Signature Printed Name
Math 508 Exam 2 Jerry L. Kazdan
December 8, 2006 12:00 – 1:20
Directions This exam has two parts, Part A has 3 shorter problems (8 points each, so 24 points),
Part B has 5 traditional problems (15 points each, so 75 points).
Closed book, no calculators – but you may use one 300 ×500 card with notes.
Part A: Short Problems (3 problems, 8 points each).
A–1. A continuous function f:R→Rhas the property that
Zx
0
f(t)dt = cos(x)e−x+C,
where Cis some constant. Find both f(x) and the constant C.
A–2. A function h:R→Rwith two continuous derivatives has the property that h(0) = 2,
h(1) = 0, and h(3)=1. Prove there is at least one point cin the interval 0 < x < 3 where
h00(c)>0 by finding some explicit m > 0 (such as m= 2/3) with h00(c)≥m.
A–3. Say a smooth function u(x) satisfies u00 −c(x)u= 0 for 0 ≤x≤1 (here c(x) is some given
contunuous function).
If c(x)>0 everywhere, show that there is no point where u(x) is both positive and has a
local maximum.
If we also knew that u(0) = 0 and u(1) = 0 , why can we conclude that u(x) = 0 for all
0≤x≤1?
Part B: Traditional Problems (5 problems, 16 points each)
B–1. Given that two functions f:R→Rand g:R→Rare differentiable at a point x=c, prove
that their product h(x) = f(x)g(x) is also differentiable at x=c.
B–2. Let α(t) and β(s) describe smooth curves in R3that do not intersect. Say the points
p=α(t0) and q=β(s0) minimize the distance between the curves. Show that the line from p
to qis perpendicular to both of these curves.
B–3. Compute lim
λ→∞ Z1
0
|sin(λx)|dx.
B–4. Consider the linear space Sof real sequences x= (x1, x2,...) with only a finite number of
non-zero terms. Let kxk:= maxj|xj|(you may use without proof that this is actually a norm).
Is this space complete with this norm? Justify your response.
