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Surname, Other Names (Print clearly) Student Number ————m_.—________
Instructor Section Page 1 of 11 QUEEN’S UNIVERSITY AT KINGSTON
DEPARTMENT OF MATHEMATICS AND STATISTICS
FACULTY OF APPLIED SCIENCE
APSC 171 FINAL EXAMINATION  DECEMBER 17, 199 HAND EN answers recorded
on question paper o If the instructor is unavailable in the examination room and if doubt exists as to the
interpretation of any question, the candidate is urged to submit with the ansWer paper
a clear statement of any assumptions made. 0 Answer in the spaces provided on the question paper. If necessary, an answer may be
continued on THE BACK OF THE PREVIOUS PAGE. 0 You may use calculators with a GOLD sticker.
0 SHOW HOW YOU REACH YOUR RESULTS. Marks are not given for a correct answer alone. State or display anSWers in an appropriate way.
0 You may write in pencil, but write clearly. Do not write in red ink. 0 Except where a decimal answer is asked for, it is preferable to leave answers in the form
7:“, e2 and so on. However, do any obvious simpliﬁcation (for example 2 + é— + :13 = 2% .1: 2
or 1%, %%=(x+l) ).
0 Marks per question or part question are shown in square brackets in the right margin
(for example [4] ). The total number of marks is 100. 0 Check that your question paper has 11 pages. FOR EXAMINER’S USE ONLY Maa Aaaaa
1 0 APSC 171 FINAL EXAMINATION  DECEMBER 1999 Page 2 of 11
1. Calculate the following limits: _ x2 + 2:: + l
(3) 113591 75:: “‘1 tb) £33 G _ e=1— 1) [5] APSC 171 FINAL EXAMINATION  DECEMBER 1999 Page 3 of 11 2. Find a if
(a) y: f “I art [5]
(b) y = (1 + was“): [s] (c) myz — 32y = 6, and evaluate the derivative at (2, 3] [5] APSC 171 FINAL EXAMINATION  DECEMBER 1999 Page 4 of 11 3. Solve the following deﬁnite or indeﬁnite integrals: (a) [2th + 1 dt (b) [3:2 cos 2:: dz: APSC 171 FINAL EXAMINATION — DECEMBER 1999 Page 5 of 11 4. A ball accelerates as it rolls down an inclined plane. The velocity v (in cm/sec) of the
ball is recorded as a function of time t (in seconds) in the following table: t 0 1 2 3 4 5 6 7 8 9 10
v 1 2.9 6.6 12.1 19.4 28.5 39.4 52.1 66.6 82.9 101 (a) What feature of the table reﬂects the fact that the ball is accelerating? (b) Using ﬁve equal subintervals and the midpoint rule, estimate the distance the ball
travels during the tensecond interval. (c) Is the estimate you obtained in (b) above or below the true value? Explain. [4] [2] APSC 171 FINAL EXAMINATION  DECEMBER 1999 Page 6 of 11 5. There are four points P on the ellipse z2+2y2 = 8 such that the normal line at P passes through (0, 1). Find these four points. (The normal line is the line perpendicular to
the tangent.) [9] APSC 171 FINAL EXAMINATION — DECEMBER 1999 Page 7 of 11 6. A student, who is 5 feet tall, is walking away from a lamppost which is 15 feet tall. If
she is moving at 2 feet per second when she is 10 feet from the base of the Iamppost,
how fast is the tip of her shadow moving? [7] APSC 171 FINAL EXAMINATION _ DECEMBER 1999 Page 8 of 11 7. The height of a tree is determined by measuring the angle of elevation to the top of
the tree from a point on the ground that is 10 meters from the base of the tree. Find
the height of the tree, and use linear approximation to estimate the maximal possible error in your calculation, given that the angle measurement is 45° with a possible error
of 1°. [8] APSC 171 FINAL EXAMINATION DECEMBER 1999 Page 9 of 11 8. (a) In each case below indicate whether the statement is true or false, and include a
brief explanation or counterexample (two or three lines) to support your answer. (i) If a function is continuous, then it is differentiable. [2] (ii) If f is a positiVe increasing function then g(x) = I: f (t) dt is concave up. [2] (iii) If f (a) < k < f (b) then there is a point e between a and I) such that f (c) = k. [2] (b) Translate the following sentence into a differential equation, indicating clearly the
meaning of each of the symbols used: A rumour spreads at a rate that is proportional to the product of the number of people
who have heard it and the number who have not. [2] APSC 171 FINAL EXAMINATION  DECEMBER 1999 Page 10 of 11 9. A gas pipeline is to be constructed from a. storage tank which is right on a road, to a
house which is 200 meters downthe road and 100 meters back from the road. Pipe
laid along the road costs $400 per meter while pipe laid oﬂ the road costs $500 per
meter. What is the minimum cost for which the pipeline can be built? [10] APSC 171 FINAL EXAMINATION— DECEMBER 1999 Page 11 of 11 10. Find the volume of the solid generated by rotating about the line y = —1 (not one of
the axes!) the region bounded by the graphs of the equations 3,: = 2:2 and y = 2 — :c. [10] ...
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