Math 142, Chapter 3
Spring 2011
Zarestky
Chapter 3 Reading Notes
In this chapter, we will learn about limits of functions and then use limits to transition into derivatives, the
first real concept of calculus. The overarching idea here is infinity. With limits, you will approach
infinity, or get infinitely close to a value, but you may never actually reach infinity or the value. It is a
new way of looking at functions and their behavior.
Section 3.1: Introduction to Limits
By the end of this section, you should be able to:
• Estimate a limit at a point from the graph of a function.
• Determine the values of x for which a limit does not exist given a graph of a function.
• Evaluate limits at a point algebraically.
Read the section Functions and Graphs: Brief Review and Limits: A Graphical Approach, through
Example 3. (About 3 pages. Examples 1 and 2 should be review. Example 3 will be new, but there isn’t
much actual work to do. Focus on understanding.) These first few pages should align with your prior
experience with functions. Pay special attention to the note in the Definition of a Limit blue box. That
will be particularly important later.
Make notes about the mathematical notation of limits.
Continue on to read the paragraphs about One-Sided Limits and Theorem 1. Make notes about the
notation and meaning of one-sided limits and the existence of a limit.
Work Example 4. There isn’t actually much work to do, but make certain you think about and
understand the book’s solution. Try the Matched Problem 4 on your own. What questions do you have?
Math 142, Chapter 3
Spring 2011
Zarestky
Make notes about Theorem 2, Properties of Limits. It is a lot of math-y formulas, but the bottom line is
that everything works the way we’d intuitively like it to.
Work Example 5. Compare your answers to the book’s solution. What questions do you have?
Continue reading. Work Examples 6, 7, and 8. In examples 7 and 8, watch out for the special left and
right notation. What questions do you have?
Read and make notes about the Indeterminate Form and the Limit of a Quotient.
Math 142, Chapter 3
Spring 2011
Zarestky
Examples 9-11 deal with the Limits of Difference Quotients. These special types of limits are going to be
very important when we get to derivatives. In the meantime, the toughest part is the algebra involved. Be
careful!
Work Example 9. Compare your answers to the book’s solution. What questions do you have?
Work Example 10. Compare your answers to the book’s solution. What questions do you have?
Work Example 11. Compare your answers to the book’s solution. What questions do you have?
Check your understanding of this section by working a few problems from the exercises on your own.
Focus on 1-15(odd), 17-20, 21-57(odd), 65-68, 71, 73, 75.
What questions do you still have?
Math 142, Chapter 3
Spring 2011
Zarestky
Section 3.2: Infinite Limits and Limits at Infinity
By the end of this section, you should be able to:
• Estimate a limit at a point and a limit at infinity from the graph of a function.
• Determine the values of x for which a limit does not exist given a graph of a function.
• Evaluate limits at a point algebraically.
• Evaluate limits at infinity for rational functions.
In this section we will look at limits for which x ! ±" and limits at vertical asymptotes. Read the pages
on Infinite Limits and Locating Vertical Asymptotes. Make notes about the definition and the special case
of rational functions. (What is a rational function? If you don’t know, go look it up. Section 2.4 in your
book.) How can you recognize when a rational function has a vertical asymptote?
Work Example 1. Compare your answers to the book’s solution. What questions do you have?
Math 142, Chapter 3
Spring 2011
Zarestky
Work Example 2. Compare your answers to the book’s solution. What questions do you have?
Read the paragraphs on Limits at Infinity. Make notes about how to read a limit at infinity from a graph,
horizontal asymptotes. Verify or Theorem 2 by graphing the functions from parts 1-4 on your graphing
calculator by choosing specific values of p and k.
Math 142, Chapter 3
Spring 2011
Zarestky
Work Example 3. Compare your answers to the book’s solution. What questions do you have?
Read Theorem 3 and the corresponding text. Be prepared to use the terminology leading term and end
behavior. If you need to review polynomials, refer to section 2.4.
Work Example 4. Compare your answers to the book’s solution. What questions do you have?
Math 142, Chapter 3
Spring 2011
Zarestky
Read the paragraphs on Finding Horizontal Asymptotes. Make notes about Theorem 4. Be prepared to
articulate the three cases in your own words.
Work Example 5. Compare your answers to the book’s solution. What questions do you have?
Work Example 6. Compare your answers to the book’s solution. What questions do you have?
Check your understanding of this section by working a few problems from the exercises on your own.
Focus on 1-8, 9-15(odd), 21-51(odd), 61-75(odd).
What questions do you still have?
Math 142, Chapter 3
Spring 2011
Zarestky
Section 3.3: Continuity
By the end of this section, you should be able to:
• Identify the intervals on which a function is continuous given its graph or its equation.
• Locate points of discontinuity for a piece-wise function.
By now you’ve seen that there are features of a graph that are analyzed by limits, such as asymptotes or
“jumps.” Those features affect the continuity of a function. Read the paragraphs on Continuity. Make
notes about reading a graph and the definition of continuity. What are some ways you can recognize that a
function is not continuous?
Work Example 1. Compare your answers to the book’s solution. What questions do you have?
Work Example 2. Compare your answers to the book’s solution. What questions do you have?
Math 142, Chapter 3
Spring 2011
Zarestky
Read the paragraphs on Continuity Properties. Make notes about continuity properties for specific functions.
