Math 508 Exam 2 Jerry L. Kazdan
December 9, 2010 9:00 – 10:20
Directions This exam has three parts, Part A asks for 3 examples (5 points each, so 15 points).
Part B has 4 shorter problems (8 points each so 32 points). Part C has 4 traditional problems (15
points each so 60 points). Total is 107 points.
Closed book, no calculators or computers– but you may use one 300 ×500 card with notes on both
sides.
Part A: Examples (3 examples, 5 points each so 15 points). Give an example having the specified
property.
1. A function f∈C1([−1,1]) but is not in C2([−1,1]) .
2. A bounded sequence akin a complete metric space Mwhere akhas no convergent subsequence.
3. A sequence of continuous functions fn(x)∈C([0,1]) that converges pointwise to zero but
R1
0fn(x)dx ≥1. [A clear sketch is adequate.]
Part B: Short Problems (4 problems, 8 points each so 32 points)
B–1. Let f(x) be a smooth function with the properties: f(−1) = 1, f(0) = 0 , and f(1) = 1.
Show that f00 (c) = 2 at some c∈(−1,1). [Suggestion: Consider g(x) := f(x)−x2.]
B–2. Let Z2x
0
f(t)dt =ecos(3x+1) +A. Find f∈C(R) and the constant A.
B–3. Let f∈C([1,3]). Compute lim
n→∞ Z3
1
f(x)e−nx dx. [Justify your assertions.]
B–4. Show that f(x) :=
∞
X
1
sin(3nx2)
n2is continuous for 0 ≤x≤π.
Part C: Traditional Problems (4 problems, 15 points each so 60 points)
C–1. Let A(t) and B(t) be n×nmatrices that are differentiable for t∈[a, b] and let t0∈(a, b).
Directly from the definition of the derivative, show that the product M(t) := A(t)B(t) is
differentiable at t=t0and obtain the usual formula for M0(t0).
C–2. Let Kbe a compact set in a complete metric space Mwith metric d(x, y) . If p∈ M is a
point not in K, let c= inf x∈Kd(p, x). Show there is a point q∈Ksuch that d(p, q ) = c.
