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University of Toronto Scarborough

Mathematics

MATA32H3

Raymond Grinnell

Fall

Description

University of Toronto at Scarborough
Department of Computer and Mathematical Sciences
FINAL EXAMINATION
MATA32 - Calculus for Management I
Examiner: R. Grinnell Date: April 21, 2012
Time: 2:00 pm
Duration: 3 hours
Provide the following information:
Lastname (PRINT):
Given Name(s) (PRINT):
Student Number :
Signature:
Read these instructions:
1. This examination has 13 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 answer space, use the back of a page or page 13. 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 (e.g. cell phone, smart phone, i-pod), extra paper, notes, textbooks, and
backpacks are forbidden at your workspace.
4. If you have brought a cell/smart phone into the exam room, it must be turned o▯ and left at
the front.
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
42 15 16 18 15 14 16 14 150
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
▯ p/q
Proﬁt = Revenue - Cost NPV =( PV )▯Initial ▯ =
dp/dq
Part A: 12 Multiple Choice Questions For each of the following, clearly print the
letter of the answer you think is most correct in the boxes on page 1. Each right answer
earns 3.5 points and no answer/wrong answers earn 0 points. No justiﬁcation is required.
1. If y = x ln(x) then y (e) equals
2 2
(A) 3e (B) 2e + e (C) 2e + e (D) 2 + e (E) none of (A) - (D)
2
2. The slope of the curve y + y +6 ln(x)=0 at (1 ,▯1) is
(A) 2 (B) 3 (C) ▯6 (D) 6 (E) a number not in (A) - (D)
▯ 1 ▯
3. The value of (3 + 6x) 2x +2 x dx is
0
(A) 12 (B) 8 (C) 6 (D) 18 (E) a number not in (A) - (D)
2 ▯ 3 ▯
x
4. If m> 0 is a constant, then the value ofx▯▯m x ▯ mx ▯ x is
2
(A) m (B) ▯m (C) m (D) 1 ▯ m (E) nonexistent
(F) a value not in (A) - (D)
5. If y = e1/x then dy equals
dx
(A) yx ▯2 (B) ▯yx ▯2 (C) yx ▯1 (D) ▯yx ▯1 (E) none of (A) - (D)
6. If a manufacturer’s total-cost function is given by 2 q +3 ,888 where q ▯ e sti
number of units produced, then the minimum average cost per unit is
(A) 18 (B) 237.56 (C) 412 (D) 432 (E) a number not in (A) - (D)
7. The least possible whole number of months for an amount to increase by 50% at 5% APR is
(A) 110 (B) 106 (C) 98 (D) 94 (E) 93 (F) none of (A) - (E)
3 cx
8. If c is a negative constant and h(x)= xe then h has
(A) a relative maximum at x =1 /c (B) a relative maximum at x = ▯1/c
(C) a relative minimum at x =1 /c (D) a relative minimum at x = ▯1/c
(E) none of properties (A) - (D)
9. If p = ▯4q + 152 is a demand function where p is the unit price and q ▯ (0,35) is the
quantity, then we have unit elasticity at
(A) q = 15 (B) q =17 .5 (C) q =19 (D) q =21 .5
(E) a value of q not in (A) - (D) (F) no value of q
▯
10. Let ▯ ▯ (0,1) be a constant and let c = q be a cost function where q> 0s iuni.
Consider the following four statements that refer to c:
(i) the marginal cost function is increasing on (0,▯)
(ii) the average cost function is increasing on (0,▯)
(iii) the marginal cost function is decreasing on (0,▯)
(iv) the average cost function is decreasing on (0,▯)
Which of the above statements are true?
(A) (i) and (ii) (B) (i) and (iv) (C) (ii) and (iii) (D) (iii) and (iv)
4 11. Let h be a continuous function on the closed interval [0,5] such that
▯ ▯ ▯ ▯
5 5 4 4
h(x) dx =8, h(x) dx =1,and h(x) dx = 4 . The value of h(x) dx is
0 3 0 3
(A) 3 (B) 1 (C))▯ D4( ▯3 (E) not in (A) - (D)
(F) unable to be found because we do not have enough information about the integrals of h
12. Exactly how many of the following four statements are always true?
▯ ▯
• If for all real x,1F (x)andF2(x) are antiderivatives of a function f(x), then 1 (x)= F 2x).
▯
• The deﬁnite integral of a function g is a function G such that g(x) dx = G(x)+ C where
C is an arbitrary constant.

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