Exam II, 2013 - UNB Department of Mathematics and Statistics

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.
1. Consider the inner product space consisting of the set of continuous function
on the interval [−1, 1] with the inner product given by
〈f, g〉 =
∫ 1
−1
x 2 f(x)g(x) dx
[6]
[6]
[8]
(a) Show that the polynomials 1 and x are orthogonal with respect to
this inner product.
(b) Find a quadratic (degree 2) polynomial orthogonal to the span of
{1, x}.
(c) Let f be the function defined by f(x) = x 2/3 , −1 ≤ x ≤ 1. Find the
projection of f onto the span of {1, x}.
2. Consider the sequence of functions (f n ) defined by f n (x) = nxe −nx , 0 ≤
x ≤ 1.
[6]
[7]
(a) Compute lim f n(x).
n→∞
(b) Does the sequence (f n ) converge uniformly
(c) Does the sequence converge with respect to the norm ‖f‖ = ∫ 1
0 |f(x)|dx
[7]
1

3. Let f n be the function defined by
⎧
1, −1 ≤ x ≤ 0,
⎪⎨
1 + nx, 0 < x < 1/n,
f n (x) =
3 − nx, 1/n ≤ x ≤ 4/n,
⎪⎩
−1, 4/n < x ≤ 4,
for positive integers n.
[3]
[3]
[3]
[4]
[3]
[4]
(a) What is the domain of f n
(b) For what values of x is f n continuous
(c) For what values of x is f n differentiable
(d) What are sup f n and inf f n .
(e) What are lim inf f n (x) and lim sup f n (x).
(f) Does (f n ) converge If so, to what limit and in what sense
[4]
[6]
[4]
[6]
4. (a) Provide a resonable definition of lim a n = ∞.
(b) Suppose that (x n ) is monotone increasing and divergent. Prove that
lim x n = ∞.
(c) Suppose a and b are positive real numbers and consider the sequence
generated by the iteration x n+1 = (b+1/x n ) −1 for n > 0 with x 1 = a.
i. Show that x n is decreasing.
ii. Prove that lim x n exists and find it.
5. Recall that a function f is continuous at the point a if given any real
number ɛ > 0, there is a real number δ so that the following is true:
|f(x) − f(a)| < ɛ, whenever |x − a| < δ.
[14]
[6]
(a) Using this definition, prove that the function f defined by f(x) =
2x + 1 is continuous at x = 1.
(b) Using this definition, prove that the function g defined by g(x) = 1/x,
for 0 < x < 1, is continuous at every point in the interval (0, 1).
2