Absolute Value:

Pythagorean Theorem:

In a right triangle with right angle at ,

Distance Formula:

Given and , then

Midpoint Formula:

Midpoint of and is

Quadratic Formula:

If , then

Translations of Functions:

Replacing with and with will translate a function units horizontally and units vertically.

translated units horizontally and units vertically will become or .

Linear Functions:

Slope-intercept:

Point-slope:

Slope:

Parallel lines have equal slopes:

Perpendicular lines have negative reciprocal slopes:

A normal line is perpendicular to a tangent line.

Circular Functions:

Given as the center of a circle and the radius, then the equation of the circle is

Area:

Circumference:

Symmetry: (Even and Odd Functions)

The graph of an equation is:

Symmetric with respect to the y-axis if replacing with gives an equivalent equation (for example ).

is even if .

Symmetric with respect to the origin if replacing with and with gives an equivalent equation (for example ).

is odd if .

Symmetric with respect to the x-axis if replacing with gives and equivalent equation (for example ).

Existence of a Limit:

if and only if and .

Limits:

Substitute . If , then factor, multiply by conjugate, rationalize, or simplify to one fraction. If , then use infinite limit properties.

Infinite Limits:

and and

Look for a constant divided by a small negative number “-” or small positive number “+”.

Limits at Infinity:

and provided that is a positive integer.

Divide by the highest power in the denominator.

Continuity at a Point:

Let be defined on an open interval containing . We say that is continuous at if:

(1) is defined

(2) exists

(3)

Intermediate Value Theorem:

If is continuous on and is any number between and , then there is at least one number in such that .

Extreme Value Theorem:

If is continuous on , then attains both a maximum value and a minimum value there. These values occur at an endpoint, where or does not exist.

Definition of a Derivative:

provided that this limit exists.

provided that this limit exists.

Differentiability Implies Continuity:

If exists, then is continuous at c.

Differential :

implies

Local Linear Approximation:

Mean Value Theorem for Derivatives:

If is continuous on a closed interval and differentiable on its interior , then there is at least one number in where

or

is the average rate of change or mean rate of change.

is the instantaneous rate of change.

Rolle’s Theorem

If is continuous on a closed interval and differentiable on its interior and if , then there is at least one number in such that .

Mean Value Theorem for Integrals:

If is continuous on a closed interval , there is a number between and such that

OR

is the average value or mean value (Average value)

Average Value of a Function:

Riemann Sums: Left, Right, Midpoint (intervals from to ):

, where and

Trapezoid Rule: Average of left hand and right hand Riemann Sum.

Definition of a Definite Integral:

, where and

Fundamental Theorem of Calculus:

, where

OR

and

Increasing and Decreasing:

implies is increasing.

implies is decreasing.

Use to determine local and/or global maximum and minimum values (points).

Critical Points: Where or does not exist.

Concave Up and Concave Down:

implies is concave up.

implies is concave down.

Use to determine points of inflection.

is concave up if is increasing.

is concave down if is decreasing.

If for some and , then is a local maximum.

If for some and , then is a local minimum.

Vertical Asymptotes:

If is undefined then evaluate the to determine function behavior near . If the or , then is a vertical asymptote.

Horizontal Asymptotes:

Evaluate the and . If either limit equals a unique number , then is a horizontal asymptote.

Differentiation Rules:

Let u and v be differentiable functions of .

Constant Multiple Rule:

Sum or Difference Rule:

Product Rule:

Quotient Rule:

Constant Rule:

Power Rule:

Chain Rule:

Trigonometric Rules:

Integration Rules:

OR , where

u-substitution