AP Calc AB Formulas

tan x =

sin x / cos x

cot x =

cos x / sin x

sec x =

1 / cos x

csc x =

1/ sin x

sin² x + cos² x =

1

sec² x - tan² x =

1

2((sin x)(cos x)) =

sin 2x

cos² x - sin² x

cos 2x

sin (-x) =

-sin x

cos (-x) =

cos x

tan (-x) =

-tan (x)

sin (A+B) =

(sin A)(cos B) + (sin B)(cos A)

sin (A-B) =

(sin A)(cos B) - (sin B)(cos A)

cos (A+B) =

(cos A)(cos B)-(sin A)(sin B)

cos (A-B) =

(cos A)(cos B) + (sin A)(sin B)

Distance between two points

Square root of (X₂-X₁)²+(Y₂-Y₁)²

Midpoint Formula

(X₁ + X₂ / 2, Y₁+Y₂ / 2)

ln (ab) =

ln (a) + ln (b)

ln (a/b)

ln (a) - ln (b)

ln (aⁿ) =

n (ln (a))

ln (1/a)

-ln (a)

lim f(x) x→a^- =

L (from the left)

lim f(x) x→a⁺ =

L (from the right)

Definition of a limit: lim f(x) x→a = L

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

lim x→a (f±g) =

lim x→a (f) ± lim x→a (g)

lim x→a (f*g)=

lim x→a f * lim x→a g

lim x→a c =

c

lim x→a (c*f) =

c * lim x→a f

lim x→a f/g =

lim x→a f / lim x→a g, lim x→a (g) ≠ 0

lim x→a f(g(x)) =

f [lim x→a (g(x))]

Definition of a Vertical Asymptote

lim x→a^- f(x)= ±∞ OR lim x→a⁺ f(x)= ±∞

Definition of Horizontal Asymptote

lim x→^-∞ f(x) = a OR lim x→∞ f(x) = a

Order Growth

ln^d a < x^c < x^c ln^da < x^c+d < a^x < x! < x^x

if f grows faster than g, then

lim x→∞ g(x)/f(x) = 0 AND lim x→∞ f(x)/g(x) = ∞

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)

IVT 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

lim x→-∞ e^x =

0

lim x→∞ e^x =

∞

lim x→0⁺ ln(x) =

-∞

lim x→∞ ln(x) =

∞

lim x→0 e^x-1/x=

1

lim x→0 sin x/ x =

1

lim x→0 1-cosx/x =

0

lim x→±∞ (1+c/x)^x =

e^c

lim x→0⁺ (1+cx)^1/x =

e^c

lim x→-∞ arctan x =

-𝜋/2

lim x→∞ arctan x =

𝜋/2

lim x→-∞ 1/x =

0

lim x→∞ 1/x =

0

Definition of a Derivative (imit of the difference quotient)

f’(x)= lim h→0 f(x+h)-f(x) / h

Definition of a Derivative (alternative form)

f’(x) = lim h→0 f(x)-f(b) / x-b

Normal Line

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

Average rate of change

slope of the secant line (change of y over change of x)

Three reasons a function f will not be differentiable

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.

Average velocity

is the same as average rate of change of the position

Chain Rule:

If h(x) = f(g(x)), then h’(x) = f’(g(x) * g’(x)

Linear approximation for f where the point of tangency is x=a

y = f(a)+f’(a)(x-a)

Derivative of a constant:

d/dx (c) =0

Constant Multiple Rule:

d/dx [cf(x)] = c(d/dx f(x)

Power Rule

d/dx (xⁿ) = nx^n-1

The sum rule (separate polynomial)

d/dx [f(x)+g(x)] = d/dx f(x) + d/dx g(x)

Difference Rule:

d/dx [f(x)-g(x)] = d/dx f(x)-d/dx g(x)

Derivative of the natural exponential function

d/dx(e^x)=e^x

Derivative of the exponential function

d/dx(a^x)=a^xlna

First Derivative

dy/dx

Second Derivative

d²y/dx²

Third Derivative

d³y/dx³

nth Derivative

dⁿy/dxⁿ