141homework.html

# Math141 (J. Wu) - Homework 1

Topics for this week -

1. Parametric equations - a way to describe curves that are more interesting than graphs of functions
2. Tangents to and areas enclosed by parametric curves
3. Arc length and surface areas for both parametric and non-parametric curves

Examples -

1. Paramnetrizations of circles and ellipses: x= a+r*cos(t), y = b+r*sin(t)
2. Cycloid - x = a*(t - sin(t)), y = a*(1-cos(t))
3. Find dy/dx and d2y/dx2 for x=sec(t), y=tan(t) when t = Pi/3.
4. Find arclength of x=t3/3, y = t2/2 for 0 < t < 1.
5. Find area of surface obtained by revolving x = et cos(t), y = et sin(t) for 0 < t < Pi/2 around the x axis
6. Understand Gabriel's horn (y=1/x revolved around x axis)
7. Find length of y = x3/2 from x=0 to 4.

Maple - Learn how to draw parametric plots in Maple. Read about it on pp 110-111 of the Lab manual. Also look at the solved problems using Maple on pp 157-160 of the Lab Manual.

First Homework Assignment - due on Friday, January 22.

1. Reading: Chapter 8, sections 8.2 and 8.3 and Chapter 9, sections 9.1, 9.2 and 9.3
2. More reading: From the Lab Manual, the section on parametric plotting (pp 110-111) and solved problems on pp 157-160.
3. Chapter 8 problems: Make certain that you can do all of the Core problems for sections 8.2 and 8.3. But write up only the following to be handed in: Section 8.2, p. 519: # 8, 14, 18 (use Maple to draw the curve and evaluate the integral numerically), 34 (draw the curves using Maple but evaluate the limit just by thinking. Section 8.3, p.526: #6, 14, 18 (use Maple to draw the surface and evaluate the integral numerically), 21, 24 [we may do these last two in class]
4. Chapter 9 problems: Make sure you can do all of the Core problems for sections 9.1, 9.2 and 9.3. But write up only the following to be handed in: Section 9.1, p. 551: #4, 8, 14, 22, 26, 38 (use Maple for the last one!) Section 9.2, p. 558: #4, 8, 10, 20 (use Maple to draw the picture), 24, 30 (draw with Maple), 32 (draw with Maple again), 40 Section 9.3, p. 563: #10 (use Maple to draw), 16, 22 (draw with Maple), 26 (ditto).

BONUS PROBLEM!! The graph of x = t2, y = (t3/3) - t has a loop. First, draw a good picture of the curve that shows the loop. Then find the arclength of the loop and the area enclosed by the loop.

When you have solved the problem, submit it to me on paper. Extra credit for the first complete correct solution I receive.