AP Calculus BC Formulas

Pythagorean Identity

sin²x + cos²x = 1

Pythagorean Identity

1 + tan²x = sec²x

Pythagorean Identity

1 + cot²x = csc²x

Quotient Identity

tan x = sin x / cos x

Quotient Identity

cot x = cos x / sin x

Reciprocal Identity

sec x = 1 / cos x

Reciprocal Identity

csc x = 1 / sin x

Double Angle Formula

sin(2x) = 2sinx cosx

Double Angle Formula

cos(2x) = cos²x − sin²x

Power Reduction Formula

cos²x = (1 + cos(2x))/2

Power Reduction Formula

sin²x = (1 − cos(2x))/2

Even-Odd Identity

sin(−x) = −sinx

Even-Odd Identity

cos(−x) = cosx

Even-Odd Identity

tan(−x) = −tanx

Sum Formula

sin(A + B) = sinA cosB + cosA sinB

Difference Formula

sin(A − B) = sinA cosB − cosA sinB

Sum Formula

cos(A + B) = cosA cosB − sinA sinB

Difference Formula

cos(A − B) = cosA cosB + sinA sinB

Product-to-Sum Formula

sinA cosB = ½[sin(A+B) + sin(A−B)]

Law of Cosines

c² = a² + b² − 2ab cosC

Distance Formula

√[(x₂ − x₁)² + (y₂ − y₁)²]

Midpoint Formula

((x₁ + x₂)/2, (y₁ + y₂)/2)

Log Property

ln(ab) = ln a + ln b

Log Property

ln(a/b) = ln a − ln b

Log Property

ln(aⁿ) = n ln a

Special Log Values

ln(1) = 0, ln(e) = 1, ln(0) undefined

Limit from the Left

lim x→a⁻ f(x) = L

Limit from the Right

lim x→a⁺ f(x) = L

Definition of a Limit

lim x→a f(x) = L iff both one-sided limits equal L

Continuity at c

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

Intermediate Value Theorem (IVT)

If f is continuous on [a,b], f(a)≠f(b), then f takes every value between f(a) and f(b)

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 VA if lim x→a⁺ f(x)=±∞ or lim x→a⁻ f(x)=±∞

Horizontal Asymptote

y=a is an HA if lim x→±∞ f(x) = a

Definition of Derivative (difference quotient)

f′(x) = lim h→0 [f(x+h)−f(x)]/h

Definition of Derivative (alternate)

f′(c) = lim x→c [f(x)−f(c)]/(x−c)

Average Rate of Change

(f(b) − f(a)) / (b − a)

Reasons f not differentiable at a

1. Discontinuous, 2. Corner/cusp, 3. Vertical tangent

Position Function

s(t) = position

Velocity Function

v(t) = s′(t)

Acceleration Function

a(t) = v′(t) = s″(t)

Particle at Rest

v(t) = 0

Speed increasing

v and a have same sign

Speed decreasing

v and a have opposite signs

Product Rule

(fg)′ = f′g + fg′

Quotient Rule

(f/g)′ = (f′g − fg′)/g²

Chain Rule

(f∘g)′(x) = f′(g(x)) ⋅ g′(x)

Linear Approximation

y ≈ f(a) + f′(a)(x − a)

Derivative of e^x

(e^x)′ = e^x

Derivative of ln x

(ln x)′ = 1/x

Derivative of logₐx

(logₐx)′ = 1/(x ln a)

Derivative of a^x

(a^x)′ = a^x ln a

Derivative of sin x

(sin x)′ = cos x

Derivative of cos x

(cos x)′ = −sin x

Derivative of tan x

(tan x)′ = sec²x

Derivative of csc x

(csc x)′ = −csc x cot x

Derivative of sec x

(sec x)′ = sec x tan x

Derivative of cot x

(cot x)′ = −csc²x

Derivative of arcsin x

(arcsin x)′ = 1/√(1−x²)

Derivative of arctan x

(arctan x)′ = 1/(1+x²)

Derivative of arcsec x

(arcsec x)′ = 1/(|x|√(x²−1))

Extreme Value Theorem (EVT)

If f is continuous on [a,b], then f has an absolute max and min on [a,b]

Critical Point

f′(c)=0 or f′(c) does not exist

Candidates Test

Check critical points and endpoints for extrema

Rolle’s Theorem

If f continuous on [a,b], differentiable on (a,b), f(a)=f(b), then ∃ c with f′(c)=0

Mean Value Theorem (MVT)

If f continuous on [a,b], differentiable on (a,b), then ∃ c with f′(c)=(f(b)−f(a))/(b−a)

1st Derivative Test

If f′ changes + to − → local max

Concavity

f″>0 concave up, f″<0 concave down