AP Calculus Master List for Equations

lim x→0 sin(x) / x =

1

lim x→0 1-cos(x) / x =

0

lim x→0 x / sin x =

1

lim x→0 cos x -1 / x = 

0

Squeeze Theorem

if g(x) ≤ f(x) ≤ h(x)
and if lim x→a g(x) = L and lim x→a h(x) = L
then lim x→a f(x) = L

Types of Discontinuities

1. Hole (removeable)
   2. Discontinuity due to a vertical asymptote
   3. Jump Discontinuity

Formal Definition of Continuity

1. f(a) must be defined at

2. lim x→a f(x) exists

3. lim x→a f(x) = f(a)

Horizontal Asymptote Rules

small / big number = 0
big number / big number = 1

Big number / small number = infinity

IVT Theorem

1. The function f(x) is continuous on an interval [a,b]

2. f(a) < f(b) or f(b) < f(a)

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

Average Rate of Change

f(a+h) -f(a) / f(a+h) -a

or

f(x) -f(a) / x-a

Instantaneous Rate of Change 

lim h→0 f(a+h) -f(a) / h 

or 

lim x→a f(x) - f(a) / x-a 

Lagrange notation

f ‘ (x) or y’

Leibniz notation 

dy/dx 

Definition of a Derivative

f ‘ (x) = lim x→0

f(x+h) - f(x) / h

Equation of the Tangent line

y - y1 = m(x - x1)

Differentiability

The derivative exists for each point in the domain. The graph must be a smooth line or curve for the derivative to exist. In other words, the graph looks like a line if you zoom in.

The derivative fails to exists when there is a discontinuity, corner or cusp, or a vertical tangent line.

power rule

f ‘ (x) = n ⋅ x^n-1

derivative rules:

Constant: D/dx(b) = 0
Constant Multiple: d/dx(bu) = b
Sum/Difference: d/dx(u+v) = du/dx +dv/dx

derivative of cos x

-sin x

derivative of sin x 

cos x 

derivative of a^x

a^x ⋅ ln(a)

derivative of e^x

e^x

derivative of loga(x) 

1/x ⋅ 1/ln(a) 

derivative of ln(x)

1/x

product rule

(f ‘ ⋅ g ) + (f ⋅ g’ )

quotient rule 

(f ‘ ⋅ g ) - (f ⋅ g’) / g²

derivative of tan x

sec²x

derivative of cot x

-csc²x

derivative of sec(x) 

sec(x) ⋅ tan(x) 

derivative of csc(x)

-csc(x) ⋅ cot(x)

chain rule

f ‘ (g(x)) ⋅ g’(x)

derivative of [ f^-1(x) ] 

1 / f ‘ (x) ⋅ f^-1(x) 

derivative of sin^-1(x)

1 / √1-x²

derivative of cos^-1(x)

-1 / √1-x³

derivative of sec^-1(x) 

1 / |x| ⋅ √x³-1

derivative of csc^-1(x)

-1 / |x| ⋅ √x³-1

derivative of tan^-1(x)

1/1+x²

derivative of cot^-1(x)

-1/1+x²

position function

s(t)

velocity function 

v(t) or s ‘ (t) 

acceleration function

a(t) = v’(t)

Velocity particle direction

v(t) < 0 means the particle is either moving up or moving left

v(t) > 0 means the particle is moving right or up.

v(t) = 0 means the particle is at rest

Average Velocity = 

Average Rate of Change 

Speed

Abs value of velocity

If velocity and acceleration have the same sign, the particle is

speeding up

If velocity and acceleration have different signs, the particle is 

slowing down 

Displacement

net change in position

If the graph is concave up, the tangent line is

an underestimate

If the graph is concave down, the tangent line is 

an overestimate 

L’Hospital’s Rule

Suppose f(a) = 0 and g(a) = 0 and the limit as x→a of f(x) / g(x) is equal to zero or infinity over infinity, you can take derivative of the top and bottom and then put x back in to solve for the limit.

Extrema

the maximum and minimum points. Extrema can be absolute or relative

Concavity 

Where the function is cupping up or down 

Points of inflection

where the second derivative is zero or DNE and changes sign

First Derivative

the first derivative is the instantaneous rate of change, or the slope of the tangent line, and can determine if the function is increasing or decreasing at a given point.

f’(x) > 0, the function is increasing
f’(x) = 0, the function is not increasing or decreasing
f(x) < 0, function is decreasing

Second Derivative

The second derivative determines concavity

f”(x) > 0 - Concave UP
f”(x) = 0 - Neither
f”(x) < 0 - Concave Down

Finding Derivatives - The First Derivative Test

1. Find all of the critical points

2. Determine whether the function is increasing or decreasing on each side of every critical point. (A chart or number line helps)

Finding Derivatives - The Second Derivative Test

1. Find all of the critical points

2. Determine whether the function is concave up or concave down at every critical point using the second derivative)

Finding Absolute Extrema on an Interval (Candidates Test) 

1. Find the critical points. The critical points are candidates as well as the endpoints of the interval. 

2. Check all of the candidates using the f(x) 

Value of a function at x 

position at time t 

First Derivative

Velocity

Second Derivative

Acceleration

f ‘ (x) > 0 (increasing function)

moving right or up 

f ‘ (x) < 0 (Decreasing function) 

Moving left or down

f ‘ (x) = 0

Not moving

Absolute max

farthest right or up

absolute minimum 

farthest left or down 

f ‘ (x) changes signs

object changes direction

f ‘ (x) and f ‘ ‘(x) have same sign

speeding up

f ‘ (x) and f ‘ ‘ (x) have different signs 

slowing down 

MVT Theorem

If a function f is continuous over the interval [a,b] and differentiable over [a,b] then there exist a point c within that open interval where the instantaneous rate of change equals the average rate of change over the interval

EVT Theorem

If a function f is continuous over the interval [a,b], then f has at least one minimum value and at least one maximum value on [a,b]

First Derivative justification

Assume c and d are critical points of a function f.

There is a minimum value at x=c because f ‘ (x) changes from negative to positive

There is a maximum value at x=d because f ‘ (x) changes from positive to negative