University of Toronto at Scarborough
Department of Computer and Mathematical Sciences
***** Solutions are not provided*****
MATA33 - Calculus for Management II
Examiners: R. Grinnell Date: April 19, 2013
E. Moore Time: 9:00 am
Duration: 3 hours
Last Name (PRINT):
Given Name(s) (PRINT):
Student Number (PRINT):
Read these instructions:
1. It is your responsibility to ensure that all 13 pages of this examination are included.
2. Put your solutions and/or rough work in the answer spaces provided. If you need extra space,
use the back of a page or blank page 13. Clearly indicate your continuing work.
3. Show all work and justify your answers. Full points are awarded for solutions that are correct,
complete, and show appropriate concepts from MATA33.
4. You may use one standard calculator that does not perform any: graphing, matrix operations,
numerical/symbolic diﬀerentiation or integration. Cell/smart phones, i-Pods/Pads, other elec-
tronic transmission/receiving devices, extra paper, notes, textbooks, pencil/pen cases, food,
drink boxes/bottles with labels, and backpacks are forbidden at your workspace.
5. You may write in pencil, pen, or other ink.
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 these boxes
A 1 2 3 4 5 6 7 8 TOTAL
30 17 14 15 16 16 11 15 16 150
1 Part A: Ten Multiple Choice Questions 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.
2 2 4 @z ▯ 83
1. If z = 8y ln(y − 3x) − then ▯ equals (A) (B) 257 (C) 79
y @y (1;2) 3
(D) 80 (E) 129 (F) none of (A) - (E)
2. If the y-intercept of a certain level curve of g(x;y) = xy + 3x − y is −1, then the equation
of that level curve is
y + 1 −y − 1
(A) x = y + 3 (B) x = y + 3 (C) x + y = −1 (D) none of (A) - (C)
3. If A is a 3 × 3 matrix such that det(A) = −2 then det det(A ) (2A) ▯1 equals
(A) −4 (B) −2 (C) −1 (D) 1 (E) 2 (F) 4
2 4. Let M be an n×n matrix where n ≥ 2. Exactly how many of the following ﬁve mathematical
properties imply that M is invertible ?
(i) det(M) > 0 (ii) MX = 0 has the trivial solution (iii) M ̸= 0
(iv) MX = B has a solution for any n × 1 matrix B
(v) CM T = I for some 3 × 3 matrix C
(A) 4 (B) 3 (C) 2 (D) 1 (E) 5
5. If R is the feasible region deﬁned by the inequality 0 ≤ x ≤ y ≤ x + 1 then for what
values of the constant ▯ does the function Z = ▯x + y not have a maximum value on R ?
(A) −1 (B) (C) 0 (D) (A) and (B) (E) (B) and (C) (F) (A) and (C)
x + ey @z
6. If z = where x = rs + se r and y = 8r + r 2 then at (r;s) = (1;4) is
2 + e 8 + e 8 + e 6 + e
(A) (B) (C) (D) (E) none of (A) - (D)
9 4 9 9
7. If the equation z = 16x − 7y deﬁnes z implicitly as a function of independent variables x
and y, then the value ofxx when x = 1; y = 0 and z = −4 is
(A) −2 (B) −1 (C) 0 (D) 1 (E) a number not in (A) - (D)
3 ∫ 4∫ 1 ( √ )
8. The value of 3 x + 2y dy dx is
(A) 20 (B) (C) 16 (D) (E) 12 (F) none of (A) - (E)
9. The critical points of f(x;y) = 3x+y+6 subject to the constraint x +y = 9 must satisfy
the equation (A) x = y (B) x = 2y (C) x = −2y (D) x = 3y
(E) y = 2x (F) x = −3y (G) none of (A) - (F)
10. For what values of the constant a will the function f(x;y) = ax −y + xy − x + y have a
relative minimum at the critical point (0;1) ?
(A) a > 0 (B) a > (C) a < (D) a < 0 (E) (A) or (C)
(F) real values of a not precisely described by any of (A) - (E) (G) no values of a
BE SURE YOU HAVE PUT YOUR ANSWERS IN THE 1-ST PAGE BOXES
4 Part B: Eight Full Solution Questions Full points are awarded only if your solutions
are correct, complete, and suﬃciently display appropriate concepts from MATA33.
9 − x − y 2 1
1. In all of this question let h(x;y) = y − 4 + xy
(a) Give an accurate, labeled sketch of the domain D of h. Use light shading to clearly
indicate D. Use solid lines for portions of the boundary included in D, and dashed lines
or small circles for portions not included in D. [6 points]
(b) Find h x2;1) and state your answer as a rational number, not a decimal. [5 points]