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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 and you've found in a table of integrals. Use a change of variable to find the answer to the original problem.

4. Evaluate 5. Evaluate 6. Evaluate 7. Evaluate where b is a constant.

8. Evaluate 9. Evaluate 10. Use integration by parts to evaluate the following integrals.   11. Evaluate Hint: Use integration by parts more than once.

12. Evaluate 13. Evaluate 14. Evaluate 15. Apply integration by parts twice to 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 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 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.