MATH 200 Study Guide - Final Guide: Contour Line, Iterated Integral, Density
The University of British Columbia
Final Examination - December 7th, 2007
Mathematics 200, joint final
Closed book examination Time: 3.0 hours
Name Signature
Student Number
Special Instructions:
- Be sure that this examination has 12 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.
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Total 100
December 2007 Math 200 Name: Page 2 out of 12
Problem 1. (11 points.)
Let Aand Bbe the points with coordinates (1,2,3) and (−1,5,1) respectively.
1. (3 points) Find symmetric equations for the line Lpassing through Aand B.
2. (3 points) Let Cbe the point (5,2,−1). Find the area of the triangle ABC.
3. (3 points) Find the angle between the sides AB and BC in the triangle ABC. You
may express your answer in terms of arccos.
4. (2 points) Find an equation for the plane passing through Cand perpendicular to L.
December 2007 Math 200 Name: Page 3 out of 12
Problem 2. (12 points) Consider a twice differentiable function f(x, y) illustrated by the
contour map on the follow page. (The numbers on the contour plot give the function values
along the contours.)
1. (2 Points.) Draw the direction of ∇fat point C on the diagram.
2. (2 Points.) Which of the 8 points in the diagram (A-H) are critical points? Classify
these points as local minima, local maxima, or saddle points.
3. Identify each of the following statements as true or false. 2 points will be given
each correct answer, -2 points for each incorrect answer, 0 points for no
answer.
(a) The derivative of fat the point C, in the direction u=h−2,−1iis positive.
(b) dy
dx at the point G along the level curve f(x, y) = 1 is negative.
(c) fyat the point G is positive.
(d) fyy at the point G is positive.
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MATH 200 Full Course Notes
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Document Summary
Be sure that this examination has 12 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. Find the area of the triangle abc: (3 points) find the angle between the sides ab and bc in the triangle abc. You may express your answer in terms of arccos: (2 points) find an equation for the plane passing through c and perpendicular to l. Problem 2. (12 points) consider a twice di erentiable function f (x, y) illustrated by the contour map on the follow page. (the numbers on the contour plot give the function values along the contours. : (2 points. ) Draw the direction of f at point c on the diagram: (2 points. ) Classify these points as local minima, local maxima, or saddle points: identify each of the following statements as true or false.
