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Solutions to Selected Practice Exercises

Solution to Exercise One Practice Problems

Exercise 1a)

ABC Children's Party Company

Maximum children attending the party Cost per Child Total Cost of Party
10 $37 $370
25 $28 $700
50 $22 $1100
100 $15 $1500

The four equations in four unknowns:

a(103)+b(102)+c(10)+d=37

a(253)+b(252)+c(25)+d=28

a(503)+b(502)+c(50)+d=22

a(1003)+b(1002)+c(100)+d=15

 

Equations in Table Form

 

This provides the flow of the solution matrix math used in a spreadsheet.
Figure 8.1 Matrix Setup for Exercise 1a

Long Description

 


Resulting Pricing Polynomial

cost=(8.52105)x3+(1.62102)x2+(1.09)x+(4.63101)

 

The matrix math and the graph of the solution illustrate the actual equation line compared to the interpolation polynomial l.
Figure 8.2 Exercise 1b

Long Description

 


Solution to Exercise Two Practice Problems

 

2a)

Newton’s Divided Difference Table is populated as follows:

Newton’s Divided Difference Table

x y b0 Linear b1(x10) Quadratic b2(x10)(x25) Cubic b3(x10)(x25)(x50)
10 37 37
37281025=0.6
25 28 28 0.6(.24)40=0.009
28222550=0.24 0.0090.00190=0.0000888
50 22 22 0.24(0.14)75=0.001
221550100=0.14
100 15 15

 

Simplifies to: 0.0000888x3+0.016548x21.0926x+46.36

2b)

2b Table

 

2b Difference Table

 

Simplifies to: -0.040x2 + 1.845x – 1.105


Solution to Chapter Three Practice Exercises

Exercise 3b)

 

The flow of the matrix multiplication and the graph compare the measured velocity to the Direct Interpolation demonstrating that direct method interpolation provides an excellent estimate within the interval of the four closest points. Note the dotted line is the cubic polynomial generated by the graphing app in the spreadsheet.
Figure 8.3 Exercise 3b

Long Description

 


Solution to Chapter Four Practice Exercise

4a)

The Setup:

Abbreviated List of weekly Dow Jones closing averages:

Weekly Closing Averages

 

 

This matrix math solution to practice problem 4a).
Figure 8.4 Matrix Solution 4a

Long Description

 

 

The graph illustrates the results of the interpolation polynomial compared to the actual Dow Jones closing averages for between January 2020 and July 2021.
Figure 8.5 Graph of Weekly DJIA

Long Description

 


Solution to Chapter Five Practice Exercises

Step One:

1a) Find the difference between each actual value and its associated value generated by the interpolative polynomial. Square the result.

1b) Find the difference between each actual value and the Mean of the actual values. Square the result.

Step Two:

2a) Sum the results from 1a

2b) Sum the results from 1b

Step Three:

Divide 2a by 2b subtracting the result from 1.

Answer:  R2=0.88270728588.3%


Solution to Chapter Six Practice Exercises

6a)

Select the Function to be approximated. Cos function centered at x=0

Derivatives of cos

f(0)=cos(0)=1

f(1)(0)=sin(0)=0

f(2)(0)=cos(0)=1

f(3)(0)=sin(0)=0

f(4)(0)=cos(0)=1

f(5)(0)=sin(0)=0

f(6)(0)=cos(0)=1

f(7)(0)=sin(0)=0

f(8)(0)=cos(0)=1

f(9)(0)=sin(0)=0

Plug derivatives into the general form of the Taylor polynomial:

 

p(0)=10!+01!(xa)+12!(xa)2+03!(xa)3+14!(xa)4+05!(xa)5+16!(xa)6+07!(xa)7+18!(xa)8+09!(xa)9

Every other term has zero in the numerator so we can drop these and condense p(0). Further since a = 0 we can simplify the binomials.

 

p(0)=10!+12!(x)2+14!(x)4+16!(x)6+18!(x)8

f(x) is generated from an app precise to 15 decimal positions. p(x) is the Taylor approximation for Cosine.

 

Points students to the table of data encouraging them to graph a solution using their favorite app.
Figure 8.6 Speech Bubble

Taylor Approximation for Cosine


Solution to Chapter Seven Practice Exercise

7a)

 

rn(x)=f(n+1)(c)(n+1)!(xa)n+1 where c is between a and x

Solving the remainder twice for 0 and π10

This will provide a range of possible values between 0 and π10

f(9)(c)=sin(c)

for c=0     r8(0)=sin(0)(9)!(00)9=0

for c=π10       r8(π10)=sin(π10)(9)!(π100)9=0.000000000025384

Drop negative as it is a matter of distance not direction.

Gives us an error possibility 0r100.000000000025384   (2.541011)

 

 

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The Art of Polynomial Interpolation Copyright © 2022 by Stuart Murphy is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License, except where otherwise noted.

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