Math 509 Exam 1 Jerry Kazdan
March 1, 2007 12:00 — 1:20
Directions This exam has two parts, Part A has 2 short answer problems (5 points each) while
Part B has 5 traditional problems (10 points each). Closed book, no calculators – but you may use
one 300 ×500 card with notes.
Part A: Proof or Counterexample (2 problems, 5 points each)
Here let fn(x) , n= 1,2,... be a sequence of continuous functions for 0 ≤x≤2 . For a counterex-
ample, a clear sketch may be completely adequate.
A–1. If fn(x) converges to zero for every x∈[0,2], then fnconverges to zero uniformly on the
interval [0,2].
A–2. If fn(x) converges uniformly to zero for xin the interval [0,2], then Z2
0
fn(x)dx →0.
Part B: Traditional Problems (5 problems, 10 points each)
B–1. Compute Za
0
x2dx (where 0 <a<∞) directly by using Riemann sums (not as the anti-
derivative). I suggest partitioning the interval 0 ≤x≤ainto segments having equal length.
You may use without proof that 12+ 22+···+k2=1
6k(k+ 1)(2k+ 1).
B–2. Let anbe a bounded sequence of real numbers. If c > 1, show that the series
∞
X
1
an
nxconverges
uniformly for x≥c.
B–3. Let f(x)∈C([0,1]) be a continuous function with the property: Z1
0
f(x)p(x)dx = 0 for
every polynomial p(x). Show that f(x)≡0.
B–4. Let
p(x) := (x−1)(x−2)(x−3)(x−4)(x−5)(x−6) = x6−21x5+···
Clearly p(4) = 0 . Denote by p(x, t) the polynomial obtained by replacing −21x5by
−(21+t)x5, with |t|small. Let x(t) denote the perturbed value of root x= 4 , so x(0) = 4.
a) Show that x(t) is a smooth function of tfor all |t|sufficiently small.
b) Compute the sensitivity of this root as one changes t, that is, compute dx(t)/dt
t=0 .
B–5. Let f(x) and h(x, y) be continuous for xand yin the interval [0,2] . Show that if λ > 0 is
sufficiently small, the equation
u(x) = f(x) + λZ2
0
h(x, y)u(y)dy
has a unique solution (that is, a solution exists and is unique).