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Mathematics

MATA32H3

Raymond Grinnell

Fall

Description

*** Sorry...no solutions are provided ***
University of Toronto at Scarborough
Department of Computer and Mathematical Sciences
FINAL EXAMINATION
MATA32 - Calculus for Management I
Examiners: R. Grinnell Date: December 9, 2011
E. Moore 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 booklet 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 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 either
by accident or intent.
Print letters for the Multiple Choice Questions in these boxes:
1 2 3 4 5 6 7 8 9 10
Do not write anything in the boxes below.
A 1 2 3 4 5 6 7 8 TOTAL
30 19 13 16 12 16 10 20 14 150
1 The following may be helpful:
∑ [ n ] [ −n]
NPV = ( PV )− Initial S = Pe rt S = R (1 + r) − 1 A = R 1 − (1 + r)
r r
[ ] [ ] n
(1 + r)+1 − 1 1 − (1 + r)n+1 ∑ n(n + 1)
S = R − R A = R + R k =
r r k=1 2
p=q
Proﬁt = Revenue − Cost Revenue = (Unit Price) × (Quantity) ▯ =
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 3 points and no answer/wrong answers earn 0 points. No justiﬁcation is required.
√
1. If h (x) = 3 x + 6x and h(1) = −15 then h(4) equals
(A) 48 (B) 44 (C) 36 (D) 34 (E) a number not in (A) - (D)
2. Let c be a positive constant. The area of the enclosed region that lies above the x-axis and
2
below the curve y = −x + cx is
c3 c3 5c3 2c3
(A) (B) (C) (D) (E) not any of (A) - (D)
6 5 6 3
√
3. If y = (x + 1)(x + 3)(2x + 3) then y (0) equals
√ √ √
(A) 6 3 (B) 3 6 (C) 2 6 (D) 2 (E) 3 (F) a number not in (A) - (E)
2 2x
4. If a is a negative constant and g(x) = axe then we may conclude that g has
(A) a relative maximum at x = 1=2 (C) a relative maximum at x = −1=2
(B) a relative minimum at x = 1=2 (D) a relative minimum at x = −1=2
(E) none of (A) - (D)
5. What amount invested now (rounded up to the nearest dollar) will be worth $40;000 in 5
years if interest is 3% APR compounding quarterly ?
(A) $9;147 (B) $13;519 (C) $32;782 (D) $34;448 (E) $35;759
(F) an amount not in (A) - (E)
6. The least whole number of months it can possibly take for an investment of $5;000 to increase
by $5;000 at 2% APR is
(A) 660 (B) 416 (C) 420 (D) 421 (E) a number not in (A) - (D)
(F) unclear because we do not know the frequency at which interest compounds annually.
128
7. If a(q) = 2 + 2 is the average cost per unit when q > 0 units are manufactured, then the
q
smallest manufacturing cost per unit is
(A) 8 (B) 12 (C) 16 (D) 28 (E) 32 (F) a number not in (A) - (E)
3 ∫1 [ ]
8. The value of 2 + (4x + 4)ex +2x dx is
0
3 3 3 3 3
(A) 2e (B) 2e − 2 (C) 4e (D) 4e + 4 (E) 2 + 2e
(F) a number not in (A) - (E)
2
9. If p = −5q + 540 is a demand function where q is quantity and 0 < q < 10 then we have
unit elasticity when q equals
√ √
(A) 4 3 (B) 4 (C) 4 2 (D) 6 (E) a number not in (A) - (D)
(F) no number since the elasticity of demand is never unit.
10. Exactly how many of the following statements are always true?
(i) The deﬁnite integral is a real number obtained by taking the limit of a special sum.
(ii) Diﬀerentiation and antidiﬀerentiation are exactly inverse processes.
(iii) A continuous function h has a local maximum at a number c if and only if c is a critical
value of h.
(iv) If 1 (x) and 2 (x) are antiderivatives of a continuous function f(x) then 1 (x) = 2 (x).
(A) 0 (B) 4 (C) 3 (D) 2 (E) 1
(Be sure you have printed the letters for your answers in the boxes on the ▯rst page)
4 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
appropria

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