MATH 200 Study Guide - Final Guide: Contour Line, Level Set, Tangent Space
The University of British Columbia
Final Examination - April 26, 2005
Mathematics 200
Instructors: Jim Bryan and Joseph Lo
Closed book examination Time: 2.5 hours
Name Signature
Student Number
Special Instructions:
- Be sure that this examination has 15 pages. Write your name on top of each page.
- No calculators or notes are permitted.
- In case of an exam disruption such as a fire alarm, leave the exam papers in the room and
exit quickly and quietly to a pre-designated location.
Rules governing examinations
•Each candidate should be prepared to produce her/his
library/AMS card upon request.
•No candidate shall be permitted to enter the examination
room after the expiration of one half hour, or to leave during
the first half hour of examination.
•Candidates are not permitted to ask questions of the in-
vigilators, except in cases of supposed errors or ambiguities
in examination questions.
•CAUTION - Candidates guilty of any of the following or
similar practices shall be immediately dismissed from the
examination and shall be liable to disciplinary action.
(a) Making use of any books, papers, or memoranda, other
than those authorized by the examiners.
(b) Speaking or communicating with other candidates.
(c) Purposely exposing written papers to the view of other
candidates.
1 14
2 10
3 6
4 6
5 10
6 12
7 10
8 10
9 12
10 10
Total 100
April 2005 Math 200 Name: Page 2 out of 15
Problem 1. Consider a twice differentiable function f(x, y) illustrated by the contour map
on the follow page.
1. (3 Points.) Draw the direction of ∇fat point H on the diagram.
2. (3 Points.) Which of the 9 points in the diagram (A-I) are critical points? Classify
these points as local minima, local maxima, or saddle points.
3. (2 Points each.)State whether the following quantities at point E are positive or
negative.
(a) derivative of fin the direction u=h−2,1i
(b) dy
dx along the level curve f(x, y) = −3
(c) fy
(d) fyy
0–2
–4
–2 0
4
2
0
42 0 –2 –4
E
H
B
A
I
D
C
F
G
0.2
0.4
0.6
0.8
1
y
0.2 0.4 0.6 0.8 1
x
51
MATH 200 Full Course Notes
Verified Note
51 documents
Document Summary
Be sure that this examination has 15 pages. Write your name on top of each page. In case of an exam disruption such as a re alarm, leave the exam papers in the room and exit quickly and quietly to a pre-designated location. Consider a twice di erentiable function f (x, y) illustrated by the contour map on the follow page: (3 points. ) Draw the direction of f at point h on the diagram: (3 points. ) Classify these points as local minima, local maxima, or saddle points: (2 points each. )state whether the following quantities at point e are positive or negative. (a) derivative of f in the direction u = h 2, 1i (b) dy dx along the level curve f (x, y) = 3 (c) fy (d) fyy. Find the global maximum and global minimum of the function f (x, y) = 2x2 + 3y2 4x 5 on the disk x2 + y2 16.
