AP Calc AB - Formulas

Continuity at a point (x = c)

1. f(x) is defined at f(c)
2. The limit as x approaches c of f(x) exists
3. The limit as x approaches c of f(x) = f(c)

Volume of a Sphere

V = 4/3 pi r^3

Surface Area of a Sphere

S = 4 pi r^2

Intermediate Value Theorem

If f(x) is continuous on a closed interval [a,b] and k is any number between f(a) and f(b), then there is at least one number c in [a,b] such that f(c) = k.

Continuity on an open interval, (a,b)

f(x) is continuous if for every point on the interval (a,b) the conditions for continuity at a point are satisfied.

Continuity on a closed interval, [a,b]

1. f(x) is continuous on the closed interval (a,b)
2. The limit from the right as x approaches a of f(x) is f(a)
3. The limit from the left as x approaches b of f(x) is f(b)

Indeterminate form

0/0

The limit as x approaches 0 of sin x / x

1

The limit as x approaches 0 of (1 - cos x) / x

0

sin π

0

cos π

-1

sin 0

0

cos 0

1

sin π/2

1

cos π/2

0

sin π/4

√2/2

cos π/4

√2/2

sin 3π/2

-1

cos 3π/2

0

sin π/6

1/2

cos π/6

√3/2

sin π/3

√3/2

cos π/3

1/2

sec x

1 / cos x

csc x

1 / sin x

cot x

1 / tan x = cos x / sin x

tan x

sin x / cos x

cos²x + sin²x

1

1 + tan²x

sec²x

1 + cot²x

csc²x

sin(2x)

2 sin x cos x

cos(2x)

cos²x - sin²x

cos²x

(1 + cos 2x) / 2

sin²x

(1 - cos 2x) / 2

Area of an equilateral triangle

√3s² / 4

Area of a circle

πr²

Circumference of a circle

2πr

Volume of a right circular cylinder

πr²h

Volume of a cone

πr²h/3

ln e

1

ln 1

0

ln (mn)

ln m + ln n

ln (m/n)

ln m - ln n

ln mⁿ

n ln m

If f(-x) = f(x)

f is an even function

If f(-x) = -f(x)

f is an odd function

How to get from precalculus to calculus

Limits

Limit Definition of a Derivative

f'(x) = lim as ∆x → 0 of [ f(x + ∆x) - f(x) ] / ∆x

Continuity & differentiability

Differentiability implies continuity, but continuity does not necessarily imply differentiability.

Alternate Limit Definition of a derivative

f'(x) = lim as x → c of [ f(x) - f(c) ] / [ x - c]

Derivative of a constant

d/dx[c] = 0

Power Rule for Derivatives

d/dx[x^n]=nx^(n-1)

d/dx[x]

1

Constant Multiple Rule for Derivatives

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

Sum and Difference Rules for Derivatives

d/dx[f(x) ± g(x)] = f'(x) ± g'(x)

d/dx[sin x]

cos x

d/dx[cos x]

-sin x

Velocity, v(t)

Derivative of Position, s'(t)

Derivative

Slope of a function at a point/slope of the tangent line to a function at a point

Average speed

∆s/∆t

Instantaneous velocity

Derivative of position at a point

Position function of a falling object (with acceleration in ft/s²)

s(t) = -16t²+ v₀t + s₀, v₀ = initial velocity, s₀ = initial height

Position function of a falling object (with acceleration in m/s²)

s(t) = -4.9t²+ v₀t + s₀, v₀ = initial velocity, s₀ = initial height

The Product Rule

If two functions, f and g, are differentiable, then
d/dx[ f(x) g(x) ] = f(x) g'(x) + g(x) f'(x)

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

f(x) g'(x) + g(x) f'(x)

The Quotient Rule

If two functions, f and g, are differentiable, then
d/dx[ f(x) / g(x) ] = [g(x)f'(x) - f(x) g'(x)] / [g(x)]²

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

[g(x)f'(x) - f(x) g'(x)] / [g(x)]²

d/dx[tan x]

sec² x

d/dx[cot x]

-csc² x

d/dx[sec x]

sec x tan x

d/dx[csc x]

-csc x cot x

d/dx[ln x]

1/x, x>0

d/dx[e^x]

e^x

d/dx[arcsin x]

1/√(1-x²)

d/dx[arccos x]

-1/√(1-x²)

d/dx[arctan x]

1/(1+x²)

d/dx[arccot x]

-1/(1+x²)

d/dx[arcsec x]

1/(|x|√(x²-1))

d/dx[arccsc x]

-1/(|x|√(x²-1))

Chain Rule: d/dx[f(g(x))] =

f'(g(x))g'(x)

Guidelines for implicit differentiation

1. Differentiate both sides w.r.t. x
2. Move all y' terms to one side & other terms to the other
3. Factor out y'
4. Divide to solve for y'

d/dx[ln u]

u'/u, u > 0

d/dx[e^u]

u' e^u

Guidelines for solving related rates problems

1. Given, Want, Sketch
2. Write an equation using variables given/to be determined
3. Differentiate w.r.t. time (using chain rule)
4. Plug in & solve

Derivative of an inverse (if g(x) is the inverse of f(x))

g'(x) = 1/f'(g(x)), f'(g(x)) cannot = 0

d/dx[a^x]

(ln a) a^x

d/dx[a^u]

u' (ln a) a^u

d/dx[log_a x]

1/((ln a) x)

d/dx[log_a u]

u'/((ln a) u)

Extreme Value Theorem

If f is continuous on the closed interval [a,b] then it must have both a minimum and maximum on [a,b].

Critical number

x values where f'(x) is zero or undefined.

Mean Value Theorem

if f(x) is continuous and differentiable, slope of tangent line equals slope of secant line at least once in the interval (a, b)

f '(c) = [f(b) - f(a)]/(b - a)

Rolle's Theorem

Let f be continuous on [a,b] and differentiable on (a,b) and if f(a)=f(b) then there is at least one number c on (a,b) such that f'(c)=0 (If the slope of the secant is 0, the derivative must = 0 somewhere in the interval).

∫0 dx

C

∫k dx

kx + C

∫xⁿ dx

xⁿ⁺¹ / (n + 1) + C

FUNdamental Theorem of Calculus

∫ f(x) dx on interval a to b = F(b) - F(a)

∫k f(x) dx

k ∫ f(x) dx

∫cos x dx

sin x + C

∫sin x dx

-cos x + C