skip to main content
skip to homework
Michael Bowen's VC Course Pages

Michael Bowen's VC Course Pages

Math V46, Spring 2017

Introduction and Announcements

Welcome to Math V46 (Applied Calculus) at Ventura College. Michael Bowen (email) will be teaching this course during the spring 2017 term.

Important note: This web page is not a substitute for attending class; regular attendance is an expectation of this course. Modifications to homework assignments, and other important news announced in class, may not appear on this page for several days. You are still responsible for all assignments and in-class announcements even if they do not appear here! If you wish to verify information on this page, please contact the instructor.

Textbook Information

The ISBN number is provided as a convenience if you wish to purchase this item online. The VC bookstore may stock a different ISBN number; either may be used for the course. If you buy from the bookstore, obtain the least expensive version you can find; do not pay extra for MyMathLab, WebAssign, or other software. If you obtain the book from another source, please be sure to obtain the correct edition, as noted below. Older editions are, of course, much less expensive, but the homework problems are different.

Select any one of the following required texts:

    • Author: R. Barnett
    • Title: Calculus for Business, Economics, Life Sciences, and Social Sciences, Thirteenth Edition
    • ISBN-13: 978-0321869838
    • Comment: Use this ISBN if you are purchasing online; this is likely to be the least expensive, provided that you do not wish to use MyMathLab for independent study.
    • Author: R. Barnett
    • Title: Calculus for Business, Economics, Life Sciences, and Social Sciences (Custom Edition for Ventura College), Thirteenth Edition
    • ISBN-13: 978-1323051573
    • Comment: This is one of the packages available in the VC bookstore; as we will not be using MyMathLab, you do not need the MyMathLab bundle unless you want to use it on your own for extra study.

If you use the Kindle or other digital version, you will want to be able to refer to the information during class, so be sure you own a compatible device (laptop, mobile phone, or e-reader) that you are willing and able to bring to class with you every day.


Classes at Ventura College will meet Monday through Thursday each week of the term, excepting only the dates listed below.

Homework Club (Office Hours) During Finals Period

Final Examination

Place/date/time:  Room SCI-351, Monday 15 May 2017, 5:00 p.m.

Be sure that your big party to celebrate the end of finals occurs after the appropriate date. Requests for administration of early or late finals that require the instructor to reschedule his work or make a special trip to campus are subject to a deduction of points, regardless of the reason for the request.

Homework Assignments

Overview: This table lists homework assignments and announces examinations. It contains four columns. First row: Column headers. Second and subsequent rows: The homework due for each section covered in the course. Details: The first row is the table header, with column headings describing the data listed in the main body of the table. The second and subsequent rows contain section numbers and titles, assigned problem numbers, and extra credit problems, if any. Column one of these rows contains a section number. Column two of these rows contains the corresponding section title. Column three lists the problem numbers for each section. Column four lists extra credit problems, if any.
§ Title Problems E.C.
Syllabus Worksheet (obtain a copy) (NOTE: This assignment is worth 15 points.)
(p. 531)
Factoring Polynomials 1–55 ODD
(p. 536)
Operations on Rational Expressions 1–33 ODD; 43; 45
See solutions for selected problems
(p. 546)
Rational Exponents and Radicals 13–59 ODD; 83; 85; 87
See solutions for selected problems
(p. 546)
Quadratic Equations (Factor and/or use the quadratic formula) 13–37 ODD; 43; 45; 47
(p. 572)
Binomial Theorem 1–31 ODD
1.1 Functions 15–20 ALL; 21–27 ODD; 29–35 ODD (graph paper is available here); 61–77 ODD; 87; 89; 91
Chapter Test 1

(Sections A.3, A.4, A.6, A.7, B.3, and 1.1)

Recommended study-guide problems

(These are sample exam-like problems for practice purposes; do not turn in with your homework)

