x × tan x

d/dx csc x

-csc x × cot x

d/dx a^x

ln a × a^x

d/dx e^x

e^x

d/dx log_ax

1 / (ln a × x)

d/dx ln x

1 / x

d/dx xⁿ

nx^n-1

derivative of inverse function g(x)

1 /  g’ (g^-1(x))

d/dx sin^-1(g)

g’ / √ 1-g²

d/dx cos^-1(g)

-g’ / √1-g²

d/dx sec^-1(g)

g’ / |g| √g²-1

d/dx csc^-1(g)

-g^’ / |g| √g²-1

d/dx tan^-1(g)

g’ / 1+g²

d/dx cot^-1(g)

-g’ / 1+g²

y = f(g(x)) (chain rule)

f’(g(x))g’(x)

y= f(x)g(x) (product rule)

f’(x)g(x) + f(x)g’(x)

y= f(x) / g(x) (quotient rule)

f’(x)g(x) - f(x)g’(x) / g(x)²

y= f(g(h(x))) (chain inside of a chain)

𝑓′(𝑔(ℎ(𝑥))⋅𝑔′(ℎ(𝑥))⋅ℎ′(𝑥)

d/dx e^g(x)

e^g(x) x g’(x)

d/dx a^g(x)

ln a x a^g(x) x g’(x)

d/dx ln (g(x))

1/ g(x) x g’(x)

d/dx log_a(g(x))

1 / (ln a x g(x)) x g’(x)

g’(x) when f(g(x)) = x

1 / f’ (g(x))

integral of xⁿ

x^n-1/n+1 +c

integral of a^x

a^x / ln a +c

integral of tan x

-ln |cos x| +c

integral of cot x

ln |sin x| +c

integral of sec x

ln |sec x + tan x| + c

integral of csc x

-ln |csc x + cot x| + c

integral of log_ax

x ln x - x / ln a +c

intermediate value theorem

if a function f is continuous on the interval [a,b] and k is a number between f(a) and f(b), then there is at least one x-value c between a and b such that f(c ) = k (a root exists)

mean value theorem

if a function is continuous and differentiable on [a,b], then for some value c in between the interval, the instantaneous rate of change at x=c will be equal to the average rate of change on [a,b]

rolle’s theorem

if a function f is continuous and differentiable on the interval [a,b] and f(a) = f(b), then there is at least one number c in (a,b) such that f’(c ) = 0.

extreme value theorem

if a function f is continuous on the closed interval [a,b], then f is guaranteed to attain an absolute minimum and absolute maximum value on [a,b]

local maximums and minimums (first derivative test)

when f’(c ) = 0 or is undefined, then it is a local maximum if it changes from positive to negative around the point, or a local minimum if it changes from negative to positive around the point.

relative maximums and minimums (second derivative test)

if f’’(c ) < 0 , f(c ) is a relative maximum. if f’’(c ) is > 0, f(c ) is a relative minimum. if f’’ (c ) = 0, the test is inconclusive.

midpoint riemann sum

adding all the rectangles where the area of the rectangle is the base times the height from the midpoint of the base.

trapezoidal riemann sum

adding all the trapezoids where the area of the trapezoid is ½h(b1 + b2).

fundamental theorem of calculus

∫_a^b f(x)dx = F(b) - F(a)…. (shows differentiation and integration are inverse operations)

second fundamental theorem of calculus

if f is a continuous function on an interval [a, b], then the derivative of its accumulation function, F(x) = d/dx (∫_a^x f(t)dt) is f(x).

an integrals with bounds of the same point are equal to

0

∫_a^bf(x)dx =

-∫_b^af(x)dx

∫_a∞ f(x)dx =

lim t → ∞ ∫_a^t f(x)dx

integration by parts formula

∫(u)(dv) = (u)(v) - ∫ v(du)

euler’s method equation

y_n+1= y_n + f’(X_n) (Δx)

logistic differential equations

dP/dt = kP(1-P/a) OR dP/dt = kP(a-P)

average value of an integral

1/b-a ∫_a^b f(x)dx

total distance traveled

∫_t1^t₂ |v(t)| dt

area between curves in terms of x (can be switched with y)

A = ∫_x1^x₂ (top - bottom) dx

disc method for horizontal rotation (switch for r(y) if vertical rotation)

V = ∫_a^b 𝝅[r(x)]² dx

washer method for horizontal rotation (switch for r(y) if vertical rotation)

𝝅∫_a^b ((R(x)²-(r(x))²) dx, where R is the midpoint height of the top curve and r is the midpoint height of the bottom curve

arc length of a curve

∫_a^b (sqrt1+(f(x))²)dx

derivative (slope) of parametric curve

dy/dx = y’(t)/x’(t)

parametric speed

s(t) = (sqrt(x’(t))² + (y’(t))²)

length of parametric curve

∫_t1^t₂ (sqrt(x’(t))² + (y’(t))²)dt

derivatives of vector-valued functions

f(t) = <x(t) , y(t)>
f’(t) = <x’(t), y’(t)>
f’’(t) = <x’’(t), y’’(t)>

total distance of vectors

∫_t1^t₂ (sqrt(x’(t))² + (y’(t))²) dt

polar to rectangular coordinates

x = rcosθ
y= rsinθ

slope of a polar curve

dy/dx = (d/dθ[y]) / (d/dθ[x])

sum of geometric series

S = a₁ / (1-r)

harmonic series divergence

by p-series test

nth term test

diverges if lim does not equal 0

p-series series

converges if p>1 and diverges if 0<p<1

geometric series

when |r| < 1, the series converges to the sum of geometric series formula, when |r|>1, the series diverges

integral test

the integral stays convergent or divergent in its sum if it is continuous, positive, and eventually decreases as x → infinity

direct comparison test

when 0<an<bn, if bn converges, an converges. if an diverges, bn diverges (an is the smaller series, bn is the bigger series)

limit comparison test

if lim n→ inf an/bn = L, where L is finite and positive, then both an and bn either converge or diverge equally.

ratio test

if lim n→ inf |a_n+1/a_n| = k, then a_n’s sum converges absolutely if k <1 and diverges if k>1

alternating series test

the sum of (-1)ⁿa_n converges if lim n→ inf a_n=0 and a_n is a positive, decreasing sequence

nth degree taylor polynomial of f about x=c

P_n(x) = f(c ) + f’(c )(x-c) + (f’’(c ) / 2!)(x-c)² +…+ (f⁽ⁿ⁾(c ) / n!) (x-c)ⁿ

maclaurin polynomial

P_n(x) = f(0) + f’(0)x + (f’’(0)/2!)x² + (f’’’(0)/3!)x³ +… (f⁽ⁿ⁾(0)/n!)xⁿ

1/1-x power series

1 + x + x² + x³ + … + xⁿ +…

e^x power series

1 + x + (x²/2!) + (x³/3!) +…+ (xⁿ/n!)+…

sin x power series

x - (x³/3!) + (x⁵/5!) - (x⁷/7!) + … + (-1)ⁿ(x²ⁿ⁺¹/(2n+1)!)+…

cos x power series

1 - (x²/2!) + (x⁴/4!) -(x⁶/6!) + …. + (-1)ⁿ(x²ⁿ/2n!)+…