Math 127C University of Nairobi Math Logic Midterm

please write the logic and answer clearly.

here are the handouts.

please finish mid and hw 3.

here are some handouts.

Math 127C, Summer Session I, 2020
Midterm
July 10, 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. Each problem is worth 10 points, for a total of 50 points.
1. Let d : R × R → R be defined to be
d(x, y) = | arctan(x) − arctan(y)|.
Show that d is a metric on R.
Math 127C, Summer Session I, 2020
Midterm
Page 2 of 5
2. Show that (R, d), where d is the metric from Problem 1, is not a complete metric space. In other
words, show that (R, d) has a Cauchy sequence that does not converge.
Cont.
Math 127C, Summer Session I, 2020
Midterm
Page 3 of 5
3. Let f : R2 → R be defined by
( 3 2
x y
if (x, y) 6= (0, 0),
4
4
f (x, y) = x +y
0
if (x, y) = (0, 0).
Determine the nonzero vectors u = (h, k) ∈ R2 for which f 0 (0; u) exists, and compute the value of
f 0 (0; u) for these vectors.
Cont.
Math 127C, Summer Session I, 2020
Midterm
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4. Let f : R2 → R2 be the function
f (x, y) = (x + 2y, (x + y)3 + x + y + 1).
Then f has an inverse g = f −1 : R2 → R2 that is everywhere differentiable. Use this information to
compute Dg(0, 1).
Cont.
Math 127C, Summer Session I, 2020
Midterm
Page 5 of 5
5. Suppose that f : R2 → R is everywhere differentiable, and let g : R3 → R be defined by
g(x, y, z) = f (3×2 + 2y − 1, y + z 3 ).
Suppose D1 f (4, 2) = 2 and D2 f (4, 2) = −6. Compute D1 g(1, 1, 1), D2 g(1, 1, 1), and D3 g(1, 1, 1).
The End.
Math 127C – Homework 3
Write complete solutions to each of the following problems. Submit your
answers in PDF format to Gradescope by Monday, July 13 at 11:59 PM.
1. Let f : R2 → R be the function defined by the formula f (x, y) = |xy|.
Show that Df (0, 0) exists but f 6∈ C 1 (U ; R) for every open set U ⊆ R2
containing (0, 0).
2. Suppose that f : A ⊆ Rm → Rn , where A is a neighborhood of a ∈ Rm .
Suppose that all directional derivatives of f exist at a, and suppose that
f 0 (a; u + v) = f 0 (a; u) + f 0 (a; v) for all u, v 6= 0 such that u + v 6= 0. Let
J denote the Jacobian of f at a, so that J is the n × m matrix whose
(i, j)-entry is Dj fi (a). Show that Ju = f 0 (a; u) for all nonzero u ∈ Rm .
3. Suppose that f : A ⊆ Rm → Rn , where A is a neighborhood of a ∈ Rm ,
and suppose that f is differentiable at a. Show that f 0 (a; u+v) = f 0 (a; u)+
f 0 (a; v) for all u, v 6= 0 such that u + v 6= 0.
4. For an n × m real matrix A, define
kAk = min{K ∈ R : kAxk ≤ Kkxk for all x ∈ Rm }.
Show that for any pair of n × m real matrices A, B, the triangle inequality
holds: kA + Bk ≤ kAk + kBk.
5. Let f : R3 → R2 be a function such that f (0, 0, 0) = (2, 4) and


1 0 2
Df (0, 0, 0) =
.
−3 1 4
Define g : R2 → R3 by the formula
g(x, y) = (x2 + y − 3, xy − 2y, 3x + 4y).
Compute D(g ◦ f )(0, 0, 0).
6. Let f : Rn → R and g : R2 → R be differentiable functions. Define
h : R2 → R by the formula
h(x, y) = f (g(x, y), . . . , g(x, y)).
1
Show that
Dj h(x, y) = Dj g(x, y)
n
X
Dk f (g(x, y), . . . , g(x, y)),
j = 1, 2
k=1
for all (x, y) ∈ R2 .
7. Let f : R2 → R2 be a differentiable function having a differentiable inverse
f −1 and such that f (0, 0) = (1, 1). Suppose D1 f1 (0, 0) = −2, D2 f1 (0, 0) =
6, D1 f2 (0, 0) = 3, and D2 f2 (0, 0) = −7. Compute Df −1 (1, 1).
2