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Untitled Page 26

  • Page ID
    148367
  • Homework Problems

    1. Graph the function y=ex-7x and get an approximate idea of where any of its zeroes are (i.e., for what values of x we have y(x)=0). Use Newton's method to find the zeroes to three significant figures of precision.

    2. The relationship between x and y is given by xy = sin y+x2y2.
    (a) Use Newton's method to find the nonzero solution for y when x=3. Answer: y=0.2231
    (b) Find dy/dx in terms of x and y, and evaluate the derivative at the point on the curve you found in part a. Answer: dy/dx=-0.0379
    Based on an example by Craig B. Watkins.

    3. Suppose you want to evaluate

    eq_8773967f.png

    and you've found

    eq_b2cc2a01.png

    in a table of integrals. Use a change of variable to find the answer to the original problem.

    4. Evaluate

    eq_28ea9c15.png

    5. Evaluate

    eq_f498f80f.png

    6. Evaluate

    eq_70a4ac95.png

    7. Evaluate

    eq_a0fc68a0.png

    where b is a constant.

    8. Evaluate

    eq_b07f8084.png

    9. Evaluate

    eq_61284c0d.png

    10. Use integration by parts to evaluate the following integrals.

    eq_8c62ef04.png

    eq_c65426fe.png

    eq_176910a4.png

    11. Evaluate

    eq_d22f8b60.png

    Hint: Use integration by parts more than once.

    12. Evaluate

    eq_7a4a5a60.png

    13. Evaluate

    eq_c692fa3a.png

    14. Evaluate

    eq_42ecb8ed.png

    15. Apply integration by parts twice to

    eq_86df2fe5.png

    examine what happens, and manipulate the result in order to solve the original integral. (An approach that doesn't rely on tricks is given in example 91 on p. 123.)

    16. Plan, but do not actually carry out the steps that would be required in order to generalize the result of example 70 on p. 91 in order to evaluate

    eq_0bb8e7bf.png

    where a and b are constants. Which is easier, the generalization from 2 to a, or the one from e to b? Do we need to introduce any restrictions on a or b? (solution in the pdf version of the book)

    17. The integral \int e^{-x2}dx can't be done in closed form. Knowing this, use a change of variable to write down a different integral that also can't be done in closed form.

    18. Consider the integral

    eq_c1e1cdba.png

    where p is a constant. There is an obvious substitution. If this is to result in an integral that can be evaluated in closed form by a series of integrations by parts, what are the possible values of p? Don't actually complete the integral; just determine what values of p will work. (solution in the pdf version of the book)

    19. Evaluate the hundredth derivative of the function
    (x2+1)/(x3-x) using paper and pencil. [Vladimir Arnol'd] (solution in the pdf version of the book)

    (c) 1998-2013 Benjamin Crowell, licensed under the Creative Commons Attribution-ShareAlike license. Photo credits are given at the end of the Adobe Acrobat version.