Download December 2009 Mathematics 100/180 Examination: Limits, Derivatives, and Applications and more Exams Calculus in PDF only on Docsity!
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 h→ 0
(3 + h)^2 − 9 h
or determine that this limit does not exist.
Answer
(b) Evaluate (^) xlim→∞(
4 x^2 + x − 2 x) or determine that this limit does not exist.
Answer
(c) Find all values of the constant c that make the function f continuous everywhere, or determine that no such value exists:
f (x) =
sin(4x) x
if x 6 = 0,
c if x = 0
Answer
(d) Find the derivative of (t^3 + 2t)et.
Answer
(e) Find the derivative of y =
sin x x^4
Answer
(f) Find f ′(x), if f (x) = ecos^ x.
Answer
(g) Find the slope of the tangent line to the curve
x + 3
y = 5 at the point (4, 1). Answer
(l) Find the absolute maximum value of f (x) = x^2 /^3 on the interval [− 1 , 2].
Answer
(m) Newton’s Method is used to approximate a solution of the equation x + ln x = 0, starting with the initial approximation x 1 = 1. Find x 2. Answer
(n) A particle is moving with velocity function v(t) = cos t − sin t and initial displacement s(0) = 0. Find the displacement at any time t. 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 bacteria culture grows with constant relative growth rate. After 2 days there are 40, bacteria and after 7 days the count is 4 billion = 4 · 109. (a) Write a differential equation satisfied by the bacteria population at any time t. Answer
(b) Find the initial population, expressed as an integer. Answer
(c) Find the population after t days. Answer
[12] 4. Let f (x) = x^5 /^3 +
x^2 /^3.
(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) (3 marks) Determine intervals where f (x) is concave upwards or downwards, and the x-coordinates of all inflection points (if any).
Question 4 continues on the next page...
Question 4 continued
(d) (2 marks) Find and verify the equations of any asymptotes (horizontal, vertical or slant), or else determine that there are no asymptotes.
(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 find f ′(x), if
f (x) =
x + 1.
You may not use derivative formulas such as the Power Rule or the Chain Rule to answer this question.
[4] 7. Determine what degree Maclaurin polynomial for ln(1−x) that should be used to approximate ln(1.1), so that the approximation is guaranteed to be accurate to within 10−^9.
Be sure that this examination has 13 pages including this cover
The University of British Columbia Sessional Examinations - December 2009
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:
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- 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
