calc formula test

Intermediate Value Theorem

if f(x) is continuous on the closed interval [a,b], every y between f(a) and f(b) must exist on f(x) for at least one x

Extreme Value Theorem

if f is continuous on a closed interval [a,b], then f has both an absolute max and absolute min on that interval

Mean Value Theorem

if f(x) is continuous on the closed and differentiable on the open, there must be some c between a and b such that [f(b) - f(a)]/(b-a) = f’(c) (average rate of change is equal to instantaneous rate of change)

Second derivative test

let f’(c) = 0 and f’’(x) exists on open interval. if f’’(x) is positive, concave up/relative min, if negative, concave down/relative max

how does displacement and total distance traveled relate to velocity

displacement is the integral of velocity, total distance traveled is the integral of absolute value of velocity

how to tell if an object is speeding up or slowing down

speeding up if acceleration and velocity are the same sign, slowing down if acceleration and velocity are opposite signs

what is average rate of change, instantaneous, and average value

avg rate of change = [f(b) - f(a)]/(b-a), instantaneous = f’(x), avg val = 1/(b-a) times integral from a to b of f(x)dx

Area of a trapezoid

½(b1 + b2)h

Disk/Washer formula

Disk: V = pi times integral from a to b of [R(x)]²dx
Washer: V = pi times integral from a to b of [R(x)² - r(x)²]dx

Volume of known cross sections

V = integral of A(x)dx
square A(x) = s²
semicircle A(x) = ½πr²
equilateral triangle A(x) = sqrt(3)/4 times s²

Arc Length of Rectangular curve

s = integral from a to b of sqrt(1 + [f’(x)]²)dx

Integration by parts formula

uv - integral of vdu

Sum of an infinite geometric series

first term divided by 1 - r

Error bound for alternating series