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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→ 1
x^2 − x x^2 − 1
or determine that this limit does not exist.
Answer
(b) Evaluate (^) xlim→∞
x^3 + 5x 2 x^3 − x^2 + 4
or determine that this limit does not exist.
Answer
(c) Find the derivative of
x 3 + ex^
Answer
(d) Find the derivative of t^3 cos t.
Answer
(e) Find the derivative of e
√x . Answer
(f) Find f ′(x), if f (x) = arctan(x^3 ). [Note: Another notation for arctan is tan−^1 .]
Answer
(g) If x^2 + xy − y^2 = 4, find dy/dx in terms of x and y.
Answer
(l) Find the absolute minimum value of f (x) = sin−^1 (x/2) on the interval [− 2 , −1]. [Note: Another notation for sin−^1 is arcsin.] Answer
(m) Newton’s Method is used to approximate a solution of the equation sin x = 1−x , starting with the initial approximation x 1 = π/2. Find x 2. Answer
(n) Given that f ′(x) = 2x − (3/x^4 ), x > 0, and f (1) = 3, find f (x).
Answer
Full-Solution Problems. In questions 2–6, justify your answers and show all your work. If a box is provided, write your final answer there. Simplification of answers is not required unless explicitly requested.
[10] 2. A freshly brewed cup of coffee initially has temperature 95◦C in a room that has fixed tem- perature 20◦C. When the coffee temperature is 70◦C, it is decreasing at a rate of 1◦C per minute. When does this occur? Assume that the temperature of the coffee in the cup satisfies Newton’s Law of Cooling. Answer
[12] 4. Let f (x) =
x^2 + 12 x − 2
(a) (1 mark) Find the domain of f (x).
(b) (4 marks) Determine intervals where f (x) is increasing or decreasing and the x- and y-coordinates of all local maxima or minima (if any).
(c) (2 marks) Determine intervals where f (x) is concave upwards or downwards, and the x-coordinates of all inflection points (if any). You may use, without verifying it, the formula f ′′(x) = 32/(x − 2)^3.
Question 4 continues on the next page...
Question 4 continued
(d) (3 marks) Find and verify the equations of any asymptotes (horizontal, vertical or slant).
(e) (4 marks) Sketch the graph of y = f (x), showing the features given in items (a) to (d) above and giving the (x, y) coordinates for all points occurring above and also all x-intercepts (if any).
[4] 6. Use the definition of the derivative to determine f ′(x), where
f (x) =
x + 5
No credit will be given for using derivative formulas.
[4] 7. The function f (x) is defined by
f (x) =
ax^2 + bx + c if x < 0 , 2 if x = 0, 2 + x^2 cos(x−^1 ) if x > 0.
Determine all values of the constants a, b, c such that f continuous at 0, or determine that no such values exist.
Be sure that this examination has 13 pages including this cover
The University of British Columbia Sessional Examinations - December 2008
Mathematics 100/ Differential Calculus with Applications to Physical Sciences and Engineering
Closed book examination Time: 2.5 hours
Surname(s): Given Name(s):
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
