# Math 545 Homework #7 Solutions

Text: CALCULUS for Biology and Medicine (3rd Edition) by Claudia Neuhauser

Office: 3135 MSB (Mathematical Sciences Building)

Phone: (530) 277-5317

My Office Hours: Announced in class or sent via e-mail

T.A. Office Hours : Math 17ABC Calculus Room in 1317 Earth & Physical Sciences Building is open 1-7 pm Wednesday through Friday

- SCANNED PROBLEMS for Chapter 10
- HW #1 ... (Section 10.1) ... p. 511: 3, 5, 6, 13 (Plot level curves for the following values of z: 0, 1, 4, 9.), 14 (Plot level curves for the following values of z; 0, root(5), root(8), 3.), 15 (Plot level curves for the following values of z: ln(1/10), ln(1/2), ln 1, ln 2, ln 3, ln 4.), 16 (Plot level curves for the following values of z: e^(0), e^(-1/4), e^(-1), e^(-4), e^(-9).), 19, 22, 24a (Read b. and c.), 25
- HW #2 ... (Section 10.2) ... p. 518: 2, 11, 15, 17, 18, 20-22, 25, 26 ... and ... Worksheet 1
- HW #3 ... (Section 10.3) ... p. 525: 1, 3, 6, 7, 9, 10, 13, 17, 19, 22, 23, 25, 30, 39, 40-42, 51
- HW #4 ... (Section 10.4) ... p. 535: 1, 4, 7, 10, 19, 21, 24, 26, 27, 29, 31, 34, 35, 37, 41, 42, 43, 46
- HW #5 ... (Section 10.5) ... p. 545: 2-4, 7, 8, 11, 14, 17, 18, 22, 25, 27, 28, 31, 34, 35, 37, 39, 42, 44
- HW #6 ... (Section 10.6) ... p. 569: (Use the Second Derivative Test Handout from class to do problems 1, 3, 4, 6, 7.) ... (Here is a Proof of the Second Derivative Test for anyone interested.) 1, 3, 4, 6, 7, 14, 15, 16 (Change f(x,y)=x^2+y^2 to f(x,y)=1-x^2+2y^3.), 17, 19, (For the next 4 problems find critical points and extreme values only. You need not verify the maximum or minimum.) 28, 29, 31, 33

EXAM 1 is Wednesday, January 27, 2016. It will cover handouts, lecture notes, and examples from class, homework assignments 1 through 6, discussion sheets 1,2, and 3, and material from sections 10.1-10.6 which was presented in lecture notes through Monday, January 25, 2016. Most of the exam questions will be similar to problems presented in class, homework problems, or discussion sheet problems. There will be an OPTIONAL EXTRA CREDIT problem on the exam.

TYPES OF QUESTIONS FOR EXAM 1 FOR WINTER 2016 (subject to unannounced changes)

- 1-- 3D-Graphing (intercepts, traces, level curves, etc.)
- 1-- Find and Classify Critical Points for z=f(x,y)
- 2-- Compute Limit or Determine that Limit Does Not Exist
- 1-- Domain and Range for z=f(x,y)
- 1-- Precise Epsilon,Delta-Proof
- 1-- Tangent Plane
- 2 or 3-- Chain Rule, Gradient Vectors, or Directional Derivative
- 1 or 2-- Others
- 1-- OPTIONAL EXTRA CREDIT

HERE ARE SOME RULES FOR EXAM 1.

- 1.) IT IS A VIOLATION OF THE UNIVERSITY HONOR CODE TO, IN ANY WAY, ASSIST ANOTHER PERSON IN THE COMPLETION OF THIS EXAM. IT IS A VIOLATION OF THE UNIVERSITY HONOR CODE TO COPY ANSWERS FROM ANOTHER STUDENT'S EXAM. IT IS A VIOLATION OF THE UNIVERSITY HONOR CODE TO HAVE ANOTHER STUDENT TAKE YOUR EXAM FOR YOU. PLEASE KEEP YOUR OWN WORK COVERED UP AS MUCH AS POSSIBLE DURING THE EXAM SO THAT OTHERS WILL NOT BE TEMPTED OR DISTRACTED. THANK YOU FOR YOUR COOPERATION.
- 2.) No notes, books, or classmates may be used as resources for this exam. YOU MAY USE A CALCULATOR ON THIS EXAM.
- 3.) You will be graded on proper use of limit notation.
- 4.) You will be graded on proper use of integral and derivative notation.
- 5.) Put units on answers where units are appropriate.
- 6.) Read directions to each problem carefully. Show all work for full credit. In most cases, a correct answer with no supporting work will receive little or no credit. What you write down and how you write it are the most important means of your getting a good score on this exam. Neatness and organization are also important.

