AP Calculus BC Formulas

sin x

Opposite / Hypotenuse

cos x

Adjacent / Hypotenuse

tan x

sin x / cos x

cot x

cos x / sin x

sec x

1 / cos x

csc x

1 / sin x

Quotient Identities

tan x = sin x / cos x
cot x = cos x / sin x

Reciprocal Identities

sec x = 1 / cos x
csc x = 1 / sin x

Pythagorean Identities

sin²x + cos²x = 1
sec²x - tan² x =1

Even-Odd Identities

sin(-x) = -sin x
cos (-x) = cos x

tan(-x) = -tan x

Double Angle Identities

sin(2x) = 2sin x cos x
cos²x - sin²x = cos2x

Power-Reducing Formulas

cos²x = (1 + cos 2x)/2
sin²x = (1-cos2x)/1

Sum & Difference (Sine)

sin(A+B) = sinA cosB + cosA sinB
sin (A-B) = sinAcosB - cosAsinB

Sum & Difference (Cosine)

cos(A+B) = cosA cosB - sinA sinB
cos(A-B) = cosAcosB + sinAsinB

Product-to-Sum Formulas

sinA cosB = (sin(A-B) + sin(A+B))/2
cosAsinB = (sin(A+B) - sin(A-B))/2
sinAsinB = (cos(A-B) - cos (A+B))/2
cosAcosB = (cos(A-B) + cos(A+B))/2

Definition of |x|

|x| = x if x ≥ 0
|x| = -x if x < 0

Law of Cosines

c² = a² + b² - 2ab cos C

Distance Formula

√((x₂-x₁)² + (y₂-y₁)²)

Midpoint Formula

((x₁+x₂)/2 , (y₁+y₂)/2)

Laws of Logarithms

ln(ab) = ln a + ln b
ln(a/b) = lna - lnb
ln(a^n) = n ln a
ln (1/a) = -lna
ln(0) → undef.
ln(1) = 0
ln(e) = 1

One-Sided Limits

from left: lim f(x) = L
from right: lim f(x) = L

Definition of a Limit

lim f(x) = L iff lim f(x) from left = L = lim f(x) from right

Limit Laws (x→a)

lim(f+g)=lim f + lim g
lim(f-g)=lim f - lim g
lim (c*f) = c lim f
lim (fg) = lim f * lim g
lim f/g = lim f / lim g
lim k = k
lim x = a
lim nth sqrt f(x) = nth sqrt (lim f(x))
lim f(g(x)) = f(lim g(x))

Continuity Definition

f is continuous at x=c if (1) f(c) exists, (2) lim f(x) exists, (3) lim f(x) = f(c)

Intermediate Value Theorem (IVT)

If f is continuous on [a,b], f(a) doesn’t = f(b), k is between f(a) and f(b), then c exists between a & B such that f(c)=k

Squeeze Theorem

If f(x) ≤ g(x) ≤ h(x) and lim f(x) = lim h(x) = L, then lim g(x) = L

Vertical Asymptote

x=a is a vertical asymptote if lim f(x)→±∞ as x→a±

Horizontal Asymptote

y=a is a horizontal asymptote if lim f(x)=a as x→±∞

Limits of the ratios of two functions

ln^d a < x^c < x^c ln^d a < x^c+d < a^x < x! < x^x
lim (x → infinity) g(x)/f(x) = 0
lim (x → infinity) f(x) / g(x) → infinity

Common Limits

lim (sin x)/x = 1

Definition of Derivative

f'(x) = lim (f(x+h)-f(x))/h as h→0

Alternate Definition of Derivative

f'(c) = lim (f(x)-f(c))/(x-c)

Average Rate of Change

(f(b)-f(a))/(b-a)

Not Differentiable Conditions

(1) Discontinuity

Velocity/Acceleration

s(t) = position

Speed Rules

At rest when v(t)=0

Product Rule

(fg)' = f'g + fg'

Quotient Rule

(f/g)' = (f'g - fg')/g²

Chain Rule

If h(x)=f(g(x)), h'(x)=f'(g(x))·g'(x)

Linear Approximation

y ≈ f(a) + f'(a)(x-a)

Differentials

dy = f'(x) dx

Inverse Function Derivative

(f⁻¹)'(x) = 1 / f'(f⁻¹(x))

L’Hôpital’s Rule

If lim f(x)/g(x) is 0/0 or ∞/∞, then lim f(x)/g(x) = lim f'(x)/g'(x)

Derivative of Constant

d/dx(c) = 0

Constant Multiple Rule

d/dx(c·f(x)) = c·f'(x)

Power Rule

d/dx(xⁿ) = n xⁿ⁻¹

Sum Rule

d/dx(f+g) = f' + g'

Difference Rule

d/dx(f-g) = f' - g'

d/dx(e^x)

e^x

d/dx(a^x)

a^x ln a

d/dx(sin x)

cos x

d/dx(cos x)

-sin x

d/dx(tan x)

sec²x

d/dx(csc x)

-csc x cot x

d/dx(sec x)

sec x tan x

d/dx(cot x)

-csc²x