The purpose of this self-assessment is to give students a flavor of math foundation for this class. Formal reasoning skills (proofs, logic) such as those taught in Math 232 are also expected.

You should feel comfortable that you already know the topics in the first section, and already know or can learn the topics in the second section with some scaffolding provided in lectures. If you don't feel comfortable about this, please come see me.

## Derivatives and Basic Probability (Prerequisites)

Here are a few basic exercises from the required prerequisite classes (Calculus I and Discrete Math). You should feel very comfortable doing these (perhaps with a bit of brushing up):

Let \(f(x) = 5x^2 + 3\). Write the derivative of \(f(x)\).

$f(x) = $

Let \(f(x) = 4e^{2x}\). Write the derivative of \(f(x)\).

$f(x) = $

A fair coin has equal probability of coming up heads or tails. If a fair coin is flipped twice, what is the probability of seeing first heads, then tails:

$(, ) = $

Consider an unfair coin that has probability \(3/4\) of coming up heads. If the coin is flipped three times, what is the probability of seeing tails each time:

$(, , ) = $

I agree to let you flip the same unfair coin one time and pay you $1 if it comes up heads and $2 if it comes up tails. Let \(X\) be the number of dollars I pay you, which is a *random variable*. What is the expected value of \(X\)?

$E[X] = $

## Partial Derivatives, Matrices and Vectors (Will be introduced; some background helpful)

These topics are covered in Multivariable Calculus and Linear Algebra, which are **not** prerequisites. Some background in these topics would be helpful. If you haven't seen these before, don't worry, we will introduce/review what you need to know about partial derivatives and linear algebra. You will be expected to come up to speed somewhat quickly.

Let \(f(x,y)= 3xy^2+2y\). Write the following partial derivative:

$ f(x,y) = $

Let \(f(x,y)= 3xy^2+2y\). Write the following partial derivative:

$ f(x,y) = $

Let \(A\) be the matrix \(\begin{bmatrix}2 & 3 \\ 5 & 1\end{bmatrix}\) and let \(\mathbf{x}\) be the (column) vector \(\begin{bmatrix}1 \\ 2\end{bmatrix}\). Let \(A^{T}\) and \(\mathbf{x}^T\) denote the *transpose* of \(A\) and \(\mathbf{x}\), respectively. Complete the following expressions:

$A =
\[\begin{bmatrix}2 & 3 \\ 5 & 1\end{bmatrix}\]
\[\begin{bmatrix}1 \\ 2\end{bmatrix}\]
= $

$A^T = $

$^T A = $