Homework Assignments for Math 132H

• Assignment 1   Due Tuesday, October 8.
  • Section 5.5 page 415:   90. Hint: Denote by I the left hand side. The suggsted substitution will result in an equation: I=[an expression involving I]. Now solve for I.
  • Section 5.5 page 415:   91.
  • Problems Plus section of Chapter 5 page 420 (page 327 on the hybrid eddition): 16. Hint: Find first one eighth of the area. You will need to find an equation for the upper boundary curve of the region. Let (x,y) be a point on this curve. Show that the distance from (x,y) to the center is equal to the distance from (x,y) to the upper edge of the square (use complete sentences and write a careful argument!!!). Express this equality as an equation in terms of x and y and solve for y as a function of x.
  • Section 6.1 page 428:   49 (include a graph and label the x and y coordinates of the leftmost and rightmost points on the loop, as well as an explicit equation of each bounding curve as a graph of some function). Hint: x(x+3)^{1/2}=(x+3)(x+3)^{1/2}-3(x+3)^{1/2}.
  • Section 6.2 page 440: 52. Support your answer with a sketch and a clear explanation!
• Assignment 2   Due Tuesday, November 5.
  • Section 7.8 page 528: 62. (This was a web assign problem).
  • Section 11.1 page 700: 47. Hint: find first the limit of ln(a_n) and then use Theorem 7. Note that n ln(...)=ln(...)/(1/n)
  • Section 11.1 page 700: 70, 79. Hint for 79: First define the sequence recursively a_{n+1}=[an expression in terms of a_n]. Next carefully prove that the sequence is convergent, by verifying the hypothesis of the monotonic sequence theorem: (i) bounded above by 2, (ii) increasing. Finally compute the limit using the limit laws.
  • Section 11.1 page 700: 92 part (b). First prove that the sequence converges as follows. Let b_n=a_{2n}, n=1,2, ... and c_n=a_{2n+1}, n=0, 1, etc. It suffices to show that both sequences {b_n} and {c_n} converge to the same limit (square root of 2), by part a. Let f(x)=(4+3x)/(3+2x).
    • Show that b_{n+1}=f(b_n) and c_{n+1}=f(c_n).
    • Show that f(x) is increasing over the positive x-axis and f(sqrt{2})=sqrt{2}.
    • Show that the sequence {c_n} is increasing and bounded above by showing that for x in the interval 0 < x < sqrt{2} the function f satisfies f(x) - x > 0 and f(x) < sqrt{2}. Explain your work with complete sentences!
    • Show that the sequence {b_n} is decreasing and bounded below by showing that if x is larger than sqrt{2}, the function f satisfies f(x) - x < 0 and f(x) > sqrt{2}.
    • Compute the limit of the sequences {b_n} and {c_n} using the limit laws.
• Assignment 3   Due Thursday, December 5.
  • Section 11.2 page 713: 87 part (a).
  • Section 11.6 page 737: 43, 44 (These problems are recommended, but are optional and will not be graded).
  • Section 11.8 page 745: 41, 42
  • Section 11.9 page 753: 41. In addition, show that the partial sum of the first 10 terms approximates pi with an error less than 0.000001 (one over a million).
  • Section 11.10 page 765: 18, 22