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Quiz 9: Directional derivatives and the gradient

Question

Question 1

Let

f (x, y) = e^{x^{2}} cos y

. What is

\nabla f (x, y) ?

Not correct. Choice (a) is false.

Not correct. Choice (b) is false.

Not correct. Choice (c) is false.

Not correct. Choice (d) is false.

Your answer is correct.

Question 2

Let

f (x, y) = \frac{1}{x + y^{2}}

. Find the gradient vector

\nabla f (1, 1)

at the point (1,1).

Your answer is correct.

Not correct. Choice (b) is false.

Not correct. Choice (c) is false.

Not correct. Choice (d) is false.

Not correct. Choice (e) is false.

Question 3

The directional derivative of

f (x, y) = x^{2} y^{3} + 2 x^{4} y

at the point

(1, - 2)

in the direction

3 i - 4 j

Not correct. Choice (a) is false.

Not correct. Choice (b) is false.

Not correct. Choice (c) is false.

Your answer is correct.

D_{u} f (1, - 2) = \nabla f (1, - 2) \cdot \hat{u}

. Here

\hat{u} = \frac{3}{5} i - \frac{4}{5} j

and

\nabla f (1, - 2) = - 32 i + 14 j

. The directional derivative is always a scalar as it is the dot product of the two vectors.

Not correct. Choice (e) is false.

Question 4

Find the direction where the directional derivative is greatest for the function

f (x, y) = 3 x^{2} y^{2} - x^{4} - y^{4}

at the point (1,2).

Not correct. Choice (a) is false.

Your answer is correct.

The gradient vector gives the direction where the directional derivative is steepest.

\nabla f (1, - 2) = 20 i - 20 j

, 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,

\frac{1}{\sqrt{2}} (i - j)

Not correct. Choice (c) is false.

Not correct. Choice (d) is false.

Not correct. Choice (e) is false.

Question 5

Find the maximum directional derivative of the function

f (x, y) = x ln y + x^{2} y^{2}

at the point (-1,1).

Not correct. Choice (a) is false.

Not correct. Choice (b) is false.

Not correct. Choice (c) is false.

Your answer is correct.

The maximum directional derivative is equal to the magnitude of the gradient vector. Here

| \nabla f | = | - 2 i + j | = | \sqrt{5} | .

Not correct. Choice (e) is false.

Question 6

Let the temperature at the point

(x, y)

in a flat plate be given by the function

T (x, y) = 3 x^{2} + 2 x y .

A tub of margarine is placed at

(3, - 6)

. In what direction should it be moved to cool most quickly?

Not correct. Choice (a) is false.

Not correct. Choice (b) is false.

Your answer is correct.

\nabla T (3, - 6) = 6 i + 6 j

so the direction of most rapid increase in

T

i + j

. For the most rapid decrease the tub of margarine must be moved in the opposite direction,

- i - j

Not correct. Choice (d) is false.

Not correct. Choice (e) is false.

Question 7

Find a vector normal to the curve

x^{2} y + ln y - 2 x = 0

at the point (2,1).

Your answer is correct.

If a planar curve in the

x y

plane is defined implicitly by

f (x, y) = c

then the vector

\nabla f

is normal to the curve. Here

\nabla f = (2 x y - 2) i + (x^{2} + \frac{1}{y}) j

Not correct. Choice (b) is false.

Not correct. Choice (c) is false.

Not correct. Choice (d) is false.

Not correct. Choice (e) is false.

Question 8

In which directions is the directional derivative of

f (x, y) = x y + cos (x^{2}) + 2 y

(0, 1)

equal to

1

Your answer is correct.

The gradient vector at

(x, y)

\nabla f (x, y) = (y - 2 x sin x^{2}) i + (x + 2) j

and so at

(0, 1)

we have

D_{u} f (0, 1) = \nabla f (0, 1) \cdot \hat{u} = (i + 2 j) \cdot \hat{u} = u_{1} + 2 u_{2},

where

\hat{u} = u_{1} i + u_{2} j

is a unit vector. Thus

D_{u} f (0, 1) = 1

u_{1} = 1 - 2 u_{2}

. Since

u_{1}^{2} + u_{2}^{2} = 1

, we obtain the conditions

u_{2} = 0

u_{2} = 4 ∕ 5

. The corresponding values of

u_{1}

are

u_{1} = 1

and

u_{1} = - 3 ∕ 5

. This gives two directions,

i

and

- \frac{3}{5} i + \frac{4}{5} j .

