Problem 1. Let . Find strictly between 0 and 1 with the property that equals the slope between the endpoints of on . For extra credit, and only if you have finished both problems, use Newton’s method to compute the first 3 significant digits of .

Since , we have that and . The slope of the line going through and is

.

The problem is therefore asking us to find a with and

.

Since , what we need to do is solve the equation . We have or

.

If we want to approximate the value of , we use that it is a solution of the equation , that can be written in the form , where . We use Newton’s method, which says that starting with a guess , we can approximate a solution to by improving the guess successively by means of the iteration given by

,

We have . Say that . Then we have:

.

.

.

.

.

.

Note that if then we have

,

so this value of is a fairly decent approximation to .

Problem 2. Find all functions with the following properties: , , .

We use that if for all , then for some constant .

If then for some constant . But then for some constant .

To find and , we use that and :

, so .

, so .

Then the only function satisfying the given conditions is .

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