THE GRADING SCALE FOR EXAM 1 FOR WINTER 2016 IS :

A+ ...... 100-108

A ...... 90-99

A-/B+ ...... 85-89

B ...... 70-84

C ...... 48-69

D ...... 35-47

F ...... 0-34

- HW #7 ... (Here is a handout for double integrals .) ... Worksheet 2
- HW #8 ... (Here are lecture NOTES for parametric equations.) ... Worksheet 3

SCANNED PROBLEMS for Chapter 11 - HW #9 ... (Section 11.1) ... p. 606: 1, 2, (On problems 5 and 7 set up a table showing the slope, direction vector, and speed at each point.) 5, 7, 9-13, 15, 17, 19, 26, 35, 37, 39, 43, 45, 52-55 ... and ... Here is an Example ... to help with ... Worksheet 3.5
- HW #10 ... (Section 11.2) ... p. 617: 9, 12, 13, 17, 18, (Use equations (11.40) and (11.41) from the text for problems 19 and 20.) 19, 20, 21ae (Assume t is years and assume x1+x2=20 for all values of t.) ... and ... Worksheet 4
- HW #11 ... (Section 9.3) ... p. 486: 63, 65, 67, 68 ... and ... (Section 11.3) ... Here is a worked out example illustrating the Analytical Approach to categorizing equilibria for 2X2 nonlinear systems of D.E.'s... Here is a worked out example with some notes illustrating the Graphical Approach to categorizing equilibria for 2X2 nonlinear systems of D.E.'s... p. 629: 1, 2, 3, (For problems 7, 9, and 11 classify each equilibrium as unstable or stable, then categorize it as a sink, source, saddle, stable spiral, or unstable spiral.) 7, 9, 11, 15, (Use this figure for 17.) 17, (Use this figure for 19.) 19, (Use this figure for 21.) 21, 17 (again, assuming that f1>0 and f2>0 near the origin, then complete the 2-dimensional sign chart and signed Jacobi Matrix), 23

EXAM 2 is Friday, February 19, 2016. It will cover handouts, lecture notes, and examples from class, homework assignments 7 through 11, discussion sheets 4, 5, and 6 and material from sections 9.3 and 11.1-11.3 in the book which was presented in lecture notes through Wednesday, February 18, 2016. Most of the exam questions will be similar to problems presented in class, homework problems, or discussion sheet problems.

TYPES OF QUESTIONS FOR EXAM 2 FOR WINTER 2016 (subject to unannounced changes)

- 2 -- Double Integrals (Area, Volume, Mass, or Average Value, etc.)
- 1 -- Describe Region Using Vertical or Horizontal Cross-Sections
- 1 -- Classify Equilibrium
- 1 -- Solve 2 by 2 Linear System of D.E.'s with Initial Conditions
- 1 -- Two Tank Mixture Problem (SET UP ONLY)
- 1 -- Graph Parametric Equations
- 1 -- Graphical Approach to Categorize Equilibrium for a Nonlinear 2 by 2 System of D.E.'s
- 1 -- Analytical Approach to Categorize Equilibria for a Nonlinear 2 by 2 System of D.E.'s
- 1 or 2 -- Others
- 1 -- OPTIONAL EXTRA CREDIT

HERE ARE SOME RULES FOR EXAM 2.

- 1.) IT IS A VIOLATION OF THE UNIVERSITY HONOR CODE TO, IN ANY WAY, ASSIST ANOTHER PERSON IN THE COMPLETION OF THIS EXAM. IT IS A VIOLATION OF THE UNIVERSITY HONOR CODE TO COPY ANSWERS FROM ANOTHER STUDENT'S EXAM. IT IS A VIOLATION OF THE UNIVERSITY HONOR CODE TO HAVE ANOTHER STUDENT TAKE YOUR EXAM FOR YOU. PLEASE KEEP YOUR OWN WORK COVERED UP AS MUCH AS POSSIBLE DURING THE EXAM SO THAT OTHERS WILL NOT BE TEMPTED OR DISTRACTED. THANK YOU FOR YOUR COOPERATION.
- 2.) No notes, books, or classmates may be used as resources for this exam. YOU MAY USE A CALCULATOR ON THIS EXAM.
- 3.) You will be graded on proper use of integral notation.
- 4.) Put units on answers where units are appropriate.
- 5.) Read directions to each problem carefully. Show all work for full credit. In most cases, a correct answer with no supporting work will NOT receive full credit. What you write down and how you write it are the most important means of your getting a good score on this exam. Neatness and organization are also important.

THE GRADING SCALE FOR EXAM 2 FOR WINTER 2016 IS :

A+ ...... 100 - 108

A ...... 90 - 99

A-/B+ ...... 84 - 89

B ...... 66 - 83

C ...... 48 - 65

D ...... 39 - 47

F ...... 0 - 38

- HW #12 ... (Section 11.4) ... Here are some notes on a Predator/Prey model. ... p. 652: 15abc and solve system using separation of variables, 16abc and solve system using separation of variables,18 and 19ab (Solve the D.E. in part a. by separation of variables; assume that N(0)=9; solve explicitly for N. Graph N=N(t).)

