Lesson 120 — Final Workshop

Compute $\displaystyle\int_0^{\pi/2} \sin^2 x \cos^3 x\, dx$.

Solve $y'' + 4y = 0$ with $y(0) = 1$, $y'(0) = 0$.

Revenue $R(q) = 120q - 2q^2$ and cost $C(q) = 200 + 40q + q^2$. Find $q^*$ that maximizes profit $L = R - C$.

Write the Taylor series of $\cos x$ centered at $x = 0$ through the $x^4$ term.

Compute $\dfrac{d}{dx} e^{x^2}$.

Compute $\displaystyle\int_1^e \frac{1}{x}\,dx$ using the FTC.

Compute $\displaystyle\lim_{x \to 0} \frac{e^x - 1 - x}{x^2}$.

Compute the volume of the solid of revolution generated by $y = \sqrt{x}$, $x \in [0,4]$, rotated about the $x$-axis.

Compute $\dfrac{d}{dx}[x^2 \sin x]$ using the product rule.

Which is the correct statement of the Fundamental Theorem of Calculus (both parts)?

Diagonalize $A = \begin{pmatrix} 3 & 1 \\ 0 & 2 \end{pmatrix}$. Find $P$ and $D$.

Compute the inverse of $A = \begin{pmatrix} 4 & 1 \\ 7 & 2 \end{pmatrix}$.

Why is every real symmetric matrix orthogonally diagonalizable? Cite the relevant theorem.

In $A = U\Sigma V^T$ (SVD), what do $U$ and $V^T$ represent geometrically?

Given the system $\begin{pmatrix}1&2\\3&4\\5&6\end{pmatrix}x = \begin{pmatrix}1\\0\\-1\end{pmatrix}$, determine if it has a solution. If so, find it.

For a matrix $A$ of size $m \times n$, what is the dimension of the null space of $A$?

Apply the 30° rotation matrix to the point $(1, 0)$.

Find the unit vector in the direction of $(3, 4)$.

$A$ symmetric $2\times 2$ with eigenvalues $\lambda_1, \lambda_2$. Show that $\operatorname{tr}(A^k) = \lambda_1^k + \lambda_2^k$ for all $k \geq 1$.

$X$ has two linearly dependent columns. Show via SVD that $X^T X$ is singular.

5 fair coin flips. Compute $P(X = 3)$ where $X$ = number of heads.

$X \sim N(0,1)$. Compute $P(X > 2)$.

Five points: $(1,2)$, $(2,3)$, $(3,5)$, $(4,4)$, $(5,6)$. Find $\hat\beta_0$ and $\hat\beta_1$ of the regression line $\hat y = \hat\beta_0 + \hat\beta_1 x$.

A/B test: conversion A = 10%, B = 12%, $n = 1000$ each. Perform the bilateral $z$-test for difference of proportions at $\alpha = 0.05$.

What is the correct difference between a 95% frequentist CI and a 95% Bayesian credible interval?

Prove that $\operatorname{Var}(X + Y) = \operatorname{Var}(X) + \operatorname{Var}(Y) + 2\operatorname{Cov}(X, Y)$.

Prior $\mu \sim \mathcal{N}(0,1)$, observations $x_i \sim \mathcal{N}(\mu, 1)$ with $\bar x = 2$, $n = 4$. Compute the posterior distribution of $\mu$.

State the Central Limit Theorem and explain intuitively why it works.

Prove that $\displaystyle\int_{-\infty}^{\infty} e^{-x^2/2}\,dx = \sqrt{2\pi}$ using polar coordinates.

Why does multiple regression with collinear features produce unstable $\hat\beta$? Explain via $X^TX$.

Damped mass-spring: $m=1$, $k=4$, $c=2$. Identify the damping type and write the general solution of $\ddot x + 2\dot x + 4x = 0$.

RC circuit with $\tau = RC = 0.1$ s. How long for the voltage to drop to 5% of initial value?

Mass-spring: $m = 1$ kg, $k = 100$ N/m, driving force $F\cos(\omega t)$, no damping. For which $\omega$ does the amplitude diverge (resonance)?

Population grows at intrinsic rate 2% per year with carrying capacity $K$ and harvesting of 1000 individuals/year. Model the ODE and identify the equilibrium points.

Use Newton-Raphson to approximate $\sqrt{2}$ starting from $x_0 = 1$. Do 3 iterations.

Markowitz portfolio: 2 assets with $\sigma_1 = 0.1$, $\sigma_2 = 0.2$, $\rho = 0.3$, equal weights. Compute the portfolio volatility.

Prove $e^{i\pi} + 1 = 0$ using Taylor series for $e^z$, $\cos\theta$, $\sin\theta$.

Prove the Fundamental Theorem of Calculus (Part 2): $\int_a^b f(x)\,dx = F(b) - F(a)$ where $F' = f$ and $f$ is continuous on $[a,b]$.

Prove: $\operatorname{Var}(X+Y) = \operatorname{Var}(X) + \operatorname{Var}(Y) + 2\operatorname{Cov}(X,Y)$.

Given $x^2 + xy + y^2 = 12$, find the points of the curve where the tangent is horizontal.