AP Calculus AB Midterm Exam

General Power Rule

d/dx (sin x)

cos x

d/dx (cos x)

-sin x

d/dx (tan x)

sec²x

d/dx (cot x)

-csc²x

d/dx (sec x)

secxtanx

d/dx (csc x)

-cscxcotx

d/dx (arcsinx)

d/dx (arccosx)

d/dx (arctanx)

d/dx (arccotx)

d/dx (arcsecx)

d/dx (arccscx)

d/dx (a^f(x))

(a^f(x))(f’(x))(ln a)

d/dx (e^x)

e^x

d/dx (log_a f(x))

d/dx (ln x)

1/x

Product Rule

Quotient Rule

Chain Rule

Squeeze Theorem

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

Intermediate Value Theorem

If a function f is continuous on a closed interval [a,b], then f takes on every value between f(a) and f(b) on the interval [a,b].

Extreme Value Theorem

If a function is continuous on [a,b], then there is an absolute max and an absolute min on [a,b].

Mean Value Theorem

If a function is continuous and differentiable on [a,b], there is a point c in between a and b such that

Continuity

1. lim f(x)_x→a exists

2. f(a) exists

3. lim f(x)_x→a = f(a)

Average Rate of Change (AROC)

The average range at which a quantity changes over a given interval

Instantaneous Rate of Change

The exact or precise rate at which a quantity is changing at an instant or specific point

Limit of a Forward Difference Quotient

Limit of a Backwards Difference Quotient

Limit of a Symmetric Difference Quotient

Limit at a Specific Point

Point-Slope Form

y-y₁ = m(x-x₁)

Normal Slope

Negative reciprocal of the tangent slope

Implicit Differentiation

Write dy/dx next to every y-variable & solve for dy/dx

Position

Original/Given Function

Velocity

f’(x) = rate of change of position (average velocity uses position)

Acceleration

f’’(x) = rate of change of velocity (average acceleration uses velocity)

Average Velocity

L’Hôpital’s Rule

If lim_x→a f(x)/g(x) yields either of the indeterminate forms 0/0 or ± ∞/∞, then lim_x→a f(x)/g(x) = lim_x→a f’(x)/g’(x)

1/∞

0

e⁰

1

Critical Points

When f’(x) = 0 or f’(x) = DNE

Absolute Maximum

Highest y-value that occurs on a closed function

Absolute Minimum

Lowest y-value that occurs on a closed function

Rolle’s Theorem

If a function is continuous and differentiable on [a,b] and f(a) = f(b) then there exists at least one value, c, in (a,b) such that f’(c) = 0 (AROC = IROC)

Local Minimum

Where f’(x) changes from negative to positive

Local Maximum

Where f’(x) changes from positive to negative

Related Rates

Multiple variables changing at one time and they are related to each other. Always taken in terms of time (dx/dt)

First Derivative Test

1. Take f’(x) and set equal to zero and DNE values to find x-values of critical points

2. Put critical point x-values on a sign chart to find where the slope of f(x) is increasing/decreasing and where the local max/local min are

Second Derivative Test

1. Take f’’(x) and set equal to zero and DNE values

2. Put x-values of f’’(x) on sign chart to find concavity and points of inflection

Point of Inflection

A point where f’’(x) changes sign & halfway between local min and local max

When f’(x) is positive, then f(x) is

Increasing

When f’(x) is negative, then f(x) is

Decreasing

When f’’(x) is positive, then f(x) is

Concave up

When f’’(x) is negative, then f(x) is

Concave down

When f’’(x) is positive, then f’(x) is

Increasing

When f’’(x) is negative, then f’(x) is

Decreasing

When f’(x) is increasing, then f(x) is

Concave Up

When f’(x) is decreasing, then f(x) is

Concave down

Linearization

A method used to approximate a complex, nonlinear function or system with a straight line near a specific point. L(x) = f(a) + f’(a)(x-a)

Riemann Sums Definition

Method for approximating the total area under a curve by splitting the area up and finding the area of each section and then adding them together.

Trapezoidal Rule

(b-a)/n

One-Sided and Two-Sided Limits

If the limit is not given in a specific direction, then you must find the limit from both directions. If the limit from both directions is the same, then the limit exists.

Differentiability

A function’s ability to have a well-defined derivative (slope) at a given point, meaning the graph is smooth, continuous, and has no sharp corners or cusps.

ln 1

0

ln e

1

Graph of y = ln x

Graph of y = e^x

Volume of a Cone

Volume of a Cylinder

Area of an Equilateral Triangle

Reciprocal Identities

Sin θ = 1/csc θ

Cos θ = 1/sec θ

Tan θ = 1/cot θ

Quotient Identities

Tan θ = sin θ/cos θ

Cot θ = cos θ/sin θ

Pythagorean Identities

sin²θ + cos²θ = 1

1 + tan²θ = sec²θ

1 + cot²θ = csc²θ

Double-Angle Identities

Sin 2θ = 2sin(θ)cos(θ)

Sin²θ = 1 - cos2θ/2

Cos²θ = 1 + cos2θ/2

Equation of a Tangent Line

Need the slope of the tangent line (derivative) and one point (x,y)

Average Value of a Function

Fundamental Theorem of Calculus Part 1

Fundamental Theorem of Calculus Part 2

Speed

If velocity and acceleration have the same sign then the particle is speeding up. If velocity and acceleration have different signs then the particle is slowing down.

Displacement

The net change in object’s position, representing the straight-line and direction from start to finish.

Total Distance Traveled

Law of Exponential Growth or Decay

The rate of change of a variable is directly proportional to its current amount.

Area Between Two Curves in Terms of f(x)

Area Between Two Curves in Terms of f(y)

Washer Method

Cross-Sections Method - Square

Cross-Sections Method - Semicircle

Cross-Sections Method - Rectangles w/ Height n Times the Base

Cross-Sections Method - Equilateral Triangle

Cross-Sections Method - Isosceles Right Triangle w/ Hypotenuse Along the Base

Cross-Sections Method - Isosceles Right Triangle w/ Side Along the Base

∫ax^n dx

∫ 0 dx

C

x ∫ dx

x(x+C)

∫ 1/x dx

ln |x| + C

∫ e^x dx

e^x + C

∫ ln x dx

x ln x - x + C

∫ cos x dx

sin x + C

∫ sin x dx

-cos x + C