Quiz 3
Time: 22 minutes
Friday, 22 Jan.
Name
Student number :
Note: Each wrong answer has 13 negative score and each right answer has 1 positive
score. If you are not sure about a question, leave it blank.
1. Assume the vector hm, n, 1i is parallel to both planes 2x−y = 5 and x+z = y −1.
What is m+n?
a) −1
b) 0
c) 3
d) 16
2. Assume Q is the plane passing through the point P (1, 2, 6) that is orthogonal to
the vector h4, 0, −1i. Which point lies on Q?
a) ( 21 , 0, 0)
b) (0, 2, 0)
c) (0, 0, 2)
d) (1, 1, 1)
3. Consider the surface z 3 x2 = z + 4z 2 + y 2 . The point (2, 1) lies on which level
curve of this surface?
a) z = −1
b) z = 0
c) z = 1
d) z = 2
4. Which plane is not parallel to the vector h−1, 1, −1i
a) x = z
b) x = y
x+y =1
d) 2x + y − z = 20
∂2f
(−1, 1).
∂y∂x
+1
d) 2e − 1
5. Let f (x, y) = exy y 2 + xy . Evaluate
a)
4
e
+1
b)
−4
e
−1
c)
2
e
6. How many saddle points does the function f (x, y) = x3 + y 3 + 3x2 − 3y 2 − 8 have?
a) none
b) one
c) two
d) three
7. According to market research, the demand curve for a local pizza restaurant
satisfies the following relation: if p is the price of a pizza (in dollars), and q is the
number of pizzas sold per day, then p2 + 4q 2 = 800. If you are the owner of this
restaurant, what price should you charge for each pizza in order to make your
daily revenue as high as possible?(use
Lagrange Multipliers)
√
a) 200
b) 10
c) 20 2
d) 20
8. Use the method of Lagrange multipliers to find the minimum value of f (x, y) =
(x + 1)2 + (y − 2)2 on the circle x2 + y 2 = 125.
a) 40
b) 80
c) 160
d) 180
9. Find the absolute maximum of g(x, y) = 2x2 − 4x + y 2 − 4y + 1 on the closed
triangular plate with vertices (0, 0), (2, 0) and (2, 2).
a) -1
b) 0
c) 1
d) 2
Quiz 3
Time: 22 minutes
Friday, 22 Jan.
10. Whatis the absolute maximum of f (x, y) = x2 + xy + 3x + 2y + 2 on the domain
D = (x, y) ∈ R2 : x2 ≤ y ≤ 4 ?
a) 16
b) 28
c) 36
d)40