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Mathematics

MATA32H3

Raymond Grinnell

Fall

Description

***** Sorry...No Solutions will be provided *****
University of Toronto at Scarborough
Department of Computer and Mathematical Sciences
FINAL EXAMINATION
MATA32 - Calculus for Management I
Examiners: R. Grinnell G. Pete Date: December 14, 2010
T. Pham B. Szegedy Duration: 3 hours
Provide the following information:
Lastname (PRINT):
Given Name(s) (PRINT):
Student Number :
Signature:
Read these instructions:
1.This examination has 14 numbered pages. It is your responsibility to ensure that at the
beginning of the exam, all of these pages are included.
2. If you need extra space for any question, use the back of a page or the blank page at the end
of the exam. Clearly indicate the location of your continuing work. You may write in pencil,
pen, or other ink.
3. You may use one standard hand-held calculator (graphing capability is permitted). All other
electronic devices, extra paper, notes, and textbooks are forbidden at your workspace.
Print letters for the Multiple Choice Questions in these boxes:
1 2 3 4 5 6 7 8 9 10 11 12
Do not write anything in the boxes below.
A 1 2 3 4 5 6 7 TOTAL
48 15 16 15 12 15 10 19 150
1 The following may be helpful:
[ n ] [ −n]
n rt (1 + r) ▯ 1 1 ▯ (1 + r)
S = P(1 + r) S = Pe S = R r A = R r
[ n+1 ] [ −n+1]
S = R (1 + r) ▯ 1 ▯ R A = R + R 1 ▯ (1 + r)
r r
∑n n(n + 1) p=q
Proﬁt = Revenue - Cost k = ▯ =
k=1 2 dp=dq
Part A: Multiple Choice Questions For each of the following, clearly print the
letter of the answer you think is most correct in the boxes on the ▯rst page. Each right
answer earns 4 points and no answer/wrong answers earn 0 points. No justiﬁcation is required.
1. If g(x) = ln(x + 2x) then the value of g (▯1) is
(a) ▯1 (b) 0 (c) 2 (d) a number not in (a) - (c) (e) undeﬁned
2. If u (t) = 6t ▯ 7 and u(4) = 35 then u(2) equals
(a) ▯2 (b) 2 (c) ▯13 (d) 13 (e) 39 (f) a number not in (a) - (e)
∫ 1 p
3. The value of (x + 1) x + 2x dx is
0
(a) p 3 (b) 3 (c) 2 3 (d) p3=2 (e) a number not in (a) - (d)
2 2
4. The equation of the tangent line to the curve y ▯y = 2ln(x) at the point (x;1) on the curve
is
(a) y = x (b) y = 3x ▯ 2 (c) y = 2x ▯ 1 (d) y = ▯x + 2
(e) an equation not in (a) - (d) (f) uncertain because we do not know the value of x
5. What is the smallest whole number of days it takes a principal to increase by 15% at 10%
APR interest compounding ﬁve times per year ? (1 year = 365 days)
(a) 511 (b) 516 (c) 517 (d) 584 (e) a number not in (a) - (d)
(f) uncertain because we do not know the value of the principal
▯
6. Let ▯ 2 (0;1) be a constant and let c = q be a cost function where q > 0 is quantity. Which
of the following statements is true?
(i) the average cost function is increasing (ii) the average cost function is decreasing
(iii) the marginal cost function is increasing (iv) the marginal cost function is decreasing
(a) (i) and (iii) (b) (i) and (iv) (c) (ii) and (iii) (d) (ii) and (iv)
(e) we cannot be sure because the exact value of ▯ is not known
3 ( ) x
e ′
7. If y = x2 then y (1) equals
(a) e=2 (b) ▯e=2 (c) 1 (d) ▯e (e) e (f) a number not in (a) - (e)
∫
8. If G(x) dx = xe −x + 5 then G(1) equals
(a) 5+(2=e) (b) 2=e (c) 0 (d) 5+(1=e)
(e) uncertain because we cannot ﬁnd the function G(x) (f) none of (a) - (e)
9. Exactly how many of the following statements are always true?
(i) If a function f is continuous at a number a then f is also diﬀerentiable at a
n xn+1
(ii) If n is a real number then the antiderivative of is + C
n + 1
(iii) A function h has a relative extremum at a number c if and only if c is a critical value of h
(iv) If F(x) and G(x) are antiderivatives of a function w(x) then F(x) = G(x)
(a) 0 (b) 1 (c) 2 (d) 3 (e) 4
4 10. A one-year computer lease consists of two parts: a one-time $599 insurance fee, plus a weekly
rental fee of $50 which is payable at the end of each of the 52 weeks in the year. The rental
fee is subject to interest of 5:2% APR compounding weekly. The insurance fee is not subject
to any interest. How much is the total lump-sum computer lease payment if it is paid at the
beginning of the year (rounded up to the nearest dollar)? Assume that ”one-year” equals 52
weeks.
(a) $3;132 (b) $3;265 (c) $3;036 (d) $3;199 (e) none of (a) - (d)
n ( )
11. The value of lim ∑ 2 ▯ k 1 is
n→∞ n n
k=1
(a) 3=2 (b) 3 (c) ▯3=2 (d) ▯2 (e) a number not in (a) - (d)
(f) impossible to ﬁnd
12. The prea of the region bounded by the x-axis, the lines x = ▯1 and y = 2, and the curve
y = x is
(a) 2=3 (b) 4 (c) 22=3 (d) 16=3 (e) 14=3
(Be sure you have printed the letters for your answers in the boxes on the ▯rst page)
5 Part B: Full-Solution Questions Write clear, full solutions in the spaces provided.
Full points will be awarded only if your solutions are correct, complete, and suﬃciently display
appropriate concepts from MATA32.
x
1. In all of this question let f(x) = e ▯ 6x + 3
(a) Find the exact values of the absolute extrema of f on [0;2]. Suﬃc

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