Homework 12 Due Wednesday March 8

I will spend some time on Chapter 5, but skip it for the moment.

Go on to Chapter 6. This Chapter is on the definition of an integral.

It will with upper and lower approximations of integrals,

similar in spirit to Archimedes’ upper and lower approximations of the circumference of the circle.

108-109 #1,11 checkout 5,6,9

Note that “strictly increasing” is the same as “strictly monotone increasing”

112-113 #7 checkout 3, 8 (Does 8 have to be an induction problem? I am not sure.)

Postscript: Problem 7, on the approximation of the second derivative f”’

f(x_{0}
+h) – 2 f(x_{0}) + f(x_{0} – h)]/h^{2} ,

turned out harder than I thought when I assigned it.

Theorem 4.24 on p. 111 is the correct tool, but it is about a function I will call g satisfying

g(x_{0})
= 0 = g’(x_{0}) when n = 2.

Of course g”(x) = f ”(x) for all x.

To get such a function, consider g(x_{0}+h) = f(x_{0}+h)
- f(x_{0}) – f ’(x_{0})h.

Then f(x_{0}+h) - f(x_{0}) – f ’(x_{0})h
= f ”(z)h^{2} for some z in (x_{0}, x_{0} + h) and

f(x_{0}-h) - f(x_{0}) + f ’(x_{0})h
= f ”(w)h^{2} for some w in (x_{0} -h, x_{0}).

See also p. 203 for the Lagrange Remainder Thm.

**Homework 13 Due ****Friday
March 8, 2006**

**p 141 #1 checkout #2**

**In #1 use the
partition {0, 1/n, 2/n, ..., 1}**

**p149 #4b. **

There is a typo in 4b: Instead of (b - a)/2 the integral
should be equal to (b^2 - a^2) / 2.