You need only hand in question 1 and one of questions 2 and 3 on this handout. (For extra credit you may hand in both of these.)

** Problem 1**

This problem deals with least squares fitting of the function

where a,b, and c are constants.

- a)
- Using Maple, derive formulas for the least squares
approximation of this function to a set of data
Recall that this involves the steps:

- Enter an expression for the total square error.
- Differentiate this with respect to a,b, and c.
- Solve the equations determining a critical point for the variables a,b, and c.

- b)
- The file
**:Math 222:lib:Ammonia Data**inside the Maple folders in the lab contains contains a table of data for Ammonia. The format is three numbers per line. The first entry is a temperature T, the second a pressure P, and the third a compressibility factor z. Use your formula from part**a)**to estimate a,b, and c from the data file.

** Problem 2**

Let .

- a)
- Using the Maple function
*mtaylor*, calculate the Taylor series of orders**k=1,2,**and**4**about . - b)
- By using Maple's
*plot3d*command for the function , estimate the maximum error in replacing the function f by each of these Taylor polynomials in the region and . - c
- ) Similarly estimate the maximum error in replacing
by in the region and
.

** Problem 3**

Let

- a)
- Using Maple (or by hand) , calculate the Taylor series of to order 2 about .
- b)
- Use
*plot3d*in Maple to estimate the maximum values of all third partial derivatives of for the region and . - c)
- Use the results of
**b)**to find a reasonable estimate of how big**L**can be so that the quadratic approximation approximates within .000001 for the region and .