Department of Mathematics and Statistics
University of New Brunswick
Math 3003
Examination #2 Solutions
Winter, 2013
Instructions: Complete any 3 of the following 5 problems. Each problem is
worth 20 marks, the marks for each subproblem are shown in the l
Department of Mathematics and Statistics
University of New Brunswick
Math 3003
Examination #2
Winter, 2013
Instructions: Complete any 3 of the following 5 problems. Each problem is
worth 20 marks, the marks for each subproblem are shown in the left margin
Math 3003, Winter 2013
Topics: Fourier Series
Assignment 8
Due Wednesday, 13 March, 2013
1. Let Tn denote the set of trigonometric polynomials:
Tn =
1
, cos(x), sin(x), cos(2x), sin(2x), . . . , cos(nx), sin(nx) .
2
The trigonometric polynomials are an o
Math 3003, Winter 2013
Topics: Functions
Assignment 2
Due Wednesday, 23 January, 2013
For the problems in this assignment, use the following denition continuity of a function.
Denition 1 (Continuous function). We say that f is continuous at the point a if
Math 3003, Winter 2013
Topics: Inner products
Assignment 6
Due Wednesday, 20 February, 2013
1. Verify that the dot product in Rn satises the axioms of an inner product.
2. Let u = (4, 3, 1) and v = (1, 1, 1), viewed as vectors in R3 .
(a) Find the project
Math 3003, Winter 2013
Topics: Orthogonal projections
Assignment 7
Due Wednesday, 27 February, 2013
1. Find an orthogonal basis for the set of vectors, (x, y, z), satisfying x y + 2z = 0.
2. Let Mn denote the space of n n matrices with real entries.
n
n
a
Math 3003, Winter 2013
Topics: Sets, Sequences and Limits
Assignment 1
Due Wednesday, 16 January, 2013
Im assuming you are familiar with the material in section 11.1 of your Calculus text (either Stewart
and Thomas). You should review that section and do
Math 3003, Winter 2013
Assignment 4
Topics: Approximations, Norms, Function Spaces
Due Wednesday, 6 February, 2013
1. Find the best rst degree polynomial approximation to x on the interval [0, 4], where best
refers to the uniform norm (the best uniform ap
Math 3003, Winter 2013
Topics: Norms, Function Spaces
Assignment 3
Due Wednesday, 30 January, 2013
1. Sketch the function dened by
(1 + x),
f (x) = 1 x,
0,
1 < x < 0,
0 < x < 1,
otherwise.
For what values of x is f continuous? dierentiable? Give a brief s
Math 3003, Winter 2013
Topics: Series
1. Show that
k=1
Assignment 5
Due Wednesday, 13 February, 2013
3
1
= .
k(k + 2)
4
2. Determine whether the following series converge or diverge.
(a)
n=1
(b)
3n
.
+1
n3
n+1
n n
n=1
(c)
n=1
(d)
n=1
9n
3 + 10n
k sin2 k
1
Math 3003, Winter 2013
Topics: Fourier Series
Assignment 9
Due Wednesday, 27 March, 2013
1. Let Tn denote the set of trigonometric polynomials:
Tn =
1
, cos(x), sin(x), cos(2x), sin(2x), . . . , cos(nx), sin(nx) .
2
The trigonometric polynomials are an o
Math 3003, Winter 2013
Topics: Series
1. Show that
k=1
Assignment 5
Due Wednesday, 13 February, 2013
3
1
= .
k(k + 2)
4
2. Determine whether the following series converge or diverge.
(a)
n=1
(b)
3n
. Converges: compare with
+1
n+1
. Diverges: compare wit
Math 3003, Winter 2013
Topics: Norms, Function Spaces
Assignment 3
Due Wednesday, 30 January, 2013
1. Sketch the function dened by
(1 + x),
f (x) = 1 x,
0,
1 < x < 0,
0 < x < 1,
otherwise.
For what values of x is f continuous? dierentiable? Give a brief s
Math 3003, Winter 2013
Assignment 4
Topics: Approximations, Norms, Function Spaces
Due Wednesday, 6 February, 2013
1. Find the best rst degree polynomial approximation to x on the interval [0, 4], where best
refers to the uniform norm (the best uniform ap
Math 3003, Winter 2013
Topics: Inner products
Assignment 6
Due Wednesday, 20 February, 2013
1. Verify that the dot product in Rn satises the axioms of an inner product.
n
Comments: Aim for clarity here. Start with x y =
i=1 xi yi and clearly state which
p
Math 3003, Winter 2013
Topics: Sets, Sequences and Limits
Assignment 1
Due Wednesday, 16 January, 2013
Im assuming you are familiar with the material in section 11.1 of your Calculus text (either
Stewart or Thomas). You should review that section and do a
Math 3003, Winter 2013
Topics: Fourier Series
Assignment 8
Due Wednesday, 13 March, 2013
1. Let Tn denote the set of trigonometric polynomials:
Tn =
1
, cos(x), sin(x), cos(2x), sin(2x), . . . , cos(nx), sin(nx) .
2
The trigonometric polynomials are an o
Math 3003, Winter 2013
Topics: Functions
Assignment 2
Due Wednesday, 23 January, 2013
For the problems in this assignment, use the following denition continuity of a function.
Denition 1 (Continuous function). We say that f is continuous at the point a if
Math 3003, Winter 2013
Topics: Orthogonal projections
Assignment 7
Due Wednesday, 27 February, 2013
1. Find an orthogonal basis for the set of vectors, (x, y, z), satisfying x y + 2z = 0.
Solution: First, note that the vectors in the plane are the vectors
Math 3003, Winter 2013
Topics: Fourier Series
Assignment 9
Due Wednesday, 27 March, 2013
1. Let Tn denote the set of trigonometric polynomials:
Tn =
1
, cos(x), sin(x), cos(2x), sin(2x), . . . , cos(nx), sin(nx) .
2
The trigonometric polynomials are an o