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The University of British Columbia Final Examination - December 13, 2008 Mathematics 104/ All Sections
Closed book examination Time: 2.5 hours
Last Name First Signature
Student Number
MATH 104 or MATH 184 (Circle one) Section Number:
Special Instructions:
No memory aids are allowed. One Sharp EL-510R calculator, WITH COVER REMOVED, may be used. Show all your work, little or no credit will be given for a numerical answer without the correct accompanying work. If you need more space than the space provided, use the back of the previous page. Where boxes are provided for answers, put your final answers in them.
Rules governing examinations
- Each candidate must be prepared to produce, upon request, a UBCcard for identification.
- Candidates are not permitted to ask questions of the invigilators, except in cases of supposed errors or ambiguities in examination questions.
- No candidate shall be permitted to enter the examination room after the expiration of one-half hour from the scheduled starting time, or to leave during the first half hour of the examination.
- Candidates suspected of any of the following, or similar, dishon- est practices shall be immediately dismissed from the examination and shall be liable to disciplinary action. (a) Having at the place of writing any books, papers or memoranda, calculators, computers, sound or image play- ers/recorders/transmitters (including telephones), or other mem- ory aid devices, other than those authorized by the examiners. (b) Speaking or communicating with other candidates. (c) Purposely exposing written papers to the view of other can- didates or imaging devices. The plea of accident or forgetfulness shall not be received.
- Candidates must not destroy or mutilate any examination mate- rial; must hand in all examination papers; and must not take any examination material from the examination room without permis- sion of the invigilator.
- Candidates must follow any additional examination rules or di- rections communicated by the instructor or invigilator.
Total 100
Page 1 of 11 pages
[42] 1. Short Problems. Each question is worth 3 points. Put your answer in the box provided and show your work. No credit will be given for the answer without the correct accompanying work.
(a) Find the number c that makes
f (x) =
x^2 − 5 x + 6 x^2 − x − 6
if x 6 = 3 c if x = 3
continuous for every x in its domain.
Answer:
(b) Evaluate lim h→ 0
4 + h − 2 h
Answer:
(c) Evaluate lim x→∞
x^3 − 5 x + 17 7 x^3 + 5x^2 + x
Answer:
(d) Find the marginal revenue at a production level of 20 units if the revenue from producing and selling q units of a product is given by R(q) = 3q − 0. 1 q^2 + ln(0. 5 q) dollars.
Answer:
(i) Find the equation of the tangent line to y = tan−^1 (x^2 ) at the point
π 4
. (Note: Another notation for tan−^1 (x) is arctan(x).)
Answer:
(j) Find the values of a and b so that
f (x) =
ax + b if x < 0 2 sin x + 3 cos x if x ≥ 0
is differentiable at x = 0.
Answer:
(k) The value of a house grows at a continuous rate of r% per year. Find the growth rate such that the value of the house doubles in 3 years.
Answer:
(l) Determine the second degree Taylor polynomial of f (x) =
x at x = 9.
Answer:
(m) Find the equation of the tangent line to x^2 y^4 = 1 at the point (4, 12 ).
Answer:
(n) The price p (in dollars) and the demand q (in thousands of units) of a commodity satisfy the demand equation 6p + q + qp = 94. Find the rate at which demand is changing when p = 9, q = 4, and the price is rising at the rate of $2 per week.
Answer:
(c) Find any asymptotes for y = f (x). [2pts]
(d) Sketch the graph of y = f (x). Identify on your graph any critical points, singular points, local maxima and local minima, and inflection points. Also, indicate any asymptotes that exist. [4pts]
[12] 3. The monthly advertising revenue A and the monthly circulation x of a magazine are related by the equation A = 6
x^2 − 400 , x ≥ 20 ,
where A is given in thousands of dollars and x is measured in thousands of copies sold. At what rate is the advertising revenue changing if x = 25 thousand copies, and the circulation is changing at the rate of 2 thousand copies per month?
Answer:
[12] 5. A furniture store expects to sell 640 sofas at a steady rate next year. The manager of the store plans to order these sofas from the manufacturer by placing several orders of the same size spaced equally throughout the year. The ordering cost for each delivery is $160, and carrying costs, based on the average number of sofas in inventory, amount to $ per year for one sofa. Determine how many sofas the manager should request each time she places an order to minimize the inventory cost (which is the sum of the ordering costs and the carrying costs).
Answer:
[10] 6. A $1000 television is purchased with a loan to be repaid in 5 monthly installments of $220. Use two iterations of the Newton-Raphson method, starting with an initial guess of i 0 = 0.03, to approximate the monthly rate of interest on the loan. You may find it useful to know the formula P i + R[(1 + i)−N^ − 1] = 0.
Answer:
The End
