assignments

1. Assignment 0 due Sat Feb 8

Compile a batch of problems that show that you've been working on bringing your Calc 1 skills back up to speed. I'll suggest a few specific questions/topics in class.

2. Assignment 1 due Mon Feb 24

• Demonstrate a grasp of the basics of integration: choose one problem from each of the four main subsections and two from the Strategy for Integration one that cover a wide range of integration problems.
• Improper Integrals: Evaluate $\int_0^{\infty} x e^{-x} \ dx$ and $\int_0^3 \frac{1}{x \sqrt{x}} \ dx$ (or show that they don't exist).
• Show off: Choose 1 or 2 problems from later in the section exercises or, probably better, Problems Plus for this section to really stretch what you can do.

3. Assignment 2 due Fri Mar 6

Two proof questions:

• Show that $\sqrt 3$ is not rational. [Hint: follow the $\sqrt 2$ argument. What's the equivalent of "even" for this situation?]
• Show that $\log_2 3$ is not rational. [Hint: Assume that $\log_2 3 = p/q$ for integers $p$ and $q$ and manipulate this to get an equation relating only positive integers. Explain why this equation cannot be true.]

From Stewart:

• Find the length of the curve $(x^2 + 2)^{3/2}/3$, for $0 \leq x \leq 1$.
• (From Problems Plus): There is a line through the origin that divides the region bounded by the parabola $y = x - x^2$ and the $x$-axis into two regions with equal area. What is the slope of that line?
• Find the volume of the solid obtained by rotating the region bounded by $y=x^2$ and $y^2 = x$ about the $x$-axis.
• Choose a question from your chosen application section and answer it.

4. Quiz 1 due Fri Mar 13

First quiz.

5. Assignment 3 due Sun Apr 19

From Stewart (numbers are from 4th Ed.):

• 10.2.34. Find the area bounded by the curve $x = t - 1/t$, $y = t + 1/t$ and the line $y = 2.5$.
• 10.4.70. Show that the curves $r = a \sin \theta$ and $r = a \cos \theta$ intersect at right angles.
• 10.5.47. Find the length of the curve $r = 2^\theta$ for $0 \leq \theta \leq 2\pi$.
• G.10, G.12 Write $3 / (4-3i)$ and $i^{100}$ in the form $a + bi$.
• G.36. Evaluate $(1-i)^8$ [Hint: don't expand the parentheses!]
• G.45. Write $e^{2+i\pi}$ in the form $a + bi$.

For fun, or for letter grades:

• Choose a question from Problems Plus from the Polar/Parametric Chapter and answer it.
• Evaluate $(cos \theta + i \sin \theta)^4$ in two different ways and hence obtain two "quadruple-angle trig identities" (analagous to the double-angle ones you are familiar with).

6. Assignment 4 due Sun May 3

From Stewart (numbers are from the 4th ed.)

• From 11.7. Test the series for convergence or divergence: $\sum_{n=1}^{\infty} \sin n$ , $\sum_{n=0}^{\infty} \frac{10^n}{n!}$, $\sum_{n=0}^{\infty} \frac{(-2)^{2n}}{n^n}$, $\sum_{n=0}^{\infty} \frac{3^n n^2}{n!}$.
• 11.10.6. Find the MacLaurin series for $f(x) = \ln(1+x)$.
• 11.10.38 Evaluate $\int \frac{\sin x}{x} \ dx$ as an infinite series.
• Problems Plus 5 on the snowflake curve (I won't write it all out---let me know if you can't find it in your book).

7. Assignment 5 due Tue May 12

From Stewart (numbers are from the 4th ed.)

• 14.1.44. Sketch a countour map and graph of the function $f(x,y) = \sqrt{36 - 9x^2 - 4y^2}$
• 14.3.11+. Find the first partial derivatives of the functions $f(x,y) = y \ln x$, $f(x,y) = x^y$ and $f(x,y) = xe^{3y}$.
• 15.2.18. Calculate $\int \int xe^{xy} \ dA$ over the region $0 \leq x,y \leq 1$.
• 15.3.7. Calculate $\int \int x^3y^2 \ dA$ over the region $0 \leq x \leq 2$ and $-x \leq y \leq x$.

8. Calc Final due Tue May 12

Optional final. See your email.

9. Final Grade due Fri May 15

Nothing for you to do. This is here so that I have a place to enter your grade where you can see it (the grades link on the left).