(*^
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Midterm Exam
Math 515-Winter 2003
;[s]
3:0,0;13,1;33,0;34,-1;
2:2,25,18,Times,1,24,0,0,0;1,15,11,Times,1,14,0,0,0;
:[font = text; inactive; preserveAspect; center; endGroup]
You must answer each question completely in order to receive full credit.  
Your work should be on the exam paper.
:[font = section; inactive; Cclosed; preserveAspect; startGroup]
Introduction
:[font = subsection; inactive; preserveAspect; startGroup]
Vectors in Euclidean space
:[font = text; inactive; Cclosed; preserveAspect; startGroup]
1.   (6 points)  Determine the 1-norm, the 2-norm and the Infinity-norm for the vector...
;[s]
9:0,0;5,2;15,0;31,1;37,0;43,1;49,0;58,1;71,0;90,-1;
3:5,13,9,Times,0,12,0,0,0;3,13,9,Times,0,12,65535,0,0;1,13,9,Times,1,12,0,0,0;
:[font = input; wordwrap; preserveAspect; endGroup]
u={1,-2,3,-4,5,-6,7};

:[font = text; inactive; preserveAspect; endGroup]
2.   (4 points)  For which values of 0<p is the unit-ball in the plane convex?  strictly convex?

;[s]
9:0,0;5,2;15,0;39,1;40,0;71,1;77,0;80,1;95,0;98,-1;
3:5,13,9,Times,0,12,0,0,0;3,13,9,Times,0,12,65535,0,0;1,13,9,Times,1,12,0,0,0;
:[font = subsection; inactive; preserveAspect; startGroup]
Function spaces
:[font = text; inactive; preserveAspect]
3.    (4 points)  Give two examples of linear function spaces.

;[s]
5:0,0;6,2;16,0;23,1;35,0;64,-1;
3:3,13,9,Times,0,12,0,0,0;1,13,9,Times,0,12,65535,0,0;1,13,9,Times,1,12,0,0,0;
:[font = text; inactive; Cclosed; preserveAspect; startGroup]
4.  a)    (6 points)  Determine the 1-norm, the 2-norm and the Infinity-norm for the function...
;[s]
9:0,0;10,2;20,0;36,1;42,0;48,1;54,0;63,1;76,0;97,-1;
3:5,13,9,Times,0,12,0,0,0;3,13,9,Times,0,12,65535,0,0;1,13,9,Times,1,12,0,0,0;
:[font = input; wordwrap; preserveAspect]
f[x_]:= x^3(1-x^2);

:[font = text; inactive; preserveAspect; endGroup; endGroup; endGroup]
on the interval [-1,1].

b)    (3 points)  Plot the functions f[x], Abs[f[x]], and f[x]^2 on the graph.  Use "dashing" to distinguish among the figures.
;[s]
3:0,0;31,1;41,0;153,-1;
2:2,13,9,Times,0,12,0,0,0;1,13,9,Times,1,12,0,0,0;
:[font = section; inactive; Cclosed; preserveAspect; startGroup]
Chapter One
:[font = subsection; inactive; preserveAspect; startGroup]
Bernstein polynomials and operator
:[font = text; inactive; Cclosed; preserveAspect; startGroup]
1.   (6 points)  Plot the all of the Bernstein polynomials of degree n=5.
;[s]
5:0,0;5,1;15,0;17,2;21,0;74,-1;
3:3,13,9,Times,0,12,0,0,0;1,13,9,Times,1,12,0,0,0;1,13,9,Times,0,12,65535,0,0;
:[font = input; wordwrap; preserveAspect; endGroup]







:[font = text; inactive; Cclosed; preserveAspect; startGroup]
2.   (4 points)  Determine the 2nd degree "Bernstein polynomial" approximation to...
;[s]
7:0,0;5,1;15,0;31,2;41,0;65,2;78,0;85,-1;
3:4,13,9,Times,0,12,0,0,0;1,13,9,Times,1,12,0,0,0;2,13,9,Times,0,12,65535,0,0;
:[font = input; preserveAspect]
Clear[f,x];
f[x_]:= x^5
:[font = text; inactive; preserveAspect; endGroup; endGroup]
 on the interval [0,1], of course.
 
a)  B[x] =


:[font = subsection; inactive; preserveAspect; startGroup]
Best uniform approximation
:[font = text; inactive; Cclosed; preserveAspect; startGroup]
3.    (10 points)  Determine the best  2nd degree "uniform polynomial" approximation to...
;[s]
7:0,0;6,2;17,0;39,1;49,0;71,1;84,0;91,-1;
3:4,13,9,Times,0,12,0,0,0;2,13,9,Times,0,12,65535,0,0;1,13,9,Times,1,12,0,0,0;
:[font = input; preserveAspect]
Clear[f,x];
f[x_]:= x^5
:[font = text; inactive; preserveAspect; endGroup]
on the interval [-1,1].  Identify the error and the alternation set.

a)  p[x] =


b)  error = 


c)   alternation set = 


;[s]
5:0,0;38,1;43,0;51,1;67,0;124,-1;
2:3,13,9,Times,0,12,0,0,0;2,13,9,Times,0,12,65535,0,0;
:[font = text; inactive; Cclosed; preserveAspect; startGroup]
4.  The Chebyshev polynomials...
:[font = text; inactive; preserveAspect; endGroup; endGroup; endGroup]
a)    (4 points)  Display the first four Chebyshev polynomials of the first kind.

b)    (4 points)  Display the first four Chebyshev polynomials of the second kind.

c)    (2 points)  What three-term recurence relation do both sets of the Chebyshev polynomials satisfy?

d)    (2 points)  With respect to what weights and on what interval, respectively, are the two sets of Chebyshev polynomials orthogonal?


