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The University of British Columbia Sessional Examinations - December 2007
Mathematics 100/ Differential Calculus with Applications to Physical Sciences and Engineering
Closed book examination Time: 2.5 hours
Last Name: First Name:
Student Number: Instructor’s Name:
Signature: Section Number:
Rules governing examinations
- Each candidate should be prepared to produce his or her library/AMS card upon request.
- Read and observe the following rules: 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 the examination. Candidates are not permitted to ask questions of the invigilators, 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. The plea of accident or forgetfulness shall not be received.
- Smoking is not permitted during examinations.
Total 100
December 2007 Mathematics 100/180 Page 2 of 12 pages
Marks
[42] 1. Short-Answer Questions. Put your answer in the box provided but show your work also. Each question is worth 3 marks, but not all questions are of equal difficulty. Full marks will be given for correct answers placed in the box, but at most 1 mark will be given for incorrect answers. Unless otherwise stated, it is not necessary to simplify your answers in this question.
(a) Evaluate lim x→ 2
x^2 + x − 6 x − 2 Answer
(b) Evaluate (^) x→−∞lim
1 − x − x^2 2 x^2 − 7 Answer
(c) Find the derivative of
x + aex^
, where a is a constant.
Answer
December 2007 Mathematics 100/180 Page 4 of 12 pages
(h) Use a linear approximation to estimate (2.001)^3. Write your answer in the form n/1000, where n is an integer. Answer
(i) Compute lim x→ 0
x + cos(2x) − ex x^2
. Hint: Use Maclaurin series (you may, however, use any method you know). Answer
(j) If f (x) = sin(x^2 ), compute f (6)(0). Hint: Use Maclaurin series. Answer
(k) Find the absolute maximum value of f (x) = x/(x^2 + 1) on the interval [0, 2]. Answer
December 2007 Mathematics 100/180 Page 5 of 12 pages
(l) The function f (x) = b/(x^2 + ax + 2) has a local maximum at x = 1, and the local maximum value f (1) equals 2. Find the values of a and b. Answer
(m) Newton’s Method is used to approximate a positive solution of the equation sin x = x^2 , starting with the initial approximation x 1 = 1. Find x 2. Remember: You do not need to simplify your answers. Answer
(n) Given that f (x) = 24x^2 + 6x + 10, f (1) = 10, and f (1) = 20, find f (x). Answer
December 2007 Mathematics 100/180 Page 7 of 12 pages
[10] 3. A baseball diamond is a square with side length 30 m, as shown in the diagram below. A batter hits the ball and runs toward 1st base with a speed of 8 m/s. When the batter is halfway to 1st base, at what rate is his distance from 2nd base changing? Remember: You do not need to simplify your answers. Answer 2nd base
1st base
. batter
30 m
December 2007 Mathematics 100/180 Page 8 of 12 pages
[16] 4. Let f (x) = x
3 − x. (a) (2 marks) Find the domain of f (x). Answer
(b) (4 marks) Determine the x-coordinates of the local maxima and minima (if any) and intervals where f (x) is increasing or decreasing.
(c) (2 marks) Determine intervals where f (x) is concave upwards or downwards, and the x-coordinates of inflection points (if any). You may use, without verifying it, the formula f (x) = (3x − 12)(3 − x)−^3 /^2 /4.
Question 4 continued on the next page...
December 2007 Mathematics 100/180 Page 10 of 12 pages
[12] 5. ABC is a right triangle, with right angle at the point B and sides AB and BC both having fixed length L. A man wants to go from point C to point A by first walking to some point D between B and C and then walking directly to the point A. He can walk with velocity v 1 from C to D and velocity v 2 from D to A. If v 1 = 2v 2 , find the distance of D from B that minimizes the time the man spends getting from C to A. Answer
B C
A
D
L
L
December 2007 Mathematics 100/180 Page 11 of 12 pages
[6] 6. Find an equation of a line that is tangent to both of the curves y = x^2 and y = x^2 − 2 x + 2 (at different points). Answer
