Math 222 Homework -- Fall 1999

These are suggested exercises. Try them ASAP then ask your TA about those you're having trouble with. All odd-numbered exercises have solutions in the back of the book, but always try to solve them on your own before looking at the solutions.

No pain, no gain...

Prerequisite: Math 221... Math 221 review problems

Chapter 7 :

• Sect. 7.1. (p. 421): 1, 3, 7, 12, 13, 14, 29, 30, 33, 34, 37-42, 52, 54, 55
• Sect. 7.2. (p. 428): 1-14, 17-23, 27, 28, 39-44, 48, 51, 53, 55, 57-62
• Sect. 7.3. (p. 434): 1, 3, 6, 13, 20, 21, 27, 35-38
• Sect. 7.4. (p. 443): 1-11 (odd only), 17, 27-30, 35, 53, 54
• Sect. 7.7. (p. 457): Use Maple to help you solve 39 to 42. If you have never used Maple, you'll need to go through the "New User's Tour" and look for Calculus and Plotting examples. Use the "leftsum", "rightsum", "trapezoid" and "middlesum" functions in the "student" package to get an approximation to the integral of sin(x)/x from 0 to Pi. Is Maple complaining? Why? How many small rectangles do you need to obtain a numerical answer that you believe to 4 decimal places? Remember "Int", "value", "evalf", "Diff",... that we saw in class.
• Sect. 7.8. (p. 465): Use Maple to help you solve 36, the pendulum problem, a basic problem in physics.
• Sect. 7.9. (p. 474): 1, 3, 5, 6, 9, 11-15, 26, 31, 42, 49-55, 57-59, 65, 66, 72, 73, 74, 75

  Warning: You will have to use Maple for the next Take-Homes. Don't wait till the last minute to become somewhat familiar with it.

Chapter 8 :

• Sect. 8.2. (p. 498): 2-4, 7, 13-16, 17, 20, 21, 23, 29-36 (in 31, replace "bomb" with "humanitarian relief" (e.g. food, medicines, ...)

Chapter 15:

• Sect 15.5. (p. 975): 1, 2, 7, 14, 21, 22, 30-32
• Sect 15.7. (p. 989): 1, 2
• App. H (p. A54): 1,3,5,7,13,19,21,23,41,43,45,48
• Sect 15.2. (p. 961): 1-9, 13, 15, 17, 19, 23, 25, 27, 28, 31, 32, 33 + solve y' + x² y = cos x with y(3)=17 (write solution in explicit integral form, all constants and limits of integration should be specified.)
• Sect 15.1. (p. 955): 1, 2, 5, 6, 8, 9, 11, 13, 25, 27, 31, 36, 37, 45, 46

Chapter 9 :

• Sect 9.1 (p. 531): 1, 4, 6, 7, 9, 11-14, 18, 21, 23, 26bcd, 27, 28, 30, 31, 32, 33, 35, 36, 39
• Sect 9.2 (p. 538): 1, 4, 7, 11, 12, 23, 24, 26, 29, 30, 32, 37, 39, 40
• Sect 9.3 (p. 543): 1, 3, 5, 6, 8, 12, 15, 16, 18. A snowboarder slides (without friction) from (0,0) to (Pi R, -2R). This takes time Pi sqrt(R/g) if the track is a cycloid (as we calculated in class). Calculate how long it takes to slide between those two points (starting from (0,0) with no velocity) if the track is (a) a straight line, (b) a sinusoid y= a sin (b x) , (c) a parabola x= c y². You may have to use Maple to evaluate some integrals. Rate the tracks from fastest to slowest. [Hint: energy is conserved so this means v²/2 + g y = Const., where g is the acceleration of gravity, v is the velocity of the snowboarder and y is pointing up, in the direction opposite to the gravitational force].
• Sect. 9.4 (p. 552): 1, 5, 7, 11, 13, 14, 21, 23, 24, 25, 27, 29, 33, 35, 36, 59-62, 77, 78, 83, 85-90.
• Sect. 9.5 (p. 558): 1, 2, 5, 7, 9, 11, 13, 14, 19, 21, 25, 33, 34, 37, 39, 41, 44, 46, 47, 51.
• Sect. 9.6 (p. 564): 17, 19, 40, 41, 42, 44-50. + 22 on page 193.

Chapter 10 :

• Sect 2.9 (p. 180): 48, 49, 55.
• Sect. 10.1 (p. 586): 2, 3, 6, 9, 10, 16, 23, 27, 31, 32, 37, 46, 49, 50, 53, 56, 59, 60. Experiment with your calculator to see whether the sequences defined by a_n+1 = (a_n+b_n)/2 and b_n+1= SQRT(a_nb_n) converge. Pick a₁=5, b₁=3. Same question for the sequence a₁ =1, a_n+1 =1/(1+a_n) .
• Sect. 10.2 (p. 596): 7, 8, 13, 14, 19-22, 31, 37, 41, 43, 51, 53, 54, 57, 58, 59, 65, 66, 70
• Sect. 10.3 (p. 603): 3, 4, 8, 13, 19, 23, 31, 32
• Sect. 10.4 (p. 608): 5, 6, 11, 13, 27, 31, 34
• Sect. 10.5 (p. 613): We are skipping the alternating series test in 10.5, but nonetheless can you solve 15 and 17 using any of your calculus tools (except the alternating series test)?
• Sect. 10.6 (p. 619): 2, 4, 8, 9, 17, 21, 25, 28, 29, 31, 33-38.
• Sect. 10.7 (p. 621): Look at all the EXAMPLES. Think about example 4 without using the alternating series test.
• Sect. 10.8 (p. 627): 1, 3, 7, 13, 25, 29, 33, 37 (posted after exam 2)
• Sect. 10.9 (p. 632): using your sharp knowledge of the GEOMETRIC series, try to solve 1, 2, 5, 21, 25. 11, 15, 30, 31 (posted after exam 2)
• Sect. 10.10 (p. 643): 1: Find the Taylor series of cos x about x=0, same question for sinx, 11, 51, 53, 54.

Chapter 11:

• Sect. 11.1 (p. 668): 5, 9, 12, 13, 15, 19, 24, 26, 31, 33, 38, 46
• Sect. 11.2 (p. 675): 1, 3, 7, 9, 17, 27, 31-34
• Sect. 11.3 (p. 680): 1, 3, 5, 7, 9, 10, 11, 15, 17, 21, 23, 25, 31, 39, 45, 47, 49, 51, 52, 53, 55, 57
• Sect. 11.4 (p. 687): 3, 5, 7, 8, 11, 21, 23, 25, 29, 31, 32, 33, 34, 39
• Sect. 11.5 (p. 696): 1, 5, 11, 15, 17, 19, 23, 27, 35, 43, 47, 51, 53, 55, 63, 65, 67, 71