Math 1b Assignments for Spring 2000

Notes:

All sections will be covering the same problems (namely, those listed below). Within any week, individual instructors may assign the problems for that week in a different order than the way they are listed here. So you should follow your section's instructions for which problems are due on which days.

Note also that the problems listed here may differ slightly from the initial list of problems handed out on the opening lecture. If any changes to the list of problems are made, you will be told about it in advance by your instructor, and the changes will be reflected on this page.

Click here if you are having trouble viewing the solutions posted below.



WeekProblems Solutions
Feb 7-11
  • §7.4: 2, 4, 8, 46
  • §11.3: 3, 4, 6, 13, 14, 20, 24, 26
  • §11.4: 2(b), 4 (determine convergence), 6, 10, 17, 23
Solutions
Feb 14-18
  • §11.6: 12, 14, 16, 18, 26, 30, 44, 46
  • §11.7: 5, 8, 10, 22, 28, 48
  • A: Find all x such that the following series converges:
    \begin{displaymath}\sum_{n=1}^{\infty} n\left( {x\over 5x+6}\right)^n\end{displaymath}
Solutions
Feb 21-25
  • §11.8: 5, 6, 8, 10, 14, 18, 30
  • §11.5: 1, 7, 10, 12, 16, 22, 24
  • A: If $\sum a_n x^n$ has radius of convergence 5, what is the radius of convergence of $\sum a_n x^{2n}$?
Solutions
Feb 28 - Mar 3
  • §11.10: 6, 16, 26, 33(a)(b), 34(a)(b)
  • A: Find the unique power series $f = \sum b_n x^n$ so that $f'(x) = \sin(x)/x$ if $x \ne 0$ and f(0) = 1. What is its radius of convergence?
  • §11.9: 3, 4, 6 (ignore ``calculating utility'' business in these problems)
  • B: Find the Maclaurin series for $\ln ((1+2x)(1+5x))$.
Solutions
March 6 - 10
  • §11.9: 16 (ignore ``calculating utility'' business in this problem), 18(a), 20(a), 22
  • A: Find the Maclaurin series for ${1\over \sqrt{1+x}}$.
  • B: Use the identity $\int {1\over \sqrt{1-x^2}}\, dx = \sin^{-1}x +C$ to find the first four nonzero terms in the Maclaurin series for $\sin^{-1}x$.
Solutions
March 13-17
  • §7.6: 6, 10, 12, 36
  • §7.3: 2, 8, 10, 14, 16, 20, 28, 50
  • §8.2: 2, 4, 12, 15, 20, 35(a)(b), 45, 46
 Solutions
March 20-24
  • §8.4: 4, 8, 10, 19*
  • §8.5: 2, 4, 8, 18
  • §8.6: 2, 4, 11, 12, 14, 17
  • §8.7: 4, 8, 12
*Note that problem 19 in §8.4 has been modified. The problem is now to do the following:
  • a) Graph the curve y=x2/3 between x=0 and x=2.
  • b) Find the arclength of this curve between x=0 and x=2.
 Solutions
March 27-31 SPRING BREAK! no solutions
April 3-7
  • §9.2: 2, 8, 10, 12, 28, 30, 32, 41
  • §9.5: 10, 12, 14, 16, 18, 20
Also, do the following problems not originally listed on the syllabus:
  • §9.5: 28, 31
  • A: Find the anti-derivative of the function
    1/(x2 + a)

    when
    1. a is less than 0
    2. a is greater than 0
    3. a is equal to 0

NB - Problem A appeared on this webpage in two different manifestations. Apologies for the inconsistency. The solutions to both problems are available to the right. 		Solutions Solutions to A (version 1) Solutions to A (version 2) Solutions to 28,31
April 10-14
  • §9.8: 1, 4, 6, 10, 14, 16, 18, 26, 49, 52(a), 55
  • A: Sketch the graph of $y = \cos(x^2)$ for $x \ge 0$, making sure to label when the graph crosses the x-axis. In terms of this picture, explain how the alternating series test can be used to show that the improper integral $\int_0^{\infty} \cos(x^2)\,{\rm {d}}x$ is convergent. Of course, you do not need to compute the value of this integral (the exact value is $(1/2)\sqrt{\pi/2}$, but the usual proofs of this requires methods based on complex numbers)

Solutions
April 17-21
  • §10.1: 2, 8, 10, 12, 16, 20, 23, 26, 28, 44, 45
  • Pages 9-10 in Supplement: 2, 4, 8, 9, 10, 14
  • A: Solve the differential equation ${dy\over dx} = y(y+1)$.
Solutions
April 24-28
  • Pages 17-18 in Supplement: 2, 6, 7, 10, 12, 22, 24
  • Pages 28-29 in Supplement: 1, 2, 3, 4
  • Page 37 in Supplement: 1
  • A: Do the following multi-part problem:
    1. Graph f(x) = (x+3)2(x-1) (justify!).

    2. Assume that P = P(t) is a function of time t which satisfies

    \begin{displaymath}{dP\over dt} = (P + 3)^2(P - 1).\end{displaymath}

    3. Graph the particular solutions P(t) with P(0) = -4, -2, 2, and label all equilibrium solutions, indicating for each whether or not it is stable.

    4. For which real numbers a is the solution P with initial value P(0) = a an increasing function of t?

    5. By using the identity

    \begin{displaymath}{1\over (x+3)^2(x-1)} = {1\over 16(x-1)} - {x+7\over 16(x+3)^2},\end{displaymath}

    find an explicit formula for the time t in terms of the value of P(t) (i.e., given the value of P at some time, give a formula which computes what time it is). Your formula should depend upon the initial condition value P(0), which you should assume is not equal to -3 or 1. How does this formula relate to the sketches in the third part of the problem? You may not use this formula in any other part of the problem.
Solutions
May 1-5
  • Page 44 in Supplement: 1, 2, 3, 4
  • A: Let the function y satisfy the differential equation y'''=-y. Suppose that y is represented by its MacLaurin series, with infinite radius of convergence, $y=\sum_{k=0}^\infty a_k x^k$.

    1) Given that y(0)=1, y'(0)=0, and y''(0)=0, what can you say about the values of a0, a1, and a2.

    2) Assuming the same initial conditions as in part 1, find the general form of the coefficient ak. (That is, write a formula for ak in terms of k.)

    3) Verify that the series with the coefficients you derived in part 2 has infinite radius of convergence.

    4) Check that the function

    \begin{displaymath}y={2\over 3} e^{x\over 2} \cos \left( {\sqrt{3}\over 2} x\right) + {1\over 3} e^{-x}\end{displaymath}

    is a solution to the differential equation y'''=-y with y(0)=1, y'(0)=0, and y''(0)=0.

    5) Use the uniqueness of the solution of the differntial equation y'''=-y with the given initial values in order to deduce the MacLaurin series for the function

    \begin{displaymath}f(x)={2\over 3} e^{x\over 2} \cos \left( {\sqrt{3}\over 2} x\right) + {1\over 3} e^{-x}.\end{displaymath}

    (Note that it would have been considerably more difficult to compute the entire MacLaurin series for this function directly.)

 Solutions



updated April 11, 2000
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