;[s]
25:0,0;6,1;16,0;36,2;40,0;70,2;80,0;89,1;99,0;119,2;123,0;153,2;164,0;173,1;183,0;190,2;219,0;278,1;288,0;311,2;318,0;330,2;339,0;397,2;407,0;411,-1;
3:13,13,9,Times,0,12,0,0,0;4,13,9,Times,1,12,0,0,0;8,13,9,Times,0,12,65535,0,0;
:[font = section; inactive; Cclosed; preserveAspect; startGroup]
Chapter Two
:[font = subsection; inactive; preserveAspect; startGroup]
Least Square approximation
:[font = text; inactive; Cclosed; preserveAspect; startGroup]
1.   (6 points)  Determine the best  2nd degree "least square polynomial" approximation to...
;[s]
7:0,0;5,1;15,0;37,2;47,0;74,2;87,0;94,-1;
3:4,13,9,Times,0,12,0,0,0;1,13,9,Times,1,12,0,0,0;2,13,9,Times,0,12,65535,0,0;
:[font = input; preserveAspect]
Clear[f,x];
f[x_]:= x^5
:[font = text; inactive; preserveAspect]
on the interval [-1,1], with weight  w[x]=1. 

a)  q[x] =


;[s]
5:0,0;7,1;15,0;29,1;35,0;60,-1;
2:3,13,9,Times,0,12,0,0,0;2,13,9,Times,0,12,65535,0,0;
:[font = text; inactive; Cclosed; preserveAspect; startGroup]
b).   (2 points)  Verify that your answer is "best" by using the characterization theorem.  Generate a random polynomial of degree at most 2, shove it in the integration querey below
to see that it is zero.
;[s]
5:0,0;6,2;16,0;65,1;89,0;207,-1;
3:3,13,9,Times,0,12,0,0,0;1,13,9,Times,0,12,65535,0,0;1,13,9,Times,1,12,0,0,0;
:[font = input; preserveAspect; endGroup; endGroup; endGroup]
Clear[q,x];
q[x_] := your approximate here
p[x_] := your random polynomial here
Integrate[(f[x]-q[x])^2 p[x],{x,-1,1}]
;[s]
5:0,0;21,1;42,0;52,1;79,0;119,-1;
2:3,7,10,Courier,1,12,0,0,0;2,7,10,Courier,1,12,65535,0,0;
:[font = subsection; inactive; preserveAspect; startGroup]
Classical orthogonal polynomials
:[font = text; inactive; Cclosed; preserveAspect; startGroup]
2.  The Legendre polynomials... 
:[font = text; inactive; preserveAspect; endGroup]
a)    (4 points)  Display the first four Legendre polynomials of the first kind.

b)    (2 points)  What three-term recurence relation do both sets of the Legendre polynomials satisfy?

c)    (2 points)  With respect to what weight and on what interval are the Legendre polynomials orthogonal?

;[s]
19:0,0;6,1;16,0;36,2;40,0;69,2;79,0;88,1;98,0;105,2;134,0;192,1;202,0;225,2;231,0;243,2;252,0;282,2;292,0;295,-1;
3:10,13,9,Times,0,12,0,0,0;3,13,9,Times,1,12,0,0,0;6,13,9,Times,0,12,65535,0,0;
:[font = text; inactive; Cclosed; preserveAspect; startGroup]
3.  The Laguerre polynomials... 
:[font = text; inactive; preserveAspect; endGroup]
a)    (4 points)  Display the first four Laguerre polynomials of the first kind.

b)    (2 points)  With respect to what weight and on what interval are the Laguerre polynomials orthogonal?

;[s]
15:0,0;6,1;16,0;36,2;40,0;69,2;79,0;88,1;98,0;121,2;127,0;139,2;149,0;178,2;188,0;191,-1;
3:8,13,9,Times,0,12,0,0,0;2,13,9,Times,1,12,0,0,0;5,13,9,Times,0,12,65535,0,0;
:[font = text; inactive; Cclosed; preserveAspect; startGroup]
4.  The Hermite polynomials... 
:[font = text; inactive; preserveAspect; endGroup; endGroup]
a)    (4 points)  Display the first four Hermite polynomials of the first kind.

b)    (2 points)  With respect to what weight and on what interval are the Hermite polynomials orthogonal?

;[s]
15:0,0;6,1;16,0;36,2;40,0;68,2;78,0;87,1;97,0;120,2;126,0;138,2;147,0;176,2;186,0;189,-1;
3:8,13,9,Times,0,12,0,0,0;2,13,9,Times,1,12,0,0,0;5,13,9,Times,0,12,65535,0,0;
:[font = subsection; inactive; preserveAspect; startGroup]
Orthogonal polynomials
:[font = text; inactive; Cclosed; preserveAspect; startGroup]
5.    (4 points)  Determine the first four orthogonal polynomials with respect to the weight ...
;[s]
5:0,0;6,2;16,0;86,1;92,0;97,-1;
3:3,13,9,Times,0,12,0,0,0;1,13,9,Times,0,12,65535,0,0;1,13,9,Times,1,12,0,0,0;
:[font = input; preserveAspect]
Clear[w,x,a,b]
w[x_] := x(1-x)

:[font = text; inactive; preserveAspect]
 and the interval [a,b]...
;[s]
3:0,0;8,1;17,0;27,-1;
2:2,13,9,Times,0,12,0,0,0;1,13,9,Times,0,12,65535,0,0;
:[font = input; preserveAspect]
a = 0;
b = 1;

:[font = text; inactive; preserveAspect; endGroup; endGroup; endGroup]
using Rivlin's construction on p. 51.

^*)