Math 107 Calculus II (Bio)

Spring 2015

Skip down to course schedule and announcements.



Course Information


  • Mona Merling
  • Email: mmerling(at)math(dot)jhu(dot)edu
  • Office hours: Wednesdays 2–4 at Krieger 311 or by appointment.


  • MWF 10:00–10:50 (Sections 1, 2, 3, 4) at Krieger 205
  • MWF 11:00–11:50 (Sections 5, 6, 7, 8) at Krieger 205


Calculus for Biology and Medicine (3rd Edition) Claudia Neuhauser, Prentice Hall


You will need an iClicker 2, which you should bring to every class. Here is a link with comprehensive information on clickers provided by the JHU Center for Educational Resources. Please register your clicker on blackboard; here are the instructions.




Time TA Room Office Hours
1 Tuesdays at  1:30pm Diego Espinoza (despino5) Krieger 300 W 12-1, Krieger 207
2 Tuesdays at  3pm Alex Grounds (aground1) Blmbrg 278 Th 3-5, Krieger 207
3 Thursdays at  3 pm Caleb Baechtold (cbaecht1) Blmbrg 278 W 1-2:30, Krieger 207
4 Thursdays at  4:30pm Kenny Co (kco1) Maryland 217 M 12-1, Krieger 207
5 Thursdays at  3pm Junyan Zhu (jzhu26) Ames 235 Th 11-1, Krieger 207
6 Tuesdays at  4:30pm Chenyun Luo (cluo5) Krieger 309 M 3-5, Krieger 207
7 Tuesdays at  3pm Chenyun Luo (cluo5) Remsen 101 M 3-5, Krieger 207
8 Thursdays at  1:30pm Junyan Zhu (jzhu26) Maryland 114 Th 11-1, Krieger 207



Homework will be posted each Friday in the course schedule below and will be due at the beginning of class the next Friday. You will receive the graded homework back in the following week's section. Sufficient practice in the homework is essential to master the material, so you are recommended to try to complete every assignment. You are allowed to work together and ask for help on the homework; however, you MUST write your own solutions. Copying is not acceptable.

You will be graded not only on your final answer, but also on the work that shows the process of how you obtained the answer. Richard Brown wrote a superb note on how to properly write up homework for this class, so that the writing process of the homework becomes a learning process, and also so that your reader can follow your thought process. The examples he gives are from math 106, so they should be familiar to you.

You must staple your homework, write your name and section number on it clearly, and write legibly. If your homework is too messy or illegible, the grader may choose not to grade it, and he may decide to take points off if the homework is not stapled.

No late homework will be accepted. On the other hand, you may miss up to two homework assignments without grade penalty, as the lowest two homework scores will be dropped from the final grade calculation. If you absolutely cannot make class, make sure someone hands in the homework for you, or make arrangements with the TA directly to get it to him before the due date.


There will be two in-class midterm examinations and a final exam.

  • Midterm 1: Monday, March 2 (week 6)
  • Midterm 2: Monday, April 13 (week 11)
  • Final: Wednesday, May 6 (finals week)

There will be no make-up exams. For excused absences, the grade for a missed exam will be a weighted average of the grades for all subsequent exams. Unexcused absences count as a 0. Documentation of illness etc. must be obtained from the Office of Academic Advising.

Class Attendance

I will not formally take attendance; however, you are encouraged to come to lectures. I will give short quizzes once in a while -- these will never be handed in; they are supposed to provide practice for the exams. Also, by attending lecture you will get a sense of what I consider important and that should help you know what to focus on studying for the exams. We will briefly talk about what to expect on each exam the class period before it takes place, so it is in your best interest to be there. If you have to miss class, you do not need to tell me; my best advice is to get notes and find out what you missed out on in class from someone who attended.

Class Rules

No cell phones and no computers, except for note taking.

Grading Scheme

The course grade will be determined as follows:

  • Homework: 10%
  • Midterm Exams: 25% each
  • Final Exam: 40%

Academic support

Check out the PILOT Learning program, and the webpage with information about academic support and tutoring. Furthermore, there is a math helproom in Krieger 213, and you are encouraged to make the best use of it.

Special Aid

Students with disabilities or other special needs who require classroom accommodations must first be registered with the disability coordinator in the Office of Academic Advising.  To arrange for testing accommodations the request must be submitted to the instructor at least 7 days (including the weekend) before each of the midterms or final exam.  You may make this request during office hours, after class or by sending an email to the instructor.

JHU Ethics Statement

The strength of the university depends on academic and personal integrity. In this course, you must be honest and truthful. Ethical violations include cheating on exams, plagiarism, reuse of assignments, improper use of the Internet and electronic devices, unauthorized collaboration, alteration of graded assignments, forgery and falsification, lying, facilitating academic dishonesty, and unfair competition.