(For students with minimal study time)
Page 531: 20; 26; 32; 38; 44; 50
Page 536: 6; 12; 18; 24; 30; 44
Page 546: 18; 24; 30; 36; 42; 48; 54; 60; 84
Page 555: 18; 24; 30; 36; 48
Page 572: 6; 12; 18; 24; 30
Page 87: 4; 5; 6; 30; 32
(For students with additional study time)
The above plus some or all of the following
Page 531: Remaining even-numbered problems not listed above from 22–48
Page 536: Remaining even-numbered problems not listed above from 8–34 and 46
Page 546: Remaining even-numbered problems not listed above from 14–58; 86; and 88
Page 555: Remaining even-numbered problems not listed above from 14–38
Page 572: Remaining even-numbered problems not listed above from 2–32
Section 1.1 22–28 EVEN; 30–36 EVEN; 62–78 EVEN; 86; 88; 90
2.1 Introduction to Limits 9–45 ODD; 51–63 EOO; 81; 83; 85; 91; 95 88
2.2 Infinite Limits and Limits and Infinity 9 (behavior of a limit at a horizontal asymptote);
13 (behavior of a limit at a vertical asymptote);
15 (behavior of a limit at a jump discontinuity);
17–23 ODD (if (A) and (B) agree, then (C) is the same; if (A) and (B) disagree or one is DNE, then (C) is DNE); 33; 35; 43–49 ODD
2.3 Continuity 15–53 ODD
2.4 The Derivative 19–41 ODD; 71; 73; 75; 79(A)(B); 81; 83
Solutions to selected homework problems are posted on Canvas.
2.5 Basic Differentiation Properties (See problems 1 through 8 for hints on how to rewrite fractions and radicals; however, these are not assigned)
9–59 ODD; 95; 97
If you are not sure how to do problems 57 and 59, I have worked a similar example (problem 60). Link to solution for problem 60.
Chapter 2 Test
(sections 2.1 through 2.5 only)

Recommended study-guide problems

(These are sample exam-like problems for practice purposes; do not turn in with your homework)

Exam date is Wed. 8 March
(For students with minimal study time)
Page 175 (Review Exercises): 4–10 ALL; 25; 26 (use difference quotient); 27 (use any method); 28 (use difference quotient); 29–31 ODD (use the properties from section 2.5); 39; 40; 43–49 ODD (use the properties from section 2.5 to obtain needed derivatives); 55–67 ODD; 73–81 ODD; 91; 97
(For students with additional study time)
The above plus
Page 175 (Review Exercises): 11–24 ALL; 30 (use any method); 44–48 EVEN (use the properties from section 2.5 to obtain needed derivatives); 56–68 EVEN; 72–82 EVEN
Unassigned problems from the ranges of the homework assignments from sections 2.1 through 2.5
2.6 Differentials 9–19 ODD; 27–31 ODD; 33(A)(C); 35(A)(C); 45; 47; 51; 54 (answers to #54: 2.6 and 1.3)
There are two ways to find the exact change $\Delta y$: either $\Delta y = f\left( {{x_2}} \right) - f\left( {{x_1}} \right)$ or $\Delta y = f\left( {x + \Delta x} \right) - f\left( x \right)$; use whichever form best matches the information given in the problem. The estimated change is given by $dy = f'\left(x\right)dx$; if $dx$ is not given, calculate it from $dx = x_{2}-x_{1}$.
2.7 Marginal Analysis in Business and Economics 9–25 EOO (all odds recommended if time permits); 33; 37; 49 ("break-even" means that profit is zero, or $P\left(x\right) = 0$)
These are really just more differential problems, so the formulas from section 2.6 still work; however, when "marginal" is used, the implied value of the run is $dx = 1$. So if a problem asks "what is the cost of the 26th widget", then assume that $x_{1} = 25$, $x_{2} = 26$, and $dx = x_{2}-x_{1}=1$.
3.1 The Constant $e$ and Continuous Compound Interest 13–21 ODD; 25–29 ODD; 33; 35; 41; 45; 47 42
3.2 Derivatives of Exponential and Logarithmic Functions 9–33 ODD; 41–53 EOO; 63; 65
3.3 Derivatives of Products and Quotients 9–33 ODD; 49–65 EOO; 77–89 EOO; 93

Typo warning: Problem 81 should read $\frac{{6\left( {\sqrt[3]{x}} \right)}}{{{x^2} - 3}}$ not $\frac{{\left( {{6^3}} \right)\sqrt x }}{{{x^2} - 3}}$
3.4 The Chain Rule 17–65 EOO; 95(A); 98
3.5 Implicit Differentiation 13–29 ODD; 35; 37; 51; 53 (for the last two problems, treat $p$ as if it were the dependent variable $y$) 48
3.6 Related Rates 9–25 ODD
3.7 Elasticity of Demand (No assignment)
4.1 First Derivative and Graphs 33–45 ODD; 85; 87; 89; 95; 97