SCANNED PROBLEMS for Chapter 12 - HW #13 ... (Section 12.1) ... p. 666: 1, 4, 5, 9, 10, 13, 14, 16, 19, 22, 24, 25, 29-31, 33-37, 39, 45-50
- HW #14 ... (Section 12.2) ... p. 677:1-12, 17, 18, 21, 22, 25-33, (See this link for a short course in X and Y chromosomes. See this link for a short course in color blindness.) 36, (See pages 674-675.),40-50, 51a
- HW #15 ... (Section 12.3) ... p. 687:1, 4-12, 15, 18-35, 37-40 (omit 37-40 for the final exam)

EXAM 3 is Friday, March 11, 2016. It will cover handouts, lecture notes, and examples from class, homework assignments 12 through 15 (only problems 1, 4-6, 8-12, and 22 from HW # 15), discussion sheets 7, 8, and 9 (problems 1-3, 8, 9a-f only) and material from sections 11.4 and 12.1-12.3 (only the topic of conditional probability from section 12.3) in the book which was presented in lecture notes through Wednesday, March 9, 2016. Most of the exam questions will be similar to problems presented in class, homework problems, or discussion sheet problems.

TYPES OF QUESTIONS FOR EXAM 3 FOR WINTER 2016 (subject to unannounced changes)

- 8 -- Counting Problems Using Fundamental Counting Principle, Combinations, or Permutations
- 2 -- Probability Problems
- 1 -- Conditional Probability (Tree Diagram) with Several Parts
- 1 -- Lotka-Voltera Predator-Prey Problem
- 1 -- OPTIONAL EXTRA CREDIT

HERE ARE SOME RULES FOR EXAM 3.

- 1.) IT IS A VIOLATION OF THE UNIVERSITY HONOR CODE TO, IN ANY WAY, ASSIST ANOTHER PERSON IN THE COMPLETION OF THIS EXAM. IT IS A VIOLATION OF THE UNIVERSITY HONOR CODE TO COPY ANSWERS FROM ANOTHER STUDENT'S EXAM. IT IS A VIOLATION OF THE UNIVERSITY HONOR CODE TO HAVE ANOTHER STUDENT TAKE YOUR EXAM FOR YOU. PLEASE KEEP YOUR OWN WORK COVERED UP AS MUCH AS POSSIBLE DURING THE EXAM SO THAT OTHERS WILL NOT BE TEMPTED OR DISTRACTED. THANK YOU FOR YOUR COOPERATION.
- 2.) No notes, books, or classmates may be used as resources for this exam. YOU MAY USE A CALCULATOR ON THIS EXAM.
- 3.) You will be graded on proper use of integral notation.
- 4.) Put units on answers where units are appropriate.
- 5.) Read directions to each problem carefully. Show all work for full credit. In most cases, a correct answer with no supporting work will NOT receive full credit. What you write down and how you write it are the most important means of your getting a good score on this exam. Neatness and organization are also important.

THE GRADING SCALE FOR EXAM 3 FOR WINTER 2016 IS :

A+ ...... 100 - 108

A ...... 90 - 99

A-/B+ ...... 87 - 89

B ...... 70 - 84

C ...... 50 - 69

D ...... 40 - 49

F ...... 0 - 39

FINAL EXAM -- SATURDAY, March 19, 2016, 3:30-5:30 p.m., in 6 Olson

BRING A PICTURE ID TO THE EXAM

AND BE PREPARED TO SHOW IT TO KOUBA OR THE TEACHING ASSISTANTS !!

The final exam will cover handouts, lecture notes, and examples from class, homework assignments 1 through 15, Discussion Sheets 1-9 and material from sections 9.3, 10.1-10.6, 11.1-11.4, and 12.1-12.3. Use your three hour exams, 15 homework assignments, and discussion sheets as a guide to your preparing for the final exam.

TYPES OF QUESTIONS FOR THE FINAL EXAM FOR WINTER 2016 (subject to unannounced changes) ...

- 2 -- Compute limits or Determine Limit Does Not Exist
- 1 -- Precise Epsilon/Delta Proof
- 1 -- Find and Classify Critical Points
- 1 -- Directional Derivative
- 1 -- Double Integral
- 1 -- Solve System of Linear Differential Equations
- 1 -- Find and Classify Equilibria for Nonlinear System of Differential Equations
- 1 -- Graph Parametric Equations
- 7 -- Various Counting and Probability Problems
- 1 -- OPTIONAL EXTRA CREDIT

HERE ARE SOME RULES FOR THE FINAL EXAM.

- 3.) You may NOT use shortcuts from the textbook for finding limits to infinity.
- 4.) You need NOT MEMORIZE trigonometry identities. A short list of identities will be provided on the front cover of the exam.
- 5.) Using only a calculator to determine the value of limits will receive little or no credit.
- 6.) You will be graded on proper use of limit notation.
- 7.) Put units on answers where units are appropriate.
- 8.) Read directions to each problem carefully. Show all work for full credit. In most cases, a correct answer with no supporting work will NOT receive full credit. What you write down and how you write it are the most important means of your getting a good score on this exam. Neatness and organization are also important.

####################################################################

####################################################################

### Your comments, questions, or suggestions can be sent via e-mail to Kouba by clicking on the following address :

kouba@math.ucdavis.edu .

## 0 comments