Master Multivariable Calculus & Linear Algebra Test

Question 1

Let $f(x,y)=x^2y + e^{xy}$. Compute $
abla f(1,0)$.

Accepted Answer

$\langle 0,1 \rangle$

Question 2

If matrix $A=\begin{pmatrix}2 & 1 \ 0 & 3\end{pmatrix}$, what are its eigenvalues?

Accepted Answer

2 and 3

Question 3

Evaluate $\int_0^1 \int_0^1 (x^2 + y^2) \, dx \, dy$. Round to 3 decimal places.

Accepted Answer

0.667

Question 4

Solve $y' - 2y = 0$ with $y(0)=3$. Find $y(1)$. Round to 3 decimal places.

Accepted Answer

22.167

Question 5

Prove that if $A$ is a symmetric matrix, then all eigenvalues of $A$ are real.

Question 6

Derive the Taylor expansion of $\ln(1+x)$ up to order $x^3$.

Question 7

Let $X$ be a continuous random variable with pdf $f(x)=2x$ on $[0,1]$. (a) Compute $E[X]$ and $Var(X)$. (b) Find $P(X>0.7)$. (c) Let $Y=X^2$. Find the pdf of $Y$.

Question 8

Consider the optimization problem: Maximize $f(x,y)=xy$ subject to $x^2+y^2=1$. (a) Use Lagrange multipliers to find critical points. (b) Classify extrema. (c) Interpret geometrically.