Report any violations you witness to the instructor. You may consult the associate dean of students and/or the chairman of the Ethics Board beforehand. Read the "Statement on Ethics" at the Ethics Board website for more information.


Course Schedule

The tentative lecture schedule and homework assignments will be updated as we go. It is highly recommended that you read the relevant sections of the book before and/or after each lecture.

Topics Sections Homework DUE
Week 1:
Jan 26, 28,30

Quick Long review of integrals and other concepts from Calc I.

§ 7.1 and older stuff Homework 1

Do the two questions on this sheet (which are the 2 exercises from class).
Update Sat morning: I added a small question to think about when you do ex 2 (a)

§ 7.1, exercises 22, 23, 30, 32, 38, 42, 48, 49, 55, 59.
(I made sure not to assign the same questions you had to do last semester)

Solutions to graded problems
Week 2:
Feb 2, 4, 6

Review of integration by parts
Rational Functions and Partial Fractions.
§ 7.2, 7.3, 7.4 Homework 2

Read this sheet and do exercise 1 (a) and (b) on it.

§ 7.2 exercises 27, 29, 32, 36, 46, 51, 64
(Again I made sure not to assign the same questions you had to do last semester)

§ 7.3 exercises 4, 16, 18, 24, 34, 51.

Solutions to graded problems
Week 3:
Feb 9, 11, 13

Unbounded integrals.
Counting Principles
§ 7.4, 12.1

Read this note that I wrote about how I found the comparison function for the example from class. It should help you with the exercises where you need comparison functions and hopefully clear confusion about what functions work and what functions don't work for the comparison test.

Read and understand ALL the examples in 12.1.4 (these things take a little bit of getting used to, so make sure you understand the worked out examples before moving on)

Homework 3

§ 7.4/ exercises 8, 15, 22, 27, 28, 31, 34 (note that 33 we did in class and 34 is similar), 40, 41, 42

Read and do the 3 exercises on this sheet.

§ 12.1/exercises 4, 9, 16, 20, 22, 33, 42, 50.

Solutions to graded book problems (updated Fri, 02/20 5pm)
Solution to extra problem 3
Week 4:
Feb 16, 18, 20

Probability and conditional probability § 12.2, 12.3.1, 12.3.2, 12.3.4

In order to see some applications of these concepts to biology, read § 12.2, example 11 and the paragraph preceding it about Mendel. As an application of Bayes's formula, read the example about hemophilia starting with the second to last paragraph of page 686 in § 12.4.

If you were in the first lecture, read this note about Bayes's formula. I wrote down the total law of probability not the way I should have in class, which I apologize for.
Homework 4

§ 12.2/exercises 9, 10, 11, 12, 18,

(optional) 20 (-- note that 19 we proved in class and 20, which I recommend you do, leads to the proof of the proposition whose proof we skipped in class, that proposition is part (c) of 20)

29 (-- for this, read example 11 first and the paragraph preceding that example.)

34, 40, 50 (--for 50, just write down a formula. You are welcome to compute it out with a calculator if you want to see what your odds would be in a game of poker.)

§ 12.3/ exercises 6, 8

2, 21, 40 (-- think about how these compare)

25, 39 (-- think about how these compare)

Solutions to graded problems
Week 5:
Feb 23, 25, 27

Probability § 12.3.3 and review

Here are the slides for the review from Friday's class.
NO HOMEWORK DUE 03/06! Hooray!

Relax after the exam.

Note that there is no homework due next Friday. The homework that you had to hand in until now covers all the sections that show up on the exam, except 12.3.3, which we discussed this week. I recommend the following problems as practice.

Recommended problems from § 12.3.3:
12.3/exercises 27, 28, 29, 30
(you don't have to hand them in; making them due next Friday after the exam would be pointless)
Week 6:
March 2, 4, 6

Midterm 1 on Monday

Discrete Random Variables
§ 12.4.1.

Read example 3 from 12.4.1 in your book; this is similar to the example of distribution function we worked out in class and graphed. However, we never got to discuss the graph and think about it, and because I want to move on next time without going over this again, here is part of your assignment (that you don't have to hand in):

Look at the graph from class and the one from this example , and write down all the characteristics you observe. Afterwards, read the paragraph on the lower half of page 691 about the description of the graph of the distribution function of a discrete random variable, see how many of those attributes you had observed yourself, and try to understand why the distribution function of a discrete random variable has those attributes.
Homework 5

§ 12.4/ 2, 4, 6, 8, 10, 12, 14 (a and b)

Optional challenge question (not to hand in): Try 14 (c).

Advice: Start looking at 14(a and b) before next class; something similar will come up in lecture when we do binomial random variables and if you have already thought about it, it will help you understand better.

Solutions to graded problems
Week 7:
March 9, 11, 13

Random Variables

§ 12.4.2, 12.4.3, 12.5.1

Read examples 20-26 from section 12.4.2 - you will see that most of these are familiar examples, we just didn't express them in terms of binomial random variables until now.