  1. A "partition number" is a value of $x$ for which either $f'\left( x \right) = 0$ or $f'\left( x \right)$ does not exist. A "critical number" is any partition number which is also in the domain of $f\left( x \right)$. So every critical number is a partition number, but not every partition number is a critical number. For example, consider $f\left( x \right) = x - 2\ln{x}$. The domain is $\left( {0,\;\infty } \right)$ because the natural logarithm is not defined for $x \leq 0$. The derivative is $f'\left( x \right) = \frac{x-2}{x}$, so the partition numbers are $x=0$ (because $f'\left( x \right)$ is undefined there) and $x=2$ (because $f'\left( x \right) = 0$ there). But only $x=2$ is a critical number, because $x=0$ is not an element of the domain of $f\left( x \right)$.
  2. Every extremum (local minimum or maximum) of a function occurs at a critical number, but not every critical number is an extremum. There must be a sign change in $f'\left( x \right)$ at the critical number for that location to be an extremum. For example, $f\left( x \right) = x^3$ has a critical number at $x=0$, but it is not an extremum because $f'\left( x \right) = 3x^2$ is positive on both the left and right sides of $x=0$.
  3. For the word problems, note that the increasing/decreasing intervals do not extend to infinity. Only evaluate increasing/decreasing behavior on the given domains. In #95, for example, only evaluate increasing/decreasing behavior on $\left( {0,\;150 } \right)$.
4.2 Second Derivative and Graphs 33; 37; 49; 53; 57; 61; 65; 69; 87; 91; 93 (graphs are optional)
4.3 L'Hopital's Rule (No assignment)
4.4 Curve-Sketching Techniques (No assignment)
4.5 Absolute Maxima and Minima (No assignment)
4.6 Optimization 9–45 EOO
Chapter 3–4 Test

Download PDF copy using link at right
Chapter 3–4 take-home exam; due Monday 17 April 2017 at 5:00 p.m. PDT
5.1 Antiderivatives and Indefinite Integrals 9–23 ODD; 43–61 ODD; 65; 67; 69; 81; 85 93
5.2 Integration by Substitution 9–41 EOO; 43; 59–69 ODD; 77; 79; 81(A) 88
5.3 Differential Equations; Growth and Decay (No assignment except for optional extra credit problems ===>) 26; 28
5.4 The Definite Integral 31–53 ODD
5.5 The Fundamental Theorem of Calculus 13–45 EOO (all ODDs recommended if time permits); 47; 49(A)–61(A) EOO (the graphs in part (B) are very optional); 71; 75; 77(A)(B)
6.1 Area Between Curves 43–57 ODD; 87; 89; 97
6.2 Applications in Business and Economics 25; 27; 41
6.3 Integration by Parts 9; 11; 15–27 ODD; 37–57 ODD; 63; 67 (review 6.2); 69 (review 6.1)
(Warning: Some of these problems may be solved by the method of $u$-substitution, as in section 5.2; you may need to experiment with both of these methods. For the problems involving integration by parts, multiple applications of the method may be necessary for #37 and some of the others)
38; 46; 56
6.4 Other Integration Methods 5–27 ODD; 31; 35; 37–61 EOO (all ODDs recommended if time permits) 78 (use your choice of a table or Simpson's rule)
Final Examination

(Chapters 5 and 6)

Recommended study-guide problems

(These are sample exam-like problems for practice purposes; do not turn in with your homework)

Exam date is Mon. 15 May or make-up on Wed. 17 May

Bring your Chapter 5/6 homework to the final for up to 20 points credit. (Not extra credit!)

Exam starts at 5:00 p.m.
(For students with minimal study time)

Page 377 (Review Exercises): 1–9 ODD; 15; 17 (use Simpson's rule with 2 rectangles instead of a right sum, and skip "calculate an error bound"; you should get close to the book's answer, but not quite the same); 18; 20; 39–57 ODD (these may require any combination of $u$-substitution, integration by parts, or integration by table)

Page 422 (Review Exercises): 5–15 ODD; 25; 27; 29; 33–41 ODD; 44; 46; 48; 49; 54 (hints: integrals may be evaluated using $u$-substitution, integration by parts, or tables; try the methods in the order listed here)

Page 329: (optional review of 5.1, if you have forgotten how to do "tricky" integrals without using $u$-substitution) 44–62 EVEN (email me if you need to check a solution, although you should be able to check 44–54 yourself by differentiating your solution)
(For students with additional study time)

The above plus

Page 377 (Review Exercises): 2–10 EVEN; 14; 16; 25–31 ODD; 38–56 EVEN (for #56, you can tell by inspection that the answer is zero, but why?); 64 (ignore the book's instructions and use Simpson's rule with 10 intervals; your answer may differ slightly from the back of the book)

Page 422 (Review Exercises): 4–16 EVEN; 24; 26; 28; 34–42 EVEN; 47(B); 52 ("fourth hour" means from $t=3$ to $t=4$); 57 (find the probability for $t=0$ to $t=2$, then subtract this result from the total probability for all possible times, which is 1)

Additional problems taken from the unassigned homework exercises

Course Handouts and Study Aids

The documents listed below are available for viewing or download. The list below provides links to download free software to read the file formats of the various documents.

Course Handouts

Study Aids

Will You Succeed or Fail in Mathematics?

This checklist is adapted from a handout prepared by math and philosophy instructor Steve Thomassin. It will allow you to compare your approach to a mathematics course to the approaches taken by successful … and unsuccessful … students.

