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LINEAR ALGEBRA AND VECTOR ANALYSIS
MATH 22B
Unit 28: Second Hourly (Practice A)
- You only need this booklet and something to write. Please stow away any other material and electronic devices. Remember the honor code.
- Please write neatly and give details. Except for problems 28.2 and 28.3, we want to see details, even if the answer should be obvious to you.
- Try to answer the question on the same page. There is also space on the back of each page.
- If you finish a problem somewhere else, please indicate on the problem page so that we find it.
- You have 75 minutes for this hourly. Archimedes sends his good luck wishes. He unfortunately can not join us as he is “busy proving a new theorem”. He just sent us his selfie. Oh well, these celebrities!
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Linear Algebra and Vector Analysis
Problems
Problem 28A.1 (10 points):
a) (4 points) Prove that if x^3 is irrational, then x is irrational.
b) (3 points) Prove or disprove: the product of two odd integers is odd.
c) (3 points) Prove or disprove: the sum of two odd integers is odd.
Linear Algebra and Vector Analysis
Problem 28A.3 (10 points) Each question is two points:
We see the level curves of a Morse function f. The circle through ABC will sometimes serve as a constraint g(x, y) = x^2 + y^2 = 1. In all questions, we only pick points from A,B,C,D,E,F,G,H,I,J,K,L,M.
a) Which points are local minima of f under the constraint g(x, y) = 1.
b) Which points are local maxima of f under the constraint g(x, y) = 1.
c) At which points do we have fx(x, y) · fy(x, y) 6 = 0?
d) At which points are |∇f (x, y)| maximal?
e) At which points are |∇f (x, y)| minimal?
0
2 1
3
(^4 )
6
7
8
9 9 9
10
10 10
11 11
(^1312 )
14 14
(^15 )
16 16
(^17 )
18 18
19 19
20 20
A
B
C
D
E
F
G
H
I J
K
L
M
x
y
Problem 28A.4 (10 points):
a) (5 points) Find the tangent plane to the surface
f (x, y, z) = x^2 y − x^3 + y^2 + z^4 xy = − 13
at the point (2, − 1 , 1).
b) (5 points) Estimate f (2. 001 , − 0. 99 , 1 .1) by linear approximation.
Problem 28A.5 (10 points):
a) (5 points) Find the quadratic approximation Q(x, y) of
f (x, y) = 5 + x + y + x^2 + 3y^2 + sin(xy) + ex
at (x, y) = (0, 0).
b) (5 points) Estimate the value of f (0. 001 , 0 .02) using quadratic approximation.
Problem 28A.6 (10 points):
a) (8 points) Classify the critical points of the function
f (x, y) = x^2 − y^3 + 2x + 3y
using the second derivative test.
b) (2 points) Does the function f (x, y) have a global minimum or global maximum?
Problem 28A.7 (10 points):
Using the Lagrange optimization method, find the parameters (x, y) for which the area of an arch f (x, y) = 2x^2 + 4xy + 3y^2
is minimal, while the perimeter
g(x, y) = 8x + 9y = 33
is fixed.