The example I did at the end of class today was example 4 from 12.5.1. Read it carefully and understand it before beginning the homework.
Homework 6

§ 12.4.2/ exercises 20, 22, 24, 25 (note that 26 we proved in class)

Also, do the exercise on this sheet.

§. 12.4.3/ 34, 36 ,48

§. 12.5.1./ 2, 4, 6, 8, 9

Solutions to graded problems
March 16-22

HAVE FUN !! No homework
Week 8:
March 23, 25, 27

Differential equations

Systems of equations

§ 8.1, 9.1.

Here is a note about the logistics equation. It's exactly what we did in class, but it points out and corrects the error from class regarding the partial fraction decomposition constants.
Homework 7

§ 8.1./ 10, 14, 16, 22, 29, 35,

39 (for part c, which asks you to plot the graphs you are welcome to use a graphing calculator or plot the graphs in wolfram alpha online to see what they look like and how they compare),

44, 48, 50, 54.

§ 9.1./ 5, 26, 28
In order to see how to do the last two, do the following reading assignment first:
Read example 5 and 8 (in 5 a system of equations is solved the way you know how to do it, in 8 the same system is solved by using the associated matrix by performing the same operations that were performed to the equations to the rows of the matrix instead. This is what we discussed in class today.)

Solutions to graded problems
Week 9:
March 30 , April 1, 3

Matrices § 9.2, 9.3 Homework 8

This will seem long, but most exercises are very short.

§ 9.2/exercises 8, 21, 22, 23, 24, 25, 26, 28, 32, 33, 38, 40, 44, 50, 51, 52, 54, 55, 58, 64

§ 9.3/ exercises 2, 29, 32, 37, 41

Also, do the exercise on this sheet (small edit made Saturday morning.)

Solutions to graded problems
Week 10:
April 6, 8, 10

Linear transformations
Eigenvalues and eigenvectors
§ 9.3

The slides from the review are posted below under announcements with some comments.
NO HOMEWORK DUE 04/17. Relax after the exam!

Recommended problems from 9.3 (not to hand in)
Find the eigenvalues and the corresponding eigenvectors for the matrices given in 9.3/ 49, 50, 51, 52.
Ignore the confusing text preceding these exercises and just find the eigenvalues starting with the definition and going through all the steps as we did in class, and then find the set of eigenvectors corresponding to each eigenvalue.
If there are no real eigenvalues, say that there are no real eigenvalues -- we are considering R^2 as a real vector space (and not a complex one), namely our set of scalars is the real numbers (and not the complex numbers).
Week 11:
April 13, 15, 17

Midterm 2 on Monday

Systems of differential equations
§ 11.1.2 Homework 9
§ 11.1/ exercises 1,2,4

For exercises 20, 22, 24, 26 do the following
a) find the general solution
b) find the solution to the initial value condition that's given (I will show you how to do this on Monday, but it would be good for you to go ahead and try, you should be able to figure out how to do it)

exercise 27 (We don't know how to find the general solution in the case of repeated eigenvalues, but in this exercise you will find some of the solutions in the case where the coefficient matrix has a repeated real eigenvalue.)

Solutions to graded problems
Week 10:
April 20, 22, 24

Systems of differential equations

Multivariable calculus
§ 11.1, § 10.1

Here is a note for Monday's lecture - it corrects a typo I made on the board at the end of the 11am lecture, and also there is a note summarizing and completing the discussion we had in class, so that you have a typed up reference.

Read example 5 on page 510 to see a biological application of level curves.
Homework 10
§ 10.1/ exercises: 4, 8, 14, 16, 18, 19-20-21-22 (these last four are all one exercise), 23

Also, do the following exercises from § 10.2 (we will learn on Monday about limits, but I wanted to include this on this homework so that you get some practice since there won't be any more homework due after this.)

§ 10.2/ exercises 2,6, 12, 16, 18, 20

Solutions to graded problems
Week 13:
April 29, 31, May 1

Multivariable calculus § 10.2, overview of 10.3 NO HOMEWORK. Review old homework and notes. I didn't assign any homework on what we did this week, but please review your notes from class.
FINAL EXAM Wednesday May 6th, 9am -- 12noon


Tue, May 5: Here are the instructions regarding which rooms to go to for the final tomorrow.

Fri, May 1: Here are the slides to the practice questions we did in class and the ones we didn't get to. Please do them on your own, and please use the review sessions next week to clarify questions about this material or any of the older material.

Wed, Apr 29: Here is a study guide for the final. Also, here are the midterms without the solutions, so that you can print them and redo them: midterm 1, midterm 2

Wed, Apr 29 Review office hours and problem sessions next week:
Monday, 3-5 I will hold review office hours in Shaffer 301.
Tuesday noon-2 (to be confirmed) there will be a review session organized by your TAs, location TBD.