Overview: This table lists typical attributes of successful and unsuccessful mathematics students. It contains three columns. First row: Column headers. Second and subsequent rows: Student attributes. Details: The first row is the table header, with column headings describing the data listed in the main body of the table. The second and subsequent rows describe specific attributes that contribute to success or failure. Column one of these rows specifies whether the attribute is related to attitude, class work, homework, or getting help. Column two of these rows contains attributes of successful students. Column three of these rows contains attributes of unsuccessful students.
Attribute Type Predictor of Success Predictor of Failure
Attitude Focus on things that are under your control. Blame things that are out of your control (the text, the instructor, or "the system") for your difficulties.
Be optimistic. Believe that you can do it. Be pessimistic. Convince yourself that you will fail.
Be positive. Find ways to make math interesting and fun. Be negative. Find ways to make math dull and painful.
Be open. See the uses, power, patterns, and magic of mathematics. Be closed. Blind yourself to math's uses and its practical and esthetic value.
Be practical. Make yourself aware of the doors that passing each math class opens to you. Be impractical. Ignore the doors that open when you pass a math class.
Class Work Attend every class. Aim for perfect attendance, even if you already know it all. Be absent often. Dig a hole so deep that you cannot climb out except by dropping the course.
Be focused. Concentrate on the math topic at hand. Be mentally elsewhere. Daydream. Talk. Distract and annoy neighboring students.
Take good notes. Solve problems along with the instructor. Avoid participating in the discussion. Just watch the instructor.
Be inquisitive. Ask questions so that the instructor knows what you would like to learn more about. Be uninterested. Make the instructor guess what it is that you might be confused about.
Homework Be regular. Always do at least some homework before the next class, and finish by the due date. Be sporadic. Do homework only when it easily fits your schedule.
Invest time. Spend double to triple the amount of in-class time. Invest little time. Spend less time doing homework than you spend in class.
Review notes; read text; do all assigned problems (maybe even more), and check the answers. Ignore notes and text explanations; try a few problems, and don't bother checking to see if they are right.
Getting Help When needed, take advantage of all opportunities: study groups, tutors, instructor office hours. Even when lost, never seek assistance.

Michael Bowen's VC Course Pages: Math V46, Spring 2017

Last modified: Monday 07 August 2017 00:56:37
Created by Michael Bowen (Professor of Mathematics)
Department of Mathematics, Ventura College, California, USA
Ventura College is an independent college within the Ventura County Community College District.
Please email comments, questions, suggestions, or a notification if you find a "broken" link.
As an alternative, you may leave a voice message at (805) 289–6256.

User agents known to render this document correctly include:

Trademarks: Adobe®, Acrobat®, Macromedia®, and Shockwave® are registered trademarks, and Acrobat Reader™ is a trademark, of Adobe Systems Incorporated in the United States and/or other countries. Macintosh® is a registered trademark, and Safari™ is a trademark, of Apple, Inc. Borland® is a registered trademark of Borland Software Corporation in the United States and other countries. This web page is an independent publication and has not been authorized, sponsored, or otherwise approved by Borland Software Corporation. Corel® and WordPerfect® are registered trademarks of Corel Corporation. The buttons, name, and corporate logo (the "CC" in a circle) of Creative Commons™ are trademarks of Creative Commons. NoteTab™ is a trademark of Fookes Software. Zip® is a registered trademark of Iomega Corporation in the United States and/or other countries. Microsoft®, Visual C++®, Visual Studio®, and Windows® are registered trademarks of Microsoft Corporation. Firefox® is a registered trademark of the Mozilla Foundation. Netscape® and Mozilla® are registered trademarks of Netscape Communications Corporation. UNIX® is a registered trademark of The Open Group. Opera Software™ is a trademark of Opera Software ASA. Oracle® and Java® are registered trademarks of Oracle and/or its affiliates. Norton AntiVirus® is a registered trademark of Symantec Corporation. Maple™ is a trademark of Waterloo Maple Inc. World Wide Web Consortium® and W3C® are registered trademarks, and Amaya™ and XHTML™ are trademarks, of the World Wide Web Consortium. All other trademarks and service marks are the property of their respective owners.

Copyrights: The orange-and-black scales-of-justice logo is in the public domain. Opera browser copyright © by Opera Software ASA. Ventura College pirate logos and "Set Your Course" logos copyright © by the Ventura County Community College District. W3C logos copyright © by the World Wide Web Consortium. WAVE copyright © 2014 by WebAIM. Remainder of page copyright © 2005–2018 by Michael Bowen. All rights reserved.

Creative Commons License

Michael Bowen's Ventura College Course Pages are licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License - external link . You are also bound by this site's Terms of Use.

Ventura College 'Set Your Course' logo
Click to return to the top of this page View the terms of use View the CSS validator home page View the W3C Validator home page View the Web Accessibility Evaluation Tool (WAVE) home page