Math 251 (Phillips) Midterm 2

Information and material covered

New topics since Midterm 1:

• All remaining differentiation rules and derivatives of standard functions, including inverse functions.
• Implicit differentiation.
• Squeeze Theorem.
• Infinite limits.
• Basic limits at infinity.
• Optimization problems, now including ones not on a closed bounded interval.
• Related rates.
• Using the derivative and second derivative to find intervals of increase and decrease, local minimums, and local maximums.

• Sections 2.1 through 2.4, now including infinite limits and limits at infinity.
• Derivatives and differentiation rules: all of Chapter 3 except Section 3.6.
• Related rates (Section 4.1).
• Maximums and minimums (Section 4.3).
• Derivatives and the shape of a graph (Section 4.5), except that we didn't yet talk about inflection points.
• The beginning of limits at infinity, up to Example 4.25 in Section 4.6.
• Maximizing and minimizing in word problems, in Section 4.7. Now also including problems not on closed intervals.

The exam will have at least one of each of the following kinds of problems:

• Optimization, maybe on a closed bounded interval, maybe not. Also maybe just a request to set the problem up, without giving a full solution.
• Related rates.
• Identifying intervals of increase and decrease, local minimums, and local maximums, possibly with the derivative already given to you.
• Implicit differentiation.
• Something involving limits.

Review session

Sample Problems

There are seven primary sources for sample problems for Midterm 2:

• Written homework. These, and their solutions, are linked on the web pages for weeks 1--8 of the course, accessible from here.
• WeBWorK homework due through Thursday 22 May 2025. Answers to WeBWorK problems are available on WeBWorK after the due date of each assignment.
• Additional practice WeBWorK problem sets. Again, answers are available on WeBWorK after the "due date" of each assignment.
• Quizzes. These, and their solutions, are linked on the web pages for weeks 1--8 of the course, accessible from here.
• In class worksheets. Warning: The hints on the worksheets won't be provided on the midterm. These, and their solutions, are linked on the web pages for weeks 1--8 of the course, accessible from here.
• Sample Midterm 2 (pdf). Solutions (pdf).
• Worksheet used at the review session. Solutions to the worksheet used at the review session. Minor corrections made 1 June 2025, mostly working improvements but several mathematical misprints. Changes are shown in the difference file (pdf); and the reverse difference file (pdf). The difference file has new text wavy underlined in blue and old text crossed out in red; the reverse difference file reverses the color scheme. More information.
• Everything posted for Midterm 1, including the real and retake real Midterm 1 themselves.

Of course, you should also know how to do all the Midterm 0 problems. Note, though, that the Midterm 2 problems will be in format much more like the problems in the first list above.

Real Midterm 2

• Midterm 2 (pdf).
• Solutions to Midterm 2 (pdf).

Reminder: Extra credit will be given for finding mistakes. You need to say what is wrong and how to fix it, you need to be one of the first two people to report the mistake to me, and you need to report it before a corrected version is posted.

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This page maintained by N. Christopher Phillips, email.

Last significant change 22 May 2025.