AP Calculus AB Theorems

limit exists

limit from the left and right equal each other

continuity at a point

limit exists (from left and right are equal)

function value exists

limit and function value are equal

discontinuities

• removable (hole)

• infinite/essential (vertical asymptote)

• jump

continuity in an interval

• (a, b): continuous at every point from a to b

• [a, b]: continuous on (a, b) and limits equal function values at endpoints

intermediate value theorem

if f is continuous on [a, b] and k is a number between f(a) and f(b), there exists at least one number c between a and b such that f(c) = k

requirements for differentiation

continuity

limit at the point exists

limit definition of the derivative (h)

limit as h approaches 0

( f(a + h) - f(a) )/h

situations where the derivative does not exist

• corner (different slopes so different derivatives from left and right)

• cusp (vertical tangent line so slope does not exist)

limit definition of the derivative (x)

limit as x approaches a

( f(x) - f(a) ) / (x - a)

chain rule

d/dx (f(g(x)) = f’(g(x)) g’(x)

speeding up/slowing down at a point

speeding up: v(t) and a(t) have the same sign

slowing down: v(t) and a(t) have opposite signs

f(x) and g(x) are inverse functions

f(x)

• (a, b)

• f’(a) = c

g(x)

• (b, a)

• g’(b) = 1/c

l’Hôpital’s rule

if f and g are differentiable functions and the limit as x approaches c of f(x)/g(x) is indeterminate (0/0 or infinity/infinity) then the limit is equal to the limit of f’(x)/g’(x)

extreme value theorem

if f is continuous in [a, b] and f has an absolute minimum and maximum in that interval, they will occur at the endpoints (x = a, b) or a critical number

relative extrema

occur at critical numbers where the derivative changes sign

• + → - : relative maximum

• - → + : relative minimum

concavity

f’’(x)

• > 0: concave up

• < 0: concave down

• changes sign: point of inflection

f’(x)

• increasing: concave up

• decreasing: concave down

• changes direction: point of inflection

second derivative test

if f is continuous and differentiable on (a, b) and there exists a point c in (a, b) such that f’(c) = 0 (critical number):

• f’’(c) > 0: relative minimum

• f’’(c) < 0: relative maximum

f’’(c) = 0: test fails

Rolle’s theorem

if f is continuous in [a, b] and differentiable in (a, b) and f(a) = f(b), there exists a point c in (a, b) such that f’(c) = 0

mean value theorem

if f is continuous in [a, b] and differentiable in (a, b), there exists a point c in (a, b) such that f’(c) = ( f(b) - f(a) ) / (b - a)

Riemann sum

small rectangles to estimate the area underneath a curve

average function value

integral from a to b of f(x) dx divided by (b - a)

fundamental theorem of the calculus (derivative)

if F is continuous in [a, b] and x is a point in [a, b]

F(x) = integral from a to x of f(t) dt

d/dx (F(x)) = f(x)

fundamental theorem of the calculus (antiderivative)

if f is continuous in [a, b] and F’(x) = f(x)

integral from a to b of f(t) dt = F(b) - F(a)

derivative of the integral from a(x) to b(x) of f(t) dt

f(b(x)) b’(x) - f(a(x)) a’(x)

area between two curves

integral from a to b of upper curve - lower curve

solids of revolution

pi times the integral from a to b of (upper)² - (lower)² dx or dy