# Solutions to Problem Set 20

Math 311-01/02

3-2-2018

[Chain Rule problems]

1. Suppose , , , and

Find .

By the Chain Rule,

2. Suppose

Find . (You may leave your answer in terms of x, y, z, and t, and you do not need to multiply it out.)

3. Suppose and are defined by

Compute and . (You may leave your answer in terms of x, y, z, u, and v, and you do not need to multiply it out.)

4. Suppose and are defined by

Compute and . (You may leave your answer in terms of x, y, z, s, and t, and you do not need to multiply it out.)

[Taylor series problems]

5. For a function ,

Write out the Taylor expansion of f at through terms of degree 2.

6. Construct the Taylor series through the order for at .

7. Let . Use a -order Taylor approximation to approximate .

I'll use a Taylor expansion at , since it's the closest "nice" point to .

The series is

Then

We must have the courage to be happy. - Henri Frederic Amiel

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