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Unformatted text preview: Faculty of Science FINAL EXAMINATION MATH 122 Calculus for Management Examiner: Dr. D. Serbin Date: December 18, 2006
Associate Examiner : Dr. L. Laayouni Time: 9 AM.  12 Noon INSTRUCTIONS There are 9 problems, altogether worth 110 marks.
Answer all questions in examination booklets.
Show all necessary steps and details in your work.
Notes and textbooks are not permitted.
Neither regular nor translation dictionaries are allowed.
You can use only non—graphing, nonprogrammable calculators.
You may keep the exam paper when finished. This exam comprises the cover and 2 pages of questions. December 18, 2006 Final Examination MATH 122 Problem 1.
(a) [5 MARKS] Find the function f which satisﬁes ow):aw”—1, fm)=1 and f%®:=_1 (b) [5 MARKS] Find the linear approximation of f (m) = ln(x) at av = 1 and use it to
approximate ln(1.25). Problem 2. The cost C(93) and revenue R(x) from the production and sale of 2: units are
given by
R(:c) = 6055 — 0.25122 and C(36) = 3x + 20. If the level of daily production is 30 units and the rate of change of production is 10 units
per day, ﬁnd (a) [3 MARKS] the rate of change of revenue with respect to time,
(b) [3 MARKS] the rate of change of cost with respect to time, (c) [4 MARKS] the rate of change of proﬁt with respect to time. Problem 3. Evaluate the following integrals (a) [4 MARKS]
/x\/:r2 + 2 dzr, (b) [5 MARKS] / ln(m) dx, :r (c) [6 MARKS] / x2 lnac dx. 1 Problem 4. (a) [5 MARKS] Evaluate
m1Wﬂ—w+nw) 39—00 (b) [5 MARKS] Find the derivative of the function me) = x210g3(4x>. Problem 5. [10 MARKS] Find the volume of the solid formed by rotating the region
inside the ﬁrst quadrant enclosed by y = x2 and y = 33: about the maxis. Problem 6. Consider the function (a) [3 MARKS] Specify the domain of f.
(b) [5 MARKS] Determine all horizontal and vertical asymptotes of f. (c) [6 MARKS] Determine the interval(s) where f increases, the interval(s) Where f
decreases, all local maxima and minima. (d) [6 MARKS] Determine the concavity of the function f.
(e) [5 MARKS] Sketch the graph of f as accurately as possible. Problem 7. Suppose that the supply function for some commodity is
S(q) : q2 — 4g + 100
and the demand function for the commodity is
D(q) = 340 — (12.
(a) [5 MARKS] Find the producers’ surplus.
(b) [5 MARKS] Find the consumers’ surplus. Problem 8. [10 MARKS] If 2700 square centimeters of material is available to make a
box with a square base and an open top, ﬁnd the largest possible volume of the box. Problem 9. Determine Whether each improper integral converges or diverges, and ﬁnd
the value of each that converges. (a) [5 MARKS] 0° 1
d .
/0 (110+1)2 11" 00 2:
f0 x2+1 (133' (b) [5 MARKS] ...
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