The purpose of this self-assessment is to give students a flavor of math background that is expected or will be introduced in this class. This assessment covers basic knowledge and skills. Formal reasoning skills (proofs, logic) such as those taught in Math 232 are also expected.

## Derivatives and Basic Probability

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

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

$$\frac{d}{dx}f(x) =$$

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

$$\frac{d}{dx}f(x) =$$

3. 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:

$$\Pr(\text{heads}, \text{tails}) =$$

4. 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:

$$\Pr(\text{tails}, \text{tails}, \text{tails}) =$$

5. 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 and Linear Algebra

These topics are covered in Multivariable Calculus and Linear Algebra, which are not prerequisites. We will introduce these topics in class. It will be a bit easier if you have seen them before.

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

$$\frac{\partial}{\partial{x}} f(x,y) =$$

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

$$\frac{\partial}{\partial{y}} f(x,y) =$$

3. 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:

$$A \mathbf{x} = \begin{bmatrix}2 & 3 \\ 5 & 1\end{bmatrix} \cdot \begin{bmatrix}1 \\ 2\end{bmatrix} =$$

$$A^T =$$

$$\mathbf{x}^T A =$$