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MATH1901 Quizzes

Quiz 12: Directional derivatives and the gradient vector
Question 1 Questions
Let f(x,y) = ex2 cosy. What is f(x,y)? Exactly one option must be correct)
a)
ex2 i + cosyj
b)
ex2 cosyi ex2 sinyj
c)
2xex2 cosy ex2 siny
d)
ex2 + cosy
e)
2xex2 cosyi ex2 sinyj

Choice (a) is incorrect
Choice (b) is incorrect
Choice (c) is incorrect
Choice (d) is incorrect
Choice (e) is correct!
Let f(x,y) = 1 x + y2. Find the gradient vector f(1,1) at the point (1,1). Exactly one option must be correct)
a)
1 4i 1 2j
b)
i 1 2j
c)
1 2i j
d)
1 4i + 1 2j
e)
i + 1 2j

Choice (a) is correct!
Choice (b) is incorrect
Choice (c) is incorrect
Choice (d) is incorrect
Choice (e) is incorrect
The directional derivative of f(x,y) = x2y3 + 2x4y at the point (1,-2) in the direction 3i 4j is Exactly one option must be correct)
a)
1 4i + 1 2j
b)
96i 56j
c)
-152
d)
-30.4
e)
32i + 14j

Choice (a) is incorrect
Choice (b) is incorrect
Choice (c) is incorrect
Choice (d) is correct!
Duf(1,2) = f(1,2) û. Here û = 3 5i 4 5j and f(1,2) = 32i + 14j. The directional derivative is always a scalar as it is the dot product of the two vectors.
Choice (e) is incorrect
Find the direction where the directional derivative is greatest for the function
f(x,y) = 3x2y2 x4 y4
at the point (1,2). Exactly one option must be correct)
a)
1 2(i + j)
b)
1 2(i j)
c)
1 2(i + j)
d)
1 5(2i + j)
e)
1 5(i j)

Choice (a) is incorrect
Choice (b) is correct!
The gradient vector gives the direction where the directional derivative is steepest. f(1,2) = 20i 20j, so any positive scalar multiple of this vector would provide an answer to this question. One such vector is the unit vector in this direction, 1 2(i j).
Choice (c) is incorrect
Choice (d) is incorrect
Choice (e) is incorrect
Find the maximum directional derivative of the function
f(x,y) = xlny + x2y2
at the point (-1,1). Exactly one option must be correct)
a)
2i + j
b)
1 5(2i + j)
c)
1
d)
5
e)
1 5

Choice (a) is incorrect
Choice (b) is incorrect
Choice (c) is incorrect
Choice (d) is correct!
The maximum directional derivative is equal to the magnitude of the gradient vector. Here
|f| = | 2i + j| = |5|.
Choice (e) is incorrect
Let the temperature at the point (x,y) in a flat plate be given by the function
T(x,y) = 3x2 + 2xy.
A tub of margarine is placed at (3,-6) in what direction should it be moved to cool most quickly? Exactly one option must be correct)
a)
6i + 6j
b)
i + j
c)
i j
d)
6i 12j
e)
(3,-6) is already the coolest point on the plate.

Choice (a) is incorrect
Choice (b) is incorrect
Choice (c) is correct!
T(3,6) = 6i + 6j so the direction of most rapid increase in T is i + j. For the most rapid decrease the tub of margarine must be moved in the opposite direction, i j.
Choice (d) is incorrect
Choice (e) is incorrect
Find a vector normal to the curve
x2y + lny 2x = 0
at the point (2,1). Exactly one option must be correct)
a)
2i + 5j
b)
5i 2j
c)
2 5i + j
d)
2i + j
e)
None of the above

Choice (a) is correct!
If a planar curve in the xy plane is defined implicitly by f(x,y) = c then the vector f is normal to the curve. Here f = (2xy 2)i + (x2 + 1 y)j.
Choice (b) is incorrect
Choice (c) is incorrect
Choice (d) is incorrect
Choice (e) is incorrect