Section 37
Marginal Analysis in Business and Economics
Goal: To use the derivative to solve business applications
Recall: The word marginal refers to an instantaneous rate of change.
Revenue vs. Profit
Illustration: Suppose the owner of a Tshirt stand b
Section 51
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?
We want a mathemati
Section 34
The Derivative
Goal: To extend the concept of slope to nonlinear functions
I. Average Rate of Change
Illustration: Consider the function y = f (x) shown in the graph below. Draw the line between the points
(a, f (a ) and (a + h, f (a + h) also
Section 53
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
3 LIMITS AND THE DERIVATIVE
EXERCISE 31
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 55
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
Section 32
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 43
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 54
Curve Sketching Techniques
Goal: To use the graphing strategy to sketch curves and interpret applications
Strategy for graphing a function y = f (x)
Step 1. Use f (x) to find the following parts of a graph:
_
Step 2. Use f ' ( x) to find the f
2 FUNCTIONS AND GRAPHS
EXERCISE 21
Things to remember:
1.
POINTBYPOINT PLOTTING
To sketch the graph of an equation in two variables, plot
enough points from its solution set in a rectangular
coordinate system so that the total graph is apparent and
the
7 ADDITIONAL INTEGRATION TOPICS
EXERCISE 71
Things to remember:
1.
AREA BETWEEN TWO CURVES
If f and g are continuous and f(x) g(x) over the interval
[a, b], then the area bounded by y = f(x) and y = g(x), for
a x b, is given exactly by:
y
A =
y = f(x)
b
Section 46
Related Rates
Goal: To solve applications involving related rates
What are related rates?
Some quantities that increase or decrease over time may be related (e.g., increase in college degrees and decrease
in poverty, increase in companys profi
Section 45
Implicit Differentiation
Goal: To use implicit differentiation to determine derivatives
Sometimes we will have an equation given in terms of x and y (i.e., does not define y as a function of x explicitly).
Rather than trying to solve the equat
Section 52
Second Derivative and Graphs
Goal: To use the second derivative to determine concavity and inflection points for function graphs
Informal Definition of Concavity
A function is said to be _ if its graph looks like it would spill water.
Similarl
Section 82
Partial Derivatives
Goal: To find partial derivatives and solve applications requiring partial derivatives
Partial Derivatives
For z = f ( x, y ) , the partial derivative of f with respect to x, denoted
z
, fx , or fx ( x, y ) is defined by
x
6 INTEGRATION
EXERCISE 61
Things to remember:
1.
A function F is an ANTIDERIVATIVE of f if F'(x) = f(x).
2.
THEOREM ON ANTIDERIVATIVES
If the derivatives of two functions are equal on an open
interval (a, b), then the functions can differ by at most a
co
Section 31
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? _
f ( x) 4.9 4.99 4
Section 11
Linear Equations and Inequalities
Goal: To solve linear equations and linear inequalities and related applications
Definition: Linear Equation
A linear equation is an equation that can be written in the form_ (standard form), a 0 .
Replacing t
5 GRAPHING AND OPTIMIZATION
EXERCISE 51
Things to remember:
1.
INCREASING AND DECREASING FUNCTIONS
For the interval (a, b):
2.
f '( x )
f (x )
Graph of f
+
Increases
Rises

Decreases
Falls
Examples
CRITICAL VALUES
The values of x in the domain of f wher
Section 85
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 41
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 35
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
1 LINEAR EQUATIONS AND GRAPHS
EXERCISE 11
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 93
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 92:
Therefore,
d
sin x = cos x
dx
d
cos x =  sin
Section 33
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 91
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 31
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