# Untitled Page 26

- Page ID
- 148367

## Homework Problems

**1**.
Graph the function *y*=*e*^{x}-7*x* 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*+*x*^{2}*y*^{2}.

(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^{-x^{2}}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

(*x*^{2}+1)/(*x*^{3}-*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.