menuicon

MATH1011 Quizzes

Quiz 9: Maxima and minima of functions of two variables
Question 1 Questions
Find the critical point and its nature for the function f(x,y) = x2 2x + 2y2 + 4y 2.
a)
(1,1), a maximum
 b)
(1,-1), a maximum
c)
(1,1), a minimum
 d)
(1,-1), a minimum

Choice (a) is incorrect
Choice (b) is incorrect
Choice (c) is incorrect
Choice (d) is correct!
f x = 2x 2 f x = 0 when x = 1.
f y = 4y + 4 f y = 0 when y = 1.
f = 5 at (1,-1).
The surface is concave up at all points.
The function f(x,y) = 1 + 2x + 8y x2 2y2 has one critical point. Determine its position and nature.
a)
(1,2), a maximum
 b)
(1,-2), a maximum
c)
(1,2), a minimum
 d)
(1,-2), a minimum

Choice (a) is correct!
f x = 2 2x f x = 0 when x = 1.
f y = 8 4y f y = 0 when y = 2.
f = 10 at (1,2).
The surface is concave down at all points.
Choice (b) is incorrect
Choice (c) is incorrect
Choice (d) is incorrect
How many critical points has the function f(x,y) = 3 4y2 + 1 24y3 1 32y4 x2 ?
a)
0
 b)
1
c)
2
 d)
3

Choice (a) is incorrect
Choice (b) is incorrect
Choice (c) is incorrect
Choice (d) is correct!
f x = 2x, f y = 3 2y + 1 8y2 1 8y3, = 1 8y(y + 3)(y 4). therefore the critical points are at (0,-3),(0,0) and (0,4).
Find the horizontal tangent plane to the surface z = 3x 2x2 + 1 3x3 y2 when x = 1.
a)
y = 0
 b)
z = 23 3
c)
z = 4 3
 d)
z = 18

Choice (a) is incorrect
Choice (b) is incorrect
Choice (c) is correct!
z y = 2y, z x = 3 4x + x2, = (3 x)(1 x). Therefore, the critical points are at (1,0) and (3,0). Thus the horizontal tangent plane at (1,0) is z = 3 2 + 1 3 0 = 4 3.
Choice (d) is incorrect
Which equation most closely corresponds to the sketch of the surface below ?
PIC
a)
z = 5 (x + 2)2 2(y + 3)2
 b)
z = 5 (x 2)2 2(y 3)2
c)
z = 5 + (x + 2)2 + 2(y + 3)2
 d)
z = 5 + (x 2)2 + 2(y 3)2

Choice (a) is incorrect
Choice (b) is correct!
There is a critical point at (2,3) with z = 5. The function needs to be concave down. Hence the correct answer is B.
Choice (c) is incorrect
Choice (d) is incorrect
What does k=152k + 1 mean ?
a)
2(1 + 2 + 3 + 4 + 5) + 1
 b)
3 + 11
c)
3 + 5 + 7 + 9 + 11
 d)
None of the above

Choice (a) is incorrect
Choice (b) is incorrect
Choice (c) is correct!
k=152k + 1 = (2 × 1 + 1) + (2 × 2 + 1) + (2 × 3 + 1) + (2 × 4 + 1) + (2 × 5 + 1) = 3 + 5 + 7 + 9 + 11.(1)
Choice (d) is incorrect
Which of the following represents 2 + 7 + 14 + 23 + 34 in summation notation ?
a)
k=15k2 + 1
 b)
k=15(k + 1)2 2
c)
k=15k + 1
 d)
k=15(k 2)2 + 1

Choice (a) is incorrect
Choice (b) is correct!
k=15(k + 1)2 2 = (4 2) + (9 2) + (16 2) + (25 2) + (36 2) = 2 + 7 + 14 + 23 + 34.(2)
Choice (c) is incorrect
Choice (d) is incorrect
An airline will only accept luggage whose “linear length” height + width + length is at most 200cm. Assume that you will choose a suitcase with a standard box shape. Which of the following correctly represents the problem of finding what is the maximum volume that your suitcase can be?
a)
Find h, w, d that maximise V (h,w,d) = hwd for 0 h 200cm, 0 w 200cm, 0 d 200cm.
 b)
Find the maximum possible value of V (h,w,d) = hwd for 0 h 200cm, 0 w 200cm, 0 d 200cm.
c)
Find the maximum of V (h,w) = hw(200 w h) where h,w 0.
 d)
Find the maximum of V (h,w) = 200hw where h,w 0

Choice (a) is incorrect
Choice (b) is incorrect
Choice (c) is correct!
Choice (d) is incorrect
An airline will only accept luggage whose “linear length” height + width + length is at most 200cm. Assume that you will choose a suitcase with a standard box shape. What is the maximum volume that your suitcase can be? Hint: The solution to 200x y2 2xy = 0 and 200y x2 2xy = 0, x,y 0 is x = y = 200 3
a)
(200 3 )3cm3 = 2.96 × 105cm3
 b)
200 3 cm3 = 66.667cm3 to three decimal places.
c)
2003 3 cm3 = 2.667 × 106cm3
 d)
200cm3

Choice (a) is correct!
Choice (b) is incorrect
Choice (c) is incorrect
Choice (d) is incorrect
An airline will only accept luggage whose “linear length” height + width + length is at most 200cm. Assume that these are measured at the widest part of the piece of luggage. Could you carry more using a spherical suitcase than using a box-shaped suitcase as described in questions 8 and 9?
a)
Yes
 b)
No

Choice (a) is incorrect
Choice (b) is correct!
A cube with sides of length 200 3 cm will have volume (200 3 )3cm3 = 296,000cm3 approximately. A sphere with the same diameter would have radius 200 6 cm, so its volume would be 4π 3 (200 6 )3 = 155,140cm3 approximately.