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).
