Signature Printed Name
Math 509 Exam 1 Jerry L. Kazdan
March 3, 2005 1:30 — 2:50
Directions This exam has two parts, Part A has 4 short answer problems (24 points) while
Part B has 6 traditional problems (72 points). Closed book, no calculators – but you may use one
3”×5” card with notes.
Part A: Proof or Counterexample (4 problems, 6 points each)
Here let fn(x) , n= 1,2, . . . be a sequence of continuous functions for 0 ≤x < ∞with fn(x) = 0
for x≥n.For a counterexample, a clear sketch may be completely adequate.
A–1. If fn(x) converges to zero for every x∈[0,1], then fnconverges to zero uniformly on the
interval [0,1].
A–2. If fn(x) converges to zero for every x∈[0,1] , then Z1
0
fn(x)dx →0.
A–3. If fn(x) converges to zero uniformly for xin the interval x∈[0,1], then Z1
0
fn(x)dx →0.
A–4. fn(x) converges to zero uniformly for xin the interval 0 ≤x < ∞, then Z∞
0
fn(x)dx →0.
Part B: Traditional Problems (6 problems, 12 points each)
B–1. Let f∈C2([0,3]) have the properties f(0) = 4, f(1) = 3 , and f(3) = 6. Show there is at
least one point z∈[0,3] where f00(z)≥const >0 and give an estimate for this constant.
B–2. Let f(x)∈C([0,2]) be a continuous function with the property: Z2
0
f(x)h(x)dx = 0 for
every function h∈C([0,2]) that is zero at the end points: h(0) = h(2) = 0 . Show that
f(x)≡0.
B–3. Let f(x)∈C([0,1]). Find lim
n→∞ nZ1
0
f(x)e−2nx dx (justify your assertions).
[continued on the next page]