Work Example 3. Compare your answers to the book’s solution. What questions do you have?
Read the paragraphs on Solving Inequalities Using Continuity Properties. Make notes about solving
inequalities and sign properties.
Work Example 4. Compare your answers to the book’s solution. Be prepared to create your own sign
charts on future problems. What questions do you have?
Check your understanding of this section by working a few problems from the exercises on your own.
Focus on 1-61(odd), 71-74, 77, 83, 85.
What questions do you still have?
Math 142, Chapter 3
Spring 2011
Zarestky
Section 3.4: The Derivative
By the end of this section, you should be able to:
• Calculate average rate of change.
• Calculate instantaneous rate of change.
• Calculate the slope of the secant line to a curve at a point.
• Write the equation of the tangent line to a curve at a point.
• Find a formula for the derivative using the limit definition of the derivative.
• Determine the values of x for which the tangent line is horizontal.
• Identify the points at which a function is nondifferentiable.
In this section, you will apply your skills with limits to find derivatives. Derivatives are used to solve
problems about tangent lines and rates of change. Read this section through Example 2. Make sure you
understand the concepts in Example 1. Make notes about average rate of change and the difference
quotient. These concepts are going to be very important!
Work Example 2. Compare your answers to the book’s solution. What questions do you have?
Math 142, Chapter 3
Spring 2011
Zarestky
Continue reading about Instantaneous Rate of Change. In this section, there is a lot of new terminology.
Describe and define average rate of change, difference quotient, instantaneous velocity, velocity, and
instantaneous rate of change. How do these terms relate to or differ from one another?
Continue reading about the Slope of the Tangent Line. Make notes about the secant line.
Work Example 3. Compare your answers to the book’s solution. What questions do you have?
Math 142, Chapter 3
Spring 2011
Zarestky
Continue reading. Make notes about the slope of a graph and tangent lines. Include a sketch of a tangent
to a graph as part of your notes.
Continue reading about the Derivative. Make notes about the definition of the derivative and its
interpretations, in particular, the three interpretations in the summary box. This is really important!
Work Example 4. Compare your answers to the book’s solution. What questions do you have?
Math 142, Chapter 3
Spring 2011
Zarestky
Work Example 5. Compare your answers to the book’s solution. What questions do you have?
Work Example 6. Compare your answers to the book’s solution. What questions do you have?
Math 142, Chapter 3
Spring 2011
Zarestky
Work Example 7. Compare your answers to the book’s solution. What questions do you have?
Read about the Nonexistence of the Derivative. What are some of the ways you can recognize when a
function is nondifferentiable for a given value of x?
Check your understanding of this section by working a few problems from the exercises on your own.
Focus on 1-29(odd), 31-38, 39, 41, 45, 47, 51-63(odd), 67.
What questions do you still have?
Math 142, Chapter 3
Spring 2011
Zarestky
Section 3.5: Basic Differentiation Properties
By the end of this section, you should be able to:
• Use short-cut rules to find a formula for the derivative of a function.
o Differentiate power functions.
o Differentiate the sum or difference of two functions.
Now that you know how to find a first derivative using the limit definition, you have the theoretical
foundation to skip that process and use short-cut rules. Congrats! Read this section through Example 2.
(Example 1 should be pretty easy.) Make notes on the constant function rule and power rule. What are the
different kinds of notation used to represent a first derivative?
Work Examples 2 and 3. Compare your answers to the book’s solutions. What questions do you have?
Math 142, Chapter 3
Spring 2011
Zarestky
Read about the Constant Multiple Property and make notes.
Work Example 4. Compare your answers to the book’s solution. What questions do you have?
Read about the Sum and Difference Property and make notes.
Work Example 5. Compare your answers to the book’s solution. What questions do you have?
Math 142, Chapter 3
Spring 2011
Zarestky
Work Example 6. Compare your answers to the book’s solution. What questions do you have?
Work Example 7. Compare your answers to the book’s solution. What questions do you have?
Check your understanding of this section by working a few problems from the exercises on your own.
Focus on 1-65(odd), 69-73(odd), 81, 83, 87, 89.
What questions do you still have?
Math 142, Chapter 3
Spring 2011
Zarestky
Section 3.7: Marginal Analysis in Business and Economics
By the end of this section, you should be able to:
• Compute marginal cost, marginal revenue, and marginal profit models.
o Determine the approximate cost, revenue, and profit for the next item.
• Compute marginal average cost, marginal average revenue, and marginal average profit models.
This section focuses on applications of the derivative as interpreted as a rate of change. Terminology is
going to be a challenge in this section, so make sure to take good notes on the new vocabulary. Read up to
Example 1 and make notes on marginal cost, marginal revenue, and marginal profit. How does the
marginal function relate to the original?
Work Example 1. Compare your answers to the book’s solution. What questions do you have?
Math 142, Chapter 3
Spring 2011
Zarestky
Continue reading. What is the relationship between marginal cost and exact cost?
Work Example 2. Compare your answers to the book’s solution. What questions do you have?
Math 142, Chapter 3
Spring 2011
Zarestky
Continue reading. Make notes about the terminology and mathematical relationships summarized in the
box on p. 199.
Work Example 3. Compare your answers to the book’s solution. What questions do you have?
Check your understanding of this section by working a few problems from the exercises on your own.
Focus on 1-19(odd), 25-37(odd), 41.
What questions do you still have?