hi, here is the exam and some handouts.

hope nice writing and clearly logic.

just ask me if anything needed.

Math 127C, Summer Session I, 2020

Final Exam

July 29, 2020

Write complete solutions to each of the following problems. If possible, write your

solutions on a copy of this template. Otherwise, please write your solutions by hand

with a separate page for each problem and with the problem statement written at the

top of each page. You may use your notes, the textbook, and all course materials

posted to Canvas, including the lectures. However, collaboration on exams is not

allowed: please do not seek outside help or ask mathematical questions about the

exam on Piazza.

1. Let (Q, d) be the metric space consisting of the set Q of rational numbers with the standard metric

d(x, y) = |x − y|. Show that the Heine-Borel theorem fails for (Q, d). In other words, show that (Q, d)

has a subset S ⊆ Q that is closed and bounded, but not compact (8 points).

Math 127C, Summer Session I, 2020

Final Exam

Page 2 of 6

2. Find a function f : R2 → R such that the partial derivative f 0 ((1, 1); u) exists for every u 6= 0, but f

is not differentiable at (1, 1). Prove that your choice of f has these properties (8 points).

Cont.

Math 127C, Summer Session I, 2020

Final Exam

Page 3 of 6

3. Let f : [0, 1] → R be uniformly continuous, so that for every > 0, there exists δ > 0 such that

|x − y| < δ =⇒ |f (x) − f (y)| < for every x, y ∈ [0, 1].
The graph of f is the set
Gf = {(x, f (x)) : x ∈ [0, 1]}.
Show that Gf has measure zero (9 points).
Cont.
Math 127C, Summer Session I, 2020
Final Exam
Page 4 of 6
4. Let f : [0, 1] × [0, 1] → R be defined by
(
1 if y = x2 ,
f (x, y) =
0 if y 6= x2 .
Show that f is integrable on [0, 1] × [0, 1]. You may take the previous problem as given (9 points).
Cont.
Math 127C, Summer Session I, 2020
Final Exam
5. Let f : R2 → R be defined by f (x, y) = |xy|e−(x
2
+y 2 )
. Evaluate
Page 5 of 6
R
R2
f , if it exists (8 points).
Cont.
Math 127C, Summer Session I, 2020
Final Exam
Page 6 of 6
6. Let S ⊆ R be the tetrahedron having vertices (0, 0, 0), (0, 1, 1), (1, 2, 3), and (−1, 0, 1). Let f : R3 → R
Rbe the function defined by f (x, y, x) = x − 2y + 3z. Using the change of variables theorem, rewrite
f as an integral over a 3-rectangle, then use Fubini’s theorem to evaluate the integral (8 points).
S
The End.