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. Prof
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 i
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 b
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 int
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
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 maxim
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 f
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
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
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
coordi
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 =
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 degr
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
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 _
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
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
interva
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
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
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
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 da
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!
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
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 w
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 f
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
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
Terminol
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