Chapter 2Limits
Limits represent one of the first big ideas of the Calculus. Limits are important since they
provide a foundation for the two main branches of the Calculusthe Differential
Calculus and the Integral Calculus. In this chapter we will examine
Chapter 1Functions
One of the most basic building blocks of the calculus is functions. In this chapter we will
explore the basic properties of a function, the graph of a function, and ways of combining
and transforming functions. Generally there are four
Math 1131
Spring 2012
Exponentials, Trig Functions, and their
Inverses
1. A variable star is one whose brightness alternately increases and decreases. For the most
visible variable star, Delta Cephei, the time between periods of maximum brightness is
5.4
Chapter 3
3.2
Derivatives
Rules of Differentiation
Up to this point you have learned how to compute a derivative using the limit definition. Now we
will learn some rules that will help you compute derivatives of certain functionsconstant
functions, power
Chapter 3Derivatives
3.10Related Rates
In related rates problems we compare the rate of change of one quantity in terms of the rate of
change of another quantity. For example, if you blow up a balloon, both the volume and the
radius of the balloon are in
Chapter 1Functions
1.2Representing Functions
In this section we will explore different ways to define and represent functions
formulas, graphs, tables, and words.
Using Formulas
There are a number of functions that typically are defined by a formuls. The
Chapter 1Functions
1.4Trigonometric Functions and Their Inverses
In this section we review some of the basic ideas and properties associated with
trigonometric functions.
Radian Measure
Angles are measured in degrees or radian. In Calculus, typically ang
Chapter 3Derivatives
3.1Introducing the Derivative
We now begin our discussion of one of the most fundamental ideas of the Calculusthe
Derivative. We will approach the Derivative from two perspectives: 1) finding the tangent line
to a curve at a particul
Chapter 3
3.3
Derivatives
The Product and Quotient Rules
Previously we learned that the derivative of a sum of functions is the sum of the derivatives of the
functions and that the derivative of the difference of functions is the difference of the deriva
Chapter 2Limits
2.7Precise Definition of Limits
The intuitive definition of a limit, that is, as x approaches a ,
f ( x) approaches some number L is a bit vague in mathematical terms. In order to be able to
prove that the limit of f ( x) is L as x appro
Chapter 2Limits
2.6Continuity
Continuity, an important mathematical concept of the Calculus, is term that is associated with
continuous functions. Essentially, continuous functions are functions that can be drawn without
any brakes, jumps, or holes in it
Chapter 2Limits
2.4Infinite Limits
An infinite limit occurs when functional values increase or decrease without bound, that is, the
values tend to +!"! . In this section we will examine such limits.
An Overview
1
(if it exists).
x 0 x 2
1
We begin by fir
Chapter 2Limits
2.5Limits at Infinity
Previously we learned about infinite limits and vertical asymptotes. Limits at infinity occur when
the independent variable
increases or decreases without bound, that is
.
In this
section we examine limits at
Chapter 2Limits
2.3Techniques for Computing Limits
Previously we learned how to estimate limits using graphs and tables. In this section we
learn analytical methods in order to calculate limits precisely. We befin with limits of
linear functions.
Limits
Chapter 2Limits
2.2Definitions of Limits
In this section we begin the process of defining the idea of a limit in a more formal
mathematical way. To begin we examine the behavior of a linear function around some
values of .
Suppose we want to investigate
Chapter 1Functions
1.3Inverse, Exponential, and Logarithmic Functions
We begin this section with a study of exponential functions. Many real-world
phenomena are modeled by exponential functions, for example, in finance, medicine,
biology, economics, and
Math 1131
Fall 2012
Section 3.4 - Derivatives of Trigonometric
Functions
1. Using the product and quotient rules, nd the derivatives of the following functions:
f (x) = 2e2x cos(x)
f ( x) =
f (x) =sin(x) xcos(x)
f (x) =
f (x) = 1+sin(x) .
1sin(x)
f ( x
Evaluating
Trig Functions
I.
Definition and Properties of the Unit Circle
a. Definition: A Unit Circle is the circle with a radius of one ( r = 1 ),
centered at the origin (0,0) .
b. Equation: x 2 + y 2 = 1
c. Arc Length
Since arc length can be found usin
Graphing Rational Function Rules
Rule for Domain
Set each factor of the denominator equal to zero. This is where the function will be undefined.
The domain for the function will be all real numbers except those that make the denominator
zero.
Rule for Ver
Determine the domain of a function
A function f is a rule that assigns to each element x in a set A (called the domain
of f ) exactly one element f (x) in a set B (called the range of f ). If a function f is given by
a formula and the domain is not specie