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MATH1001 Quizzes

Quiz 9: Directional derivatives and the gradient
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)
i 4 + j 2
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
In which directions is the directional derivative of f(x,y) = xy + cos(x2) + 2y at (0,1) equal to 1 ? Exactly one option must be correct)
a)
3i + 4j and i
b)
±(i 25j)
c)
±(2i j)
d)
2i + j and i
e)
3i 4j and j

Choice (a) is correct!
The gradient vector at (x,y) is f(x,y) = (y 2xsinx2)i + (x + 2)j and so at (0,1) we have Duf(0,1) = f(0,1) û = (i + 2j) û = u1 + 2u2, where û = u1i + u2j is a unit vector. Thus Duf(0,1) = 1 if u1 = 1 2u2. Since u12 + u22 = 1, we obtain the conditions u2 = 0 or u2 = 45. The corresponding values of u1 are u1 = 1 and u1 = 35. This gives two directions, i and 3 5i + 4 5j.
Choice (b) is incorrect
Choice (c) is incorrect
Choice (d) is incorrect
Choice (e) is incorrect
Find the maximum rate of change of f(x,y) = y2x at (2,3), and the direction in which it occurs. Exactly one option must be correct)
a)
214, in the direction of (94)i 3j
b)
214, in the direction of (94)i + 3j
c)
154, in the direction of (94)i + 3j
d)
154, in the direction of (94)i + 3j
e)
94, in the direction of (94)i + 3j

Choice (a) is incorrect
Choice (b) is incorrect
Choice (c) is incorrect
Choice (d) is correct!
The gradient vector at (x,y) is f(x,y) = (y2x2)i + (2yx)j and so at (2,3) we have f(2,3) = (94)i + 3j. This is the direction of maximum rate of change of f at the point (2,3). The actual maximum rate of change is |f(2,3)| = |(94)i + 3j| = (94)2 + 32 = 154.
Choice (e) is incorrect
Find the direction in which the function g(x,y) = x4y x2y3 decreases fastest at the point (2,3). Exactly one option must be correct)
a)
12i 92j
b)
12i + 92j
c)
12i 92j
d)
12i + 92j

Choice (a) is incorrect
Choice (b) is incorrect
Choice (c) is incorrect
Choice (d) is correct!
The gradient vector at (x,y) is g(x,y) = (4x3y 2xy3)i + (x4 3x2y2)j and so at (2,3) we have g(2,3) = 12i 92j. This is the direction of fastest increase of g at the point (2,3). The direction of fastest decrease is the opposite direction, namely 12i + 92j.