The exam will take place on Wednesday, May 6 - I will soon post an announcement about which room you have to go to, we have several rooms reserved in Krieger.

People who need extra time can schedule their exam with the SDS office at 9am on Wednesday (if you want to do this, schedule your exam asap). Alternatively, they can take the exam as instructed by Richard Brown at 8am in Krieger 308.

Wed, Apr 29 There was a typo in the solution to homework 9 (the computation of the initial value problem was done for a different initial value than the question was asking in exercise 20). It's now fixed.

Mon, Apr 20: I posted a note above regarding some details from today's lecture.

Mon, Apr 13: Here are solutions to the second midterm. Please read them carefully and try to understand what you did wrong, or why your solution was incomplete whevever it was. (If you find typos or errors, please alert me.) The exams will be returned in section, and the rules about returning them are the same as for the first midterm. Here are the instructions for that again. I will be happy to talk with anyone who wants to discuss their exam. You still have time to improve on the final, which counts the most!

Sun, Apr 12: Typo fixed in the note about the logistics equation. Throughout the whole note I was considering a function y of t, but in the very first line I had written dx/dy - it's now fixed to dy/dt, which is what is in agreement with the notation used throught the rest of the note. Thanks to Josh for pointing it out.

Sat, Apr 11: Typo fixed in the study guide for the second midterm. I had written dy/dt =f(x)g(y), where I meant dy/dt=f(t)g(y) -- f and y should be functions of the same variable. Now fixed, thanks to Ivory for pointing that out.

Fri, Apr 10: Here are the slides to the practice questions we did in class and the ones we didn't get to. Please do them on your own. I wrote the solution to exercise 4 more clearly -- please make sure you understand why it is what it is. If you are confused, please come to my office hours today (or Diego's review session tomorrow, or both).

I think my solution to exercises 4, part 2, was correct after all (we didn't get to this in the first lecture). However, I've hastily wrote these exercises, so if you find errors, please alert me so that I can fix them.

Thu, Apr 9: I will have extra office hours on Friday 4-6pm in Krieger 204, and Diego will be holding a review session on Saturday at 7pm in Remsen 101.

Mon, Apr 6: Here is a study guide for the second midterm, which will take place on Monday, April 13.

Here are the instructions and logistics for the second exam. The instructions are the same as for the first exam. I have only rewritten in this handout the instructions about classrooms and absences. Please refer back to the instructions guide for the first exam if you want to reread the extra info about how the exam looks like and how your solutions should look like.

Wed, Mar 25: I have posted a note about the logistics equation above in the course schedule. I corrected the small error I had made in class. The note is basically what we did in class typed up.

Tue, Mar 3: Here are solutions to midterm 1. Exams will be returned in section, here are the instructions for that.

Fri, Feb 27: At 11:30pm, I have fixed a typo in the file with the 3 solutions to the question from last class. Thanks Diego for catching it -- one of the fractions in the last line was flipped. It's fixed now.

Fri, Feb 27: I have posted above in the course schedule the slides for the review from class today. Please do it again on your own. I believe I have fixed all the typos; if you find more typos, alert me.

Thu, Feb 26: I have typed up 3 different solutions to the problem I left for you to do at the end of last class. I won't go over it in class anymore. Please do it and then read the solutions. Hopefully at least one of them will make sense to you. :)

Thu, Feb 26: Here are solutions to the practice problems on Bayes's formula. Please try to do the problems on your own first and then consult the solutions in order to compare answers.

Thu, Feb 26: Here are the instructions and logistics for the first exam. PLEASE read carefully and make sure you know which classroom you have to go to to take the exam. There are clear instructions about how the students will be split up into the classrooms that we were assigned.

Tue, Feb 24: I will have extra office hours on Friday 3-5pm. For this I have reserved the lounge on the 4th floor of Krieger because I am hoping more people will come than my office can fit.

Also, Diego Espinoza has graciously volunteered to hold a review session, not just for his section, but for all the students in the class. This will take place on Saturday 12-2 in Hodson 210.

Mon, Feb 23: Each student needing special accomodations for the exam needs to go into AIM and request to take the exam in the student disability office, and needs to have this request approved by them. Some of you have already done so. If you haven't and need accommodations, make sure to do so.

Mon, Feb 23: Here is a practice sheet with some problems that require using Bayes's formula. Typo fixed Monday at 10:48pm.

Sat, Feb 21: Here is a study guide for the first midterm, which will take place on Monday, March 2.

Fri, Feb 20: For the 10am lecture -- I posted above a note to read about Bayes's formula, because I didn't write it the way I should have in class. I apologize for that. We will just go through the example with the Monty hall problem next time, and I'll make sure to end in the second lecture at the same point where I end in the first to sync up the lectures.