# Handout 6 - Due Friday Ocober 20

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.
Use our sample least squares worksheet as a reference here.
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.

Here z is the compressibility factor (), T is the temperature and P is the pressure. (Thus for an ideal gas, z would just be 1.) This formula is known as Berthelot's approximation to the equation of state.

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 .