This test contains 19 multiple choice questions about quantitative methods. It will be administered on Friday April 25, 2008 for 60 minutes and is worth a total of 35 marks. The test covers topics such as derivatives, optimization, and linear programming.

1. University of KwaZulu-Natal
School of Mathematical Sciences
MATH134: Quantitative Methods I
Test 2
Date: Friday April 25, 2008 Max Time Allowed: 60 min Total Marks: 35
This test contains 19 multiple choice questions. For each question, indicate your
answer by marking the corresponding letter on the MCQ answer sheet provided.
The mark allocation for each question is given in square brackets [ ]. Negative
marking will not be applied.
dy
Q1. If y  x 2 − 6, then  1
dx
a. 2x − 6
b. 2x − 1
c. x
d. 2x
e. none of the above.
Q2. Find the derivative of fx  x4 . 2
3x − 1
3
a. 4x
3
4x 3 3x − 1 − 3x 4
b.
3x − 1 2
c. 4x − 32
3
3x − 1
d. x 2 − 4x 3
4x 3 3x − 1  3x 4
e.
3x − 1 2
Q3. If P  4x 2 − x 10 , then the derivative dP equals: 2
dx
9
a. 104x − x 8x − 1
2
b. 108x − 1 9
c. 104x 2 − x8x − 1 9
d. 80x 19 − 10x 9
e. none of the above.
Q4. The definition of the derivative is 1
fx − fx  Δx
a. f ′ x  lim
Δx→0 Δx
fx  Δx  fx
b. f ′ x  lim
Δx→0 Δx
′ fx  Δx − fx
c. f x  lim
Δx→0 Δx
fx  Δx − fx
d. f ′ x  lim
Δx→0 Δx

2. e. none of the above.
Δy
Q5. If fx  1 , the difference quotient, , is: 2
x Δx
a. − 1
xx  Δx
1  Δx
b. −
xx  Δx
c. 1  Δx
xx  Δx
d. 1
xx  Δx
1 − x − Δx
e.
Δxx  Δx
Q6. The demand function for a gas cookers is pQ  360 − 3Q. What is the total
revenue function? 1
a. −3
b. 360Q − 3Q 2
360 − 3Q
c.
Q
d. 360Q 2 − 3Q 3
e. 360 − 6Q
Q7. What is the marginal revenue function for the demand function in question 6 (above)
when the demand is 40 gas cookers? 2
a. 240
b. 9600
c. 6
d. 3
e. 120
Q8. Solve for x, to 2 decimal places, if 3  e x  17. 2
a. x  1. 73
b. x  2. 64
c. x  1. 15
d. x  2. 58
e. none of the above

3. Q9. Solve for x, to 2 decimal places, if x 7  −20 2
a. No real solution exists.
b. x  −1. 28  10 9
c. x  −23. 67
d. x  −1. 28
e. x  −1. 53
dy
Q10. Find if y  3e 4x  e x  e. 2
dx
a. 3e 4x  e x  1
b. 12e 4x  e x  e
c. 3 4x   1
1
x
d. 12e  e
4x x
e. 3 4x   1  1
1
x
Q11. What is the derivative of y  log e x 5 ? 2
a. y ′  5x
b. y ′  5log e x 4 e x 
4
c. y ′  5 1 x
d. y ′  5log e x 4 1 x
e. none of the above.
Q12. Solve f ′ x  0 if fx  x 3 − 27x. 2
a. x   27
b. x  0
c. x  0 and x  9
d. x  3
e. None of these are solutions.
Q13. What is the local maximum value for fx  x 3 − 27x? 2
a. 9
b. 108
c. −54
d. 18
e. None of these.
Q14. Farmer Brown wants to build a new rectangular chicken pen on his farm. He needs:
 the area of the pen to be 60 m 2
 the pen to border the farm fence so that he only has to fence 3 sides of the chicken
pen
 to use a minimum amount of fencing.
Give a formula for the amount of fencing needed. [1]
a. xy
b. x 2 y
c. x  y

4. d. 2x  y
e. 2xy
Q15. What dimensions will minimise the amount of fencing needed for Farmer Brown’s
chicken pen? [2]
a. x  6, y  10
b. x  5, y  10
c. x  30 , y  60
30
d. x  60 , y  60
60
e. x  , y  60
30
Q16. Find and simplify ∂P if P  xy log e x .
∂x
[2]
xy log e x
a. x logx 2 y
b. log e xxy log e x−1
c. y log e x logx 2 y
1
d. y x
e. xy log e x y  y log e x
Q17. The Big-M method is being used to solve a maximisation linear programming
problem. The following step has been reached:
cj −2 −4 0 0 −M −M
cB Basis x 1 x2 x3 x4 x5 x6 Solution Ratio
−2 x1 1 1/4 −1/4 0 1/4 0 3
−M x6 0 9/2 1/2 −1 −1/2 1 12
zj
cj − zj
Complete the last two rows and the “Ratio" column of the tableau. Only one of the
following statements is true. Which one is it? [2]
(a) x 2 enters the basis and x 1 leaves the basis.
(b) x 2 enters the basis and x 6 leaves the basis.
(c) x 3 enters the basis and x 1 leaves the basis.
(d) x 3 enters the basis and x 6 leaves the basis.
(e) x 5 enters the basis and x 6 leaves the basis.
Q18. Consider the following linear programming problem:

5. Maximize P  3x 1  x 2
subject to x1  x2 ≥ 3
2x 1  x 2 ≤ 4
x1, x2 ≥ 0
Which one of the following partially completed simplex tableaux is the correct initial
simplex tableau?
[2]
(a) cj 3 1
c B Basis x 1 x 2 Solution Ratio
3 x1 1 1 3
1 x2 2 1 4
zj
cj − zj
(b) cj 3 1 −M 0
cB Basis x 1 x 2 x 3 x 4 Solution Ratio
−M x3 1 1 −1 0 3
0 x4 2 1 0 1 4
zj
cj − zj
(c) cj 3 1 −M 0
cB Basis x 1 x 2 x 3 x 4 Solution Ratio
−M x3 1 1 1 0 3
0 x4 2 1 0 1 4
zj
cj − zj
(d) cj 3 1 0 −M 0
cB Basis x 1 x 2 x 3 x4 x 5 Solution Ratio
−M x4 1 1 −1 1 0 3
0 x5 2 1 0 0 1 4
zj
cj − zj
(e) None of the above

6. Q19. The simplex method is being used to solve a maximisation linear programming
problem. The following step has been reached:
cj 3 2 0 0 0
c B Basis x 1 x 2 x 3 x 4 x 5 Solution Ratio
0 x3 0 1 1 0 −1 5
0 x4 0 2 0 1 −1 21
3 x1 1 0 0 0 1 15
zj
cj − zj
Proceed with the simplex method until an optimal tableau is reached. The optimal
solution is [3]
(a) x 1  5, x 2  11, x 3  15, x 4  0, x 5  0, P  55
(b) x 1  15, x 2  0, x 3  5, x 4  11, x 5  0, P  55
(c) x 1  15, x 2  5, x 3  0, x 4  11, x 5  0, P  55
(d) x 1  11, x 2  15, x 3  0, x 4  0, x 2  5, P  55
(e) x 1  0, x 2  0, x 3  5, x 4  11, x 5  15, P  55