Section 5-1
First Derivative and Graphs
Goal: To use the first derivative to determine when functions are increasing or decreasing and the local extrema of
functions
Question: What does a graph that is increasing look like? Decreasing?
An increasing graph

5 GRAPHING AND OPTIMIZATION
EXERCISE 5-1
2. (b, c); (c, d); (f, g)
4. (a, b); (d, f); (g, h)
6. x = b, g
8. x = d, g
10. f has a local maximum at x = d, and a local minimum at x = b; f does
not have a local extremum at x = a or at x = c.
12. (b)
14.
(h)
1

Section 3-1
Introduction to Limits
Goal: To evaluate limits graphically and algebraically
Concept of Limits
Suppose you were given the following table of values:
x
1.9 1.99 1.999 2 2.001 2.01 2.1 What would you guess that f (2) equals? 5
f ( x) 4.9 4.99 4

Section 9-1
Trigonometric Functions Review
Goal: To convert measures of angles into radians or degrees, find exact values of trig functions for special angles,
graph sine and cosine functions
Terminology:
The distance around a circle is called its _circum

Section 3-3
Infinite Limits and Limits at Infinity
Goal: To determine infinite limits, evaluate limits at infinity, and locate vertical and horizontal asymptotes
Infinite Limits
Question: If lim f ( x) does not exist, can f (x) be continuous at x = a? Exp

Section 9-3
Integration of Trigonometric Functions
Goal: To integrate sine and cosine functions and to solve applications requiring integration of these functions
Integrals of Sine and Cosine
Recall from 9-2:
Therefore,
d
sin x = cos x
dx
d
cos x = - sin

1 LINEAR EQUATIONS AND GRAPHS
EXERCISE 1-1
Things to remember:
1.
FIRST DEGREE, OR LINEAR, EQUATIONS AND INEQUALITIES
A FIRST DEGREE, or LINEAR, EQUATION in one variable x is an
equation that can be written in the form
STANDARD FORM: ax + b = 0, a 0
If th

Section 3-5
Power Rule and Basic Differentiation Properties
Goal: To use the constant function rule, power rule, constant multiple property, and sum & differences property to
determine the derivative functions
Today, our work for calculating the derivativ

Section 3-6
Differentials
Goal: To use differentials to approximate increments
Recall from 3-4: The average rate of change can be interpreted graphically as the slope of a secant line.
If the secant line passes through 2 points ( x1 , y1 ) and ( x 2 , y 2

Section 3-6
Differentials
Goal: To use differentials to approximate increments
Recall from 3-4: The average rate of change can be interpreted graphically as the slope of a secant line.
If the secant line passes through 2 points ( x1 , y1 ) and ( x 2 , y 2

Section 4-7
Elasticity of Demand
Goal: To solve problems involving relative rate of change and elasticity of demand
Definition: Relative and Percentage Rates of Change
The relative rate of change of a function f (x) is
f' ( x )
.
f( x )
Multiplying this n

Section 6-3
Differential Equations; Growth and Decay
Goal: To solve basic differential equations and applications involving such equations
Definition. An equation that involves an unknown function and one or more of its derivatives is called a
differentia

Section 3-7
Marginal Analysis in Business and Economics
Goal: To use the derivative to solve business applications
Recall: The word marginal refers to an _ rate of change.
Revenue vs. Profit
Illustration: Suppose the owner of a T-shirt stand buys his T-sh

Section 4-4
Chain Rule
Goal: To use the chain rule to determine derivatives of composite functions
Definition. A function m is a composite of functions f and g if _
The domain of m is the set of all x such that _
Example. Write the following functions as

Section 7-2
Applications in Business and Economics
Goal: To solve applications involving probability density functions, continuous income streams, and consumers and
producers surplus
Probability Density Function
Suppose x is a possible outcome, where c x

APPENDIX B
SPECIAL TOPICS
EXERCISE B-1
Things to remember:
1.
SEQUENCES
A SEQUENCE is a function whose domain is a set of successive
integers. If the domain of a given sequence is a finite set, then
the sequence is called a FINITE SEQUENCE; otherwise, the

Section 4-1
The Constant e and Continuous Compound Interest
Goal: To solve exponential equations and applications involving exponential growth/decay
In applications, the constant e appears frequently! e = lim (1 + 1n ) . e _
n
n
Quick Review of Logarithms

Section 8-5
Method of Least Squares
Goal: To use the method of least squares to find elementary functions to model data
The method of least squares is used to find the best function to fit a set of data points.
Theorem 1: Least Squares Approximation
For a

Section 4-3
Derivatives of Products and Quotients
Goal: To use the product and quotient rules to determine function derivatives
The product and quotient rules are not as simple as your intuition might first lead you to believe.
Product Rule
If y = f ( x)

Section 3-2
Continuity
Goal: To determine if a function is continuous & to construct a sign chart
Explain. In your own words, describe what it means for a function to be continuous.
Consider the two functions whose graphs are shown below. Which is continu

Section 5-5
Absolute Maxima and Minima
Goal: To use the derivative to identify absolute extrema
Absolute Extrema vs. Local Extrema
Illustration: Identify all local maxima/minima and all absolute maxima/minima from the graph shown below.
Formal Definition

3 LIMITS AND THE DERIVATIVE
EXERCISE 3-1
Things to remember:
1.
LIMIT
We write
lim f(x) = L or f(x) L as x c
x !c
if the functional value f(x) is close to the single real number
L whenever x is close to but not equal to c (on either side of c).
[Note: The

Section 5-3
LHpitals Rule
Goal: To use LHpials Rule to evaluate limits
Recall from Chapter 3: The limit of a polynomials as x can be reduce to the limit of the _
of the polynomial.
If n is an even integer, the limit of ( x c) n as x approaches c (or ) beh