Using Long Division with Polynomials
Division for polynomials follows the same procedures as long division with numbers. To
divide polynomials, follow the same steps you used for elementary school long division.
When setting up your polynomial division pr

Unit: Special Functions
Module: Logarithmic Functions
[page 1 of 3]
Evaluating Logarithmic Functions
Remember: Change of base theorem:
loga x
logb x =
loga b
allows revising a logarithm problem to be in a base that is easier to use in solving
the problem.

Unit: Special Functions
Module: Exponential Functions
Graphing Exponential Functions
An exponential function has the variable in the exponent, not the base.
Exponential functions cannot have negative bases. Exponential functions with
positive bases less t

Fundamental Trigonometric Identities
An identity is an equation that is true for all meaningful values of the variable(s) involved.
Four basic trigonometric identities:
sin
cos
2
cos + sin2 = 1 (the Pythagorean identity)
1 + tan2 = sec 2
1 + cot 2 = csc

Unit: The Basics
Module: Overview
The Two Questions of Calculus
Use calculus to find instantaneous rates of change and areas of exotic shapes.
Average rate of change: R =
D.
T
Calculus creates a connection between two very
different problems.
The first pr

Unit: The Basics
Module: Precalculus Review
Functions
A function pairs one object with another. A function will produce only one object
for any pairing.
A function can be represented by an equation. To evaluate the function for a
particular value, substit

Unit: Computational Techniques
Module: The Product and the Quotient Rule
The Product Rule
The derivative of a product of two functions is not necessarily equal to the product
of the two derivatives.
The product rule states that if p( x ) = f ( x ) g( x )

Unit: Practical Application of the Derivative
Module: Position and Velocity
Acceleration and the Derivative
Velocity is the rate of change of position. Acceleration is the rate of change of
velocity.
Tangent lines can be used to approximate functions that

Unit: Special Functions
Module: Exponential Functions
[page 1 of 2]
Derivatives of Exponential Functions
Studying the slopes of tangent lines to the graph of a function can help you
determine the derivative.
The derivative of an exponential function is th

Everything you
always wanted to
know about trig*
Explained by
John Baber, A.B.,Cd.E.
*but were afraid to ask.
Any figure drawn on a piece of paper with straight edges can be split up into right triangles. Because of this, studying squares, rectangles, rho

Unit: Special Functions
Module: Logarithmic Functions
The Derivative of the Natural Log Function
The derivative of the natural exponential function is itself. Use the chain rule to
find the derivative of the composition of the natural exponential function

The Pebble Problem
Related rate problems involve using a known rate of change to find an associated
rate of change.
The three steps to problem solving are understanding what you want, determining
what you know, and finding a connection between the two.
If

Unit: Elementary Functions and Their Inverses
Module: Inverse Trigonometric Functions
The Inverse Sine, Cosine, and Tangent Functions
The standard trigonometric functions do not have inverses. Only by restricting the
domain can you make them one-to-one fu

Unit: Elementary Functions and Their Inverses
Module: Calculus of Inverse Functions
Derivatives of Inverse Functions
You can calculate the derivative of an inverse function at a point without
determining the actual inverse function.
The inverse of a funct

Unit: Implicit Differentiation
Module: Applying Implicit Differentiation
Using Implicit Differentiation
Find the derivative of a relation by differentiating each side of its equation implicitly
and solving for the derivative as an unknown. This process is

Unit: Implicit Differentiation
Module: Implicit Differentiation Basics
Introduction to Implicit Differentiation
The definition of the derivative empowers you to take derivatives of functions, not
relations.
Leibniz notation is another way of writing deriv

Unit: Elementary Functions and Their Inverses
Module: Inverse Functions
The Exponential and Natural Log Functions
Raising the number e to the power x produces the exponential function. It is a
positive, increasing function.
Taking the log to the base e of

Unit: Elementary Functions and Their Inverses
Module: Inverse Functions
The Basics of Inverse Functions
Inverse functions undo each other.
In inverse functions, the dependent variable and independent variable switch
roles. The graph of an inverse function

Investments
You can use systems of linear equations to solve many real-world problems.
Suppose that you invested a total of $3,000 in stocks
and bonds. Given the rates of return for stock and
bonds and the total interest you earned after the first
year of

Vertical Asymptotes
A vertical asymptote is a vertical line that the graph of a function approaches but
never touches.
To find the vertical asymptotes of a rational function:
1. Cancel any common factors in the numerator and denominator to put the fract

Systems of Inequalities
An Introduction to Systems of Inequalities
A system of inequalities means that you have more than one inequality,
or an inequality mixed in with regular equations.
When graphing inequalities, start by drawing the equation normally.

Solving Rational Inequalities
Solving inequalities:
Factor where possible.
Determine the points where each factor equals zero.
Determine the points that cause the denominator to equal zero and must be excluded
from the domain for x.
Determine what in

Finding the Vertex by Completing the Square
Standard form of a parabola: f (x) = ax2 + bx + c.
Vertex of a parabola: (h,k).
h = -b/2a.
k = f (h).
Standard form for parabola showing vertex: f (x) = a (x - h)2 + k.
Completing the square is one method for

Determining Intervals Over Which a Function is Increasing
! A function is decreasing if the graph drops as the x-values get larger.
! A function is increasing if the graph rises as the x-values get larger.
! A function is constant if it is neither increas

Using Operations on Functions
Using operations with functions is a useful way to create new functions for new
purposes.
All four of the basic operations can be used to combine two or more functions into a new
function.
Remember to watch signs carefully

Finding the Center-Radius Form of the Equation of a Circle
A circle is the set of all points on a plane which are the same distance from a given
point, the center of the circle.
The center of a circle is usually given as the ordered pair (h,k) where h i

Presenting the Rational Zero Theorem
A rational number is a fraction.
If the polynomial f ( x ) = an x n + an 1 x n 1 + .a1 x + a0 has integer coefficients, then every
p
where p and q have no common factors other than 1,
q
p is a factor of a0 , and q is

Finding the Domain and Range of a Function
The domain includes the set of all real numbers that can be used for x.
The range of a function is the collection of all the y-values or output that exist.
Values excluded from the domain include negatives und

Writing an Equation of a Line
Standard equation of a line: Ax + By = C. A linear equation will always have an x to
the first power, a y, and some constant.
Slope-intercept equation of a line: y = mx + b. The constant, m, represents the
slope or rate of