AP Calculus AB Vocabulary

Limit

value a function approaches as the input approaches a certain point

One

sided limit

Continuity

a function is continuous if the limit equals the value at that point

Removable discontinuity

hole in the graph where limit exists but function is not defined or mismatched

Jump discontinuity

function has different left and right limits

Infinite discontinuity

function approaches infinity at a point

Derivative

instantaneous rate of change or slope of tangent line

Differentiability

function is differentiable if it is continuous and smooth at that point

Power rule

d/dx of xⁿ is n times x to the n minus 1

Product rule

d/dx of uv is u prime v plus u v prime

Quotient rule

d/dx of u over v is u prime v minus u v prime over v squared

Chain rule

d/dx of f of g of x is f prime of g of x times g prime of x

Critical point

point where derivative is zero or undefined

Increasing function

function with positive derivative

Decreasing function

function with negative derivative

Relative maximum

point where function changes from increasing to decreasing

Relative minimum

point where function changes from decreasing to increasing

Inflection point

point where concavity changes

Concave up

second derivative is positive

Concave down

second derivative is negative

First derivative test

uses sign of first derivative to determine relative extrema

Second derivative test

uses sign of second derivative to confirm concavity and relative extrema

Velocity

derivative of position function

Acceleration

derivative of velocity function or second derivative of position

Speed

absolute value of velocity

Mean value theorem

if function is continuous and differentiable there exists c such that f prime of c equals f of b minus f of a over b minus a

Rolle’s theorem

if f of a equals f of b and function is continuous and differentiable then f prime of c equals 0 for some c

Antiderivative

function whose derivative is given function

Indefinite integral

general antiderivative plus constant C

Definite integral

gives net area under curve between two x values

Fundamental theorem of calculus part 1

if F is antiderivative of f then integral from a to b of f of x dx equals F of b minus F of a

Fundamental theorem of calculus part 2

d/dx of integral from a to x of f of t dt equals f of x

Integration by substitution

method to simplify integrals by changing variables

Area under a curve

definite integral of function between two points

Area between curves

integral from a to b of top function minus bottom function dx

Volume by disks

pi times integral from a to b of radius squared dx

Volume by washers

pi times integral from a to b of outer radius squared minus inner radius squared dx

Volume by shells

two pi times integral from a to b of radius times height dx

Slope of tangent line

value of derivative at point

Linear approximation

f of x is approximately f of a plus f prime of a times x minus a

Differential

dy equals f prime of x times dx

Logarithmic differentiation

taking natural log of both sides to differentiate complicated products or quotients

L’Hôpital’s rule

if limit gives indeterminate form use derivative of top and bottom

Related rates

word problems involving rates of change with respect to time

Separable differential equation

can be written as dy over dx equals g of x times h of y and solved by integrating both sides

Initial condition

used to solve for C in general solution of differential equation

Slope field

graphical representation of a differential equation using small line segments

Rectilinear motion

motion along a straight line using position velocity and acceleration functions

Total distance traveled

integral from a to b of absolute value of velocity dx

Displacement

integral from a to b of velocity dx

n times x to the n minus 1

d/dx of xⁿ

cos x

d/dx of sin x

minus sin x

d/dx of cos x

sec squared x

d/dx of tan x

sec x tan x

d/dx of sec x

minus csc x cot x

d/dx of csc x

minus csc squared x

d/dx of cot x

e to the x

d/dx of e to the x

a to the x times ln a

d/dx of a to the x

1 over x

d/dx of ln x

1 over x ln a

d/dx of log base a of x

1 over square root of 1 minus x squared

d/dx of inverse sin x

minus 1 over square root of 1 minus x squared

d/dx of inverse cos x

1 over 1 plus x squared

d/dx of inverse tan x

Constant multiple rule

∫ c f of x dx equals c times ∫ f of x dx

sum and difference rule

∫ f plus or minus g dx equals ∫ f dx plus or minus ∫ g dx

1

xⁿ⁺¹ over n plus 1 plus C for n not equal to

ln absolute value of x plus C

∫ 1 over x dx

e to the x plus C

∫ e to the x dx

a to the x over ln a plus C

∫ a to the x dx

minus cos x plus C

∫ sin x dx

sin x plus C

∫ cos x dx

tan x plus C

∫ sec squared x dx

minus cot x plus C

∫ csc squared x dx

sec x plus C

∫ sec x tan x dx

minus csc x plus C

∫ csc x cot x dx