Final Practice Exam

Hey All,

I posted a final practice exam. I created problems which seemed to match the description of the exam. I'm missing out on the end of the exam at the moment and wouldn't know how closely what I wrote up resembles the real exam. At any given time, you should make sure you have the most up-to-date version number.

Note that the copies in the Dropbox will get updated dynamically. While the copies in my Course Files folder will get updated periodically.

Current Version Numbers and Links to Files:
Final Practice Exam [v1.2.4.2]
Final Practice Exam Solutions [v1.2.4.2]

You can ask me questions via e-mail or posting a comment here. Cheers.

Pictures

These pictures give a rough idea of how well (or not well) the normal distribution approximates the binomial distribution for increasing values of $n$. Specifically, we have $n$ going from 1, 2, 3, 4, 5, 10, 20, 30, 40, 50, 100, 200, 300, 400, 500. Note that the $y$-axis does not necessarily intersect the origin and that the range of values of both axes are changing as well.

$n=1$

Using the central limit theorem

We apply the central limit theorem to integer-valued random variables. In that situation, we assign to every integer $k$ it's own little interval $\left(k-\frac{1}{2},k+\frac{1}{2}\right)$. Thus instead of $P(X=5)$ we'd be looking for $P(4.5 \leq X \leq 5.5)$.

Calculus II for Bio Q&A

Finals are around the corner. Ask your questions! (either for credit or just because)

The ever useful overlapping circles...

Here's a list of things to make your comments look cool! I suggest you create your comment in a text file and copy and paste over to the comment field here, or just send me an email.

Answering Student Questions

Images in this post were generated using Mathematica 8.0.0.0

"In a compartment model, what doe a determinant of 0 mean?"
On page 612, the book explains that there are three possibilities when the determinant is 0. Matter gets stuck in compartment 1, matter gets stuck in compartment 2, or the matter in the system is a constant.

How do you compute the Hessian matrix of a function $f$?
The formula for the Hessian is on page 551 and the components of the matrix are second derivatives of $f$. If we write $\mathrm{Hess}f(x,y)=\begin{bmatrix} a & b \\ c & d \end{bmatrix} $, then compute $a$ by taking two derivatives in $x$, compute $d$ by taking two derivatives in $y$, and compute $b$ and $c$ by taking one derivative in $x$ and then taking a second derivative in $y$.

"One of the exercises generated a value of 2 for every boundary. Does that mean it is a max and min, or there is none?"
You have to compare the value 2 to the value obtained at the critical point on the interior. If the value of the critical point is greater than 2, then the minimum is attained all along the boundary and the maximum was attained at the critical point. If the value of the critical point is less than 2, then the opposite. In the situation that there are multiple critical points, then you must compare all the values attained.

Calculus II for Bio

Ask a math question of medium difficulty. Try looking through some and see if you can work out the answers. You'll need a writing utensil and some paper. Don't be lazy.

Here are two problems to get things started:
(Without looking in your book) Give an example of two $2\times 2$ matrices which do not commute.

If there was a $2\times 2$ matrix $A$ such that $A \cdot B = B \cdot A $ for all $2\times 2$ matrices $B$, what does $A$ look like?

Section 3 Comments 20130919

This section has a relaxed atmosphere which is conducive to learning.
This section is cool.
: Great! I definitely want students to come to a conducive-learning environment.

I don't like the quizzes. The homework is going to give me gray hairs at age 18. Tim/John*, you're a cool TA though.
*Inside joke
The section is helpful, but I don't like the quizzes. We should do more exercises.
Do the quizzes count as a grade?
: I don't like the quizzes either, but you probably need them.
: Yes, the quizzes are factored into your grade.

The differentials involving population are quite difficult.
: We'll have to talk about that some time.

I hate trigonometry.
: There's a saying: "Keep your friends close, and your enemies closer." You'll need trigonometry to answer calculus questions. So while you don't have to like it, you have to use it.