BC Calc Memorization

(d/dx) (log_b)x

1/(xlnb)

(d/dx) b^x

(b^x)lnb

(d/dx) arcsinx

1/sqrt(1-x²)

(d/dx) arccosx

-1/sqrt(1-x²)

(d/dx) arctanx

1/(1+x²)

(d/dx) arcsecx

1/(|x|sqrt(1-x²))

limit definition of derivative

lim(h→0)(f(x+h)-f(x))/h and lim(x→a)(f(x)-f(a))/(x-a)

derivatives of inverse functions

(f^-1)’(x) = 1/(f’(f^-1(x))

antiderivative of a^x dx

(a^x)/(lna) + C

L’Hopital’s Rule

If lim(x—>a) f(x)/g(x) = 0/0 or inf/inf, then lim(x—>a) f(x)/g(x) = lim(x—>a) f’(x)/g’(x)

volume: disc

V = pi(integral from a to b) r² dx

volume: washer

V = pi(integral from a to b) (R² - r²) dx

volume: shell

V = 2pi(integral from a to b) rh dx

volume: cross section

V = (integral from a to b) A dx

first fundamental theorem

(d/dx) (integral from a to g(x)) f(t) dt = f(g(x)) * g’(x)

second fundamental theorem

(integral from a to b) f(t) dt = F(b)-F(a) where F’(x) = f(x)

velocity

(d/dt) pos

acceleration

(d/dt) velocity

speed

|velocity|

alternating series error bound

error <= |a_(n+1)|

Lagrange error

error<= | (f^{n+1} (c) (x-a)^(n+1))/(n+1)! | where |f^{n+1} (c)| is the maximum value of f^{n+1} (x) on [a,b]

logistic differential equation

(dP/dt) = kP(M-P) where M is the carrying capacity. Fastest growth at P=M/2. P = M/(1+Ce^(-kt))

mclaurin series e^x

sum of n=0 to infinity of (x^n)/n!, interval of convergence R

mclaurin series sinx

sum of n=0 to infinity of ((-1)^n*x^(2n+1))/(2n+1)!, interval of convergence R

mclaurin series cosx

sum of n=0 to infinity of ((-1)^n*x^(2n))/(2n)!, interval of convergence R

mclaurin series 1/(1-x)

sum of n=0 to infinity of x^n, interval of convergence (-1, 1)

Taylor series

f(x) = f(a) + f’(a)(x-a) + (f’’(a)(x-a)²)/2! + (f’’’(a)(x-a)³)/3!+…+(f^{n}(a)(x-a)^n)/n!+…

Euler’s method table

x, y, dy/dx, delta x; add delta x * dy/dx to y for next value

start + accumulation

f(b) = f(a) + (integral from a to b) f’(x) dx

average rate of change

(f(b)-f(a))/(b-a)

instantaneous rate of change

f’(c)

mean value theorem

f’(c) = (f(b)-f(a))/(b-a) must be cont. on [a, b] and differentiable on (a, b)

average value

((integral from a to b) f(x) dx)/(b-a)

intermediate value theorem

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

extreme value theorem

A function f that is continuous on [a, b] has both an absolute min and an absolute max on the interval

continuity

lim (x—>c^-) f(x) = lim (x—>c^+) f(x) = f(c)

squeeze theorem

if f(x)<=g(x)<=h(x) for all x≠c in some interval containing c and if lim (x—>c) f(x) = L = lim (x—>c) h(x), then lim (x—>c) g(x) = L

rolle’s theorem

if f(a) = f(b), then f’(c) = 0

arc length cartesian

(integral from a to b) sqrt(a + (f’(x))²) dx

arc length parametric

(integral from t1 to t2) sqrt((dx/dt)² + (dy/dt)²) dt

speed |v(t)|

sqrt((dx/dt)²+(dy/dt)²)

polar area

(1/2)(integral from theta1 to theta2) r² dtheta

parametric derivatives

(dy/dx) = (dy/dt)/(dx/dt), (d²y/dx²) = (d/dt)(dy/dx)/(dx/dt)

nth term test

diverges if lim n→infinity a_n ≠ 0

geometric series test

(sum from n=0 to infinity) ar^n converges if |r|<1, diverges if |r|>=1, S = a/(1-r)

p-series test

(sum from 1 to infinity) 1/n^p converges if p>1, diverges if p<=1

alternating series test

decreasing terms and lim (n—>inf) a_n = 0 then converges (check for conditionally or absolutely)

integral test

f(x) must be positive, continuous, and decreasing a_n = f(n) (sum from 1 to infinity) a_n converges if (integral from 1 to infinity) f(x) dx converges, (sum from 1 to infinity) a_n diverges if (integral from 1 to infinity) f(x) dx diverges

ratio test

lim(n—>inf) |(a_(n+1))/a_n| < 1 converges, lim(n—>inf) |(a_(n+1))/a_n| > 1 diverges, inconclusive if = 1

direct comparison test

both series positive, a series with terms smaller than a known convergent series also converges, a series with terms larger than a known divergent series also diverges

limit comparison test

if lim (n—>inf) a_n/b_n is finite and positive both series converge or both diverge