I prepared a study sheet for the final exam.

You might benefit from the previous study sheets for exam 1 and exam 2.

Keep in mind the Library has an archive of previous final exams.

Here is the Sage code to produce a direction field for a 2×2 system of ODEs.


# v' = Av, A is 2x2 matrix
A = matrix([[-1/2,1],[-1,-1/2]]);
print 'You entered: A = ';
print A;
x,y = var('x,y');
intermediate_step = (A*vector([x,y])).normalized();
v_prime(x,y) = (intermediate_step[0],intermediate_step[1]);
plot_vector_field(v_prime(x,y), (x,-2,2), (y,-2,2))


If you haven’t already, go to sagenb.org, log in (you might should click the Google or Yahoo icon), open a new worksheet, copy paste this into a cell, and click “evaluate”. You should get a beautiful direction field.

Note, I’m finding that sagenb.org isn’t working the way it used to. If having trouble, you might try https://cloud.sagemath.com/ and create a new project and create a new Sage worksheet, then copy, paste, and run the code.

I compiled a list of topics and sections for exam 2, which will be April 1.

We talked about forced spring vibrations. You can download a Sage worksheet that I used to generate the plots. Also, check out the disastrous consequences of resonance and the “beats” caused by amplitude modulation, both below.

(Note: for the sage worksheet to unpack properly, I had to use Stuffit Expander on a Mac. The build in archive utility didn’t work. You should get a .sws file after unzipping)

Resonance:

Beats:

Spring-mass systems (i.e. vibrations) bring this 90s gem to mind:

…And this older one:

Consider the ODE \(y” – 3y’ -4y = t^2 +2\). In class we came up with a particular solution: \( Y(t) = -1/4*t + 2/8*t – 29/32\). Here’s how you can get Sage to check quickly that you are correct.

t = var('t');
Y = -1/4*t^2 + 3/8*t - 29/32;
Y.diff(2) - 3*Y.diff() -4*Y

If you enter this in Sage, your output will be \(t^2 + 2\), which means \(Y(t)\) is a solution to the ODE given above.

The first midterm exam is February 18. I’ve created a study sheet that lists the sections and topics covered and the skills you need to know for the exam. The first two bullet points under skills are calculus 1, 2, and 3 skills. Keep in mind that solutions to all homeworks and quizzes are in BlackBoard.

A group of students asked me after class, “How much time do you intend for us to spend on homework?”

I didn’t know what to say to that. (See below for what I told them. In between you will read what I have thought about since they asked me the question).

I have never thought about homework from a time perspective. As a teacher, I make an assignment that covers a representative sample of the material (to make sure students know how to solve the different types of problems that will be on the quizzes and tests). I am a firm believer that you learn mathematics by doing mathematics.  If you sat in class and watched me solve problems without doing the homework, how would you do on a quiz? My answer to that question is “poor.” My philosophy is that each homework problem is a benefit to you because it prepares you for quizzes and exams, but also for your life later on. Homework is practice.

I read some online and see a general guideline that for every hour spent in lecture, a college student should expect to spend 2-3 hours studying. In a literature class, that 2-3 hours might be spent reading a novel. In math class, it might be spent solving problems.

Let’s take 2.5 hours of study to 1 hour of lecture as our ratio. The average student takes, say, 15 credit hours (this means 15 hours of lecture per week). So according to these guidelines, you sit in class for 15 hours per week and study for 38, for a total of 53 hours per week devoted to academics. Being a student is a full-time job, which makes 53 hours seem pretty reasonable. If this seems reasonable to you, then I hope you would agree that for a 3 credit hour class, 7.5 hours per week of study time (including homework) is reasonable.

(My answer to the students was 6-8 hours on ODE homework).

I skipped an integration by parts problem on the “periodic pollution” example.

Here is how to figure out \(e^{t/2} \, \sin(2t)\,dt\):

Some have asked me, “Why is my homework grade so low? I worked hard on it.”

Let me explain how the homework is graded and discuss why you might be getting a lower grade than expected.

The homework grader has a very limited amount of time to do the grading. Due to this constraint, an efficient and fair grading method was devised. Each problem is worth 2 4 points (a problem with parts a b and c is worth 6 12 points). The full 2 4 points is earned when you write a complete and correct solution (showing all work). No points will be given for a problem not attempted. Partial credit will be given everything in between: started on a problem but didn’t finish, a solution with a correct answer but missing work, etc.

(As a side note, each WeBWorK problem is worth 3 4 points in the grade book).

Here are some tips to get the full 4 points:

• Read the directions carefully. It is easy to omit part of the solution when you haven’t read all the instructions.
• Show all of your work. If you used a calculator to solve an equation, say so. That’s legal.
• Beware: the answers in the back of the book are surprisingly incomplete. Compare problem 1.1.6’s answer in the back of the book to the full solution (in BlackBoard under Content). There is a big difference between the two! You won’t get full credit just because your answer matches the back of the book.
• Some of the problems require time but aren’t too difficult. Start working on the assignment in advance.
• Pay attention to the length of the solutions I provide you. If my solutions are 12 pages long and yours are 2, there might be a problem.
• Write the problem and instructions on your paper. This will help internalize what you are asked to do on that problem.

The grader has been advised to leave comments when possible. Before asking me why you got a low score on a problem, please:

• Reread the problem and instructions in the book
• Read my solution in BlackBoard -> Content
• Compare your solution to mine
• Absorb the grader’s comments
• Determine if you still want to ask about the points

Sometimes running through the above checklist will save you and me time. Keep in mind that I’m human and can easily make mistakes when creating the solutions. Also keep in mind that I’m here to help, so don’t be shy to ask questions.

One thing you need to know how to do in 414 is to integrate. Fundamental theorem of calculus, u-substitution, integration by parts, etc. I found an integration review online with answers! I’m happy to answer questions you may have.

Please note that this is a collection of many types of integration problems for another course at another university. There might be some very tricky problems on it. If you want to practice integration by parts, say, you might do better to look in your calculus book in the IBP section and try some homework problems in that section.