AP Calc AB Formulas

sin²x + cos²x =

1

1 + tan²x =

sec^2 x

1 + cot²x =

csc^2 x

sin(A + B) =

sinA cosB + cosA sinB

sin(A - B) =

sinA cosB - cosA sinB

cos (A + B) =

cosA cosB - sinA sinB

cos (A - B)

cosA cosB + sinA sinB

sin(2x) =

2sinx cosx

cos(2x)

cos^2 x - sin^2 x

sin(-x) =

-sinx

cos(-x) =

cosx

tan(-x) =

-tanx

Definition of |x|

|x| = {
x if x >= 0
-x if x < 0

distance between two points

sqrt( (x2 - x1)^2 + (y2 - y1)^2 )

midpoint formula

((x1 + x2) / 2, (y1 + y2) / 2)

ln(ab) =

lna + lnb

ln(a/b) =

lna - lnb

ln(a^n) =

n * lna

ln(1/a) =

-lna

sec(x) =

1 / cos x

csc(x) =

1 / sin x

tan(x) =

sin x / cos x

cot(x) =

cos x / sin x

Definition of a limit

lim x -> a f(x) = L iff lim x -> a^- f(x) = L = lim x -> a^+ f(x)

lim x -> a f +- lim x -> a g

lim x -> a f * lim x -> a g

c

c * lim x -> a f

lim x -> a f / lim x -> a g for lim x -> a g != 0

Definition of Vertical Asymptote:

The line x = a is called a vertical asymptote iff lim x -> a^- f(x) = +-infinity or lim x -> a^+ f(x) = +-infinity

Definition of Horizontal Asymptote: (use “a” and f(x) to answer)

The line y = a is called a horizontal asymptote iff lim x -> -infinity f(x) = a or lim x -> infinity f(x) = a

Limits of the ratios of two functions
Consider these:
x^c
ln a
x^c * ln a
a^x
x!
x^x
x^(c + d)

ln a < x^c < x^c * ln a < x^(c + d) < a^x < x! < x^x

if f grows faster than g, then

lim x -> infinity
g(x) / f(x) = 0
f(x) / g(x) = infinity

Definition of continuity
a function f is continuous at x = a iff

1. f(a) exists
2. lim x -> a f(x) exists
3. lim x -> a f(x) = f(a)

Intermediate Value Theorem (IVT): If, then

If

1. f is continuous on the closed interval [a,b]

2. f(a) != f(b)

3. k is between f(a) and f(b)

Then there exists a number c between a and b for which f(c) = k

Squeeze Theorem

if f(x) <= g(x) <= h(x) and as x -> a, f(x) -> L and h(x) -> L, then g(x) -> L

0

lim x -> infinity e^x =

infinity

-infinity

infinity

1

1

0

e^c = (2 equations of common limits)

lim x -> +- infinity (1 + c/x)^x
lim x -> 0^+ (1 + cx)^(1/x)

-pi / 2

pi / 2

0

0

derivative standard formula/ limit of the difference quotient

f’(x) = lim h -> 0 ( f(x + h) - f(x) ) / h

Derivative alternative formula

f’(x) = lim x -> c ( f(x) - f(c) ) / (x - c)

Normal Line

The line perpendicular to the tangent line at the point of tangency.

Average rate of change of f from x = a to x = b is (definition and equation)

the slope of the secant line between the 2 points
( f(b) - f(a) ) / ( b - a )

average rate of change =

delta f / delta x
delta y / delta x
( f(b) - f(a) ) / (b - a)

Three reasons a function, f, will not be differentiable at a point x = a

1. f is not continuous at x = a

2. The graph of f has a “corner” or “cusp” at x = a

3. The graph of f has a vertical tangent at x = a

s(t) =

position

v(t) =

velocity = s’(t)

|v(t)| =

speed

a(t) = 

acceleration = v’(t) = s’’(t)

particle at rest when v(t) =

0

speed is increasing if

v and a have the same sign

speed is decreasing if

v and a have opposite signs

average velocity is the same as 

average rate of change of the position

differentiation rules (for things like chain rule, product, and quotient rule) (functions f and g, and what c is)

f and g are functions of x
c is a constant

0

n * x^(n-1)

or d/dx [cf(x)] =

c * d/dx f(x)

d/dx f(x) +- d/dx g(x)

e^x

a^x * ln(a)

1/x

1/(x * lna)

Chain Rule: if h(x) = f(g(x), then

h’(x) = f’(g(x) * g’(x)

Product Rule

f’ * g + f * g’

Quotient Rule

(f’ * g - f * g’) / g^2

cosx

-sinx

sec^2 x

-csc^2 x

secx * tanx

-cscx * cotx

The equation for Linear approximation of f for values of x near x = a

y = f(a) + f’(a) (x - a)

Linear approximation of f for values of x near x = a
What is a?

a is the x-value at the point of tangency

(tangent line at x = a)

Inverse Function (if then equation)

f must be differentiable and one-to-one on the interval, then
1 / ( f’(f^-1(x)) )

L’Hospital’s Rule (When, then)

This rule can only be used directly when the limit evaulates to the indeterminate forms 0/0 and infinity / infinity.
lim x -> a f’(x) / g’(x)

1 / sqrt(1 - x^2)

-1 / sqrt(1 - x^2)

1 / (x^2 + 1)

-1 / |x| sqrt(x^2 - 1)

1 / ( |x| * sqrt(x^2 - 1) )

-1 / (x^2 + 1)

Extreme Value Theorem (EVT)

If f is continuous on a closed interval [a,b]
Then f has an absolute max and an absolute min on the interval [a,b]

Definition of critical points: A critical point of f is a number c (x value) such that either

f’(c) = 0 or DNE

Candidates Test for extreme values on a closed interval

1. Find the y-values of critical points in the interval (a,b)
2. Find the y values of endpoints, a and b
3. The largest y-value is the maximum and the smallest y-value is the minimum

Rolle’s Theorem

If
1. f is continuous on [a,b]
2. differentiable on (a,b)
3. f(a) = f(b)

Then there is at least one number c on (a,b) such that f’(c) = 0

Mean Value Theorem

If
1. f is continuous on [a,b]
2. differentiable on (a,b)

Then there exists a number c between a and b such that f’(c) = ( f(b) - f(a) ) / (b - a)

Test for increasing and decreasing respecitively

f’ > 0 means increasing
f’ < 0 means decreasing

Relative Extrema

relative (or local) mins/maxes must occur at critical points

1st derivative test for relative extrema ( f(c) must exist)

1. if f’ changes from positive to negative at c, then f(c) is a relative max
2. if f’ changes from negative to positive at c, then f(c) is a relative min
3. if f’ does not change signs at c, then f(c) is neither a max or a min

Test for concavity

f’’ > 0 means f is concave up
f’’ < 0 means f is concave down

Definition of point of inflection (POI)

A point on the graph of f where the concavity of f changes