Math 445 and 447 - PRACTICE FINAL EXAM V2
Part D: Long Answer.
Math 445 students: Complete 7 problems from the following list.
Math 447 students: Complete 8 problems from the following list.
1. Define fn on (0,∞) by
fn(x) =
nx+ 1
1 + n2x2
.
Show that fn converges pointwise but not uniformly on (0,∞).
2. Define gn by
gn(x) =
nx+ n2
n+ n4x3
.
Show that
∑
gn converges uniformly on any interval of the form [a,∞) where a > 0.
3. Let X be the set of continuous functions on [0, 1]. Let d(f, g) = ‖f − g‖∞ and
d1(f, g) =
∫ 1
0
|f(x)− g(x)|dx,
both of which define metrics on X. Show that the identity function from (X, d) to (X, d1) is continuous.
4. Show that the identity function from (X, d1) to (X, d) is not continuous where X,X1, d, d1 are from
problem 3.
5. Calculate the operator norm for the matrix
A =
1 00 1
1 −1
 .
6. Calculate the operator norm for the linear transformation T : C([0, 12])→ C([0, 12]) where
Tf(x) =
∫ x
0
f(t)dt.
7. Show that the following function
f(x, y) =
{
x3y
x6+y2
: (x, y) 6= (0, 0)
0 : (x, y) = (0, 0)
has the property that all directional derivatives at (0, 0) exist, but f is not differentiable at (0, 0).
8. Suppose f : R2 → R is differentiable. For a unit vector v = (v1, v2) ∈ R2, show that
Dvf(x, y) = Dxf(x, y)v1 +Dyf(x, y)v2.
HINT: In the definition of Df , replace h with tv and let t→ 0.
9. Let fn(x) =
x2
(1+x2)n
. Evaluate the sum f(x) =
∑∞
n=0 fn(x) and show it is uniformly convergent on [a,∞)
for a > 0.
10. Suppose an is a sequence in R so that
∑ |an| <∞. Show that the series∑ an cos(nx) converges uniformly
on R.
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