Not correct. Choice (b) is false.

Not correct. Choice (c) is false.

Not correct. Choice (d) is false.

Not correct. Choice (e) is false.

Question 9

Find the maximum rate of change of

f (x, y) = y^{2} ∕ x

(2, 3),

and the direction in which it occurs.

Not correct. Choice (a) is false.

Not correct. Choice (b) is false.

Not correct. Choice (c) is false.

Your answer is correct.

The gradient vector at

(x, y)

\nabla f (x, y) = (- y^{2} ∕ x^{2}) i + (2 y ∕ x) j

and so at

(2, 3)

we have

\nabla f (2, 3) = (- 9 ∕ 4) i + 3 j .

This is the direction of maximum rate of change of

f

at the point

(2, 3)

. The actual maximum rate of change is

| \nabla f (2, 3) | = | (- 9 ∕ 4) i + 3 j | = \sqrt{{(- 9 ∕ 4)}^{2} + 3^{2}} = 15 ∕ 4 .

Not correct. Choice (e) is false.

Question 10

Find the direction in which the function

g (x, y) = x^{4} y - x^{2} y^{3}

decreases fastest at the point

(2, - 3)

Not correct. Choice (a) is false.

Not correct. Choice (b) is false.

Not correct. Choice (c) is false.

Your answer is correct.

The gradient vector at

(x, y)

\nabla g (x, y) = (4 x^{3} y - 2 x y^{3}) i + (x^{4} - 3 x^{2} y^{2}) j

and so at

(2, - 3)

we have

\nabla g (2, - 3) = 12 i - 92 j .

This is the direction of fastest increase of

g

at the point

(2, - 3)

. The direction of fastest decrease is the opposite direction, namely

- 12 i + 92 j .

ABN: 15 211 513 464. CRICOS number: 00026A. Phone: +61 2 9351 2222.

Authorised by: Head, School of Mathematics and Statistics.

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Questions:

right first attempt
right
wrong

a)	$e^{x^{2}} i + cos y j$	b)	$e^{x^{2}} cos y i - e^{x^{2}} sin y j$
c)	$2 x e^{x^{2}} cos y - e^{x^{2}} sin y$	d)	$e^{x^{2}} + cos y$
e)	$2 x e^{x^{2}} cos y i - e^{x^{2}} sin y j$

a)	$- \frac{1}{4} i - \frac{1}{2} j$	b)	$- i - \frac{1}{2} j$
c)	$- \frac{1}{2} i - j$	d)	$\frac{1}{4} i + \frac{1}{2} j$
e)	$i + \frac{1}{2} j$

a)	$\frac{i}{4} + \frac{j}{2}$	b)	$- 96 i - 56 j$
c)	$- 152$	d)	$- 30.4$
e)	$- 32 i + 14 j$

a)	$\frac{1}{\sqrt{2}} (- i + j)$	b)	$\frac{1}{\sqrt{2}} (i - j)$
c)	$\frac{1}{\sqrt{2}} (i + j)$	d)	$\frac{1}{\sqrt{5}} (2 i + j)$
e)	$- \frac{1}{\sqrt{5}} (i - j)$

a)	$- 2 i + j$	b)	$\frac{1}{\sqrt{5}} (- 2 i + j)$
c)	$1$	d)	$\sqrt{5}$
e)	$\frac{1}{\sqrt{5}}$

a)	$6 i + 6 j$	b)	$i + j$
c)	$- i - j$	d)	$6 i - 12 j$
e)	(3,-6) is already the coolest point on the plate.

a)	$2 i + 5 j$	b)	$5 i - 2 j$
c)	$- \frac{2}{5} i + j$	d)	$2 i + j$
e)	None of the above

a)	$- 3 i + 4 j$ and $i$	b)	$\pm (i - 2 ∕ 5 j)$
c)	$\pm (2 i - j)$	d)	$2 i + j$ and $i$
e)	$3 i - 4 j$ and $j$

a)	$21 ∕ 4$ , in the direction of $(- 9 ∕ 4) i - 3 j$	b)	$21 ∕ 4$ , in the direction of $(- 9 ∕ 4) i + 3 j$
c)	$15 ∕ 4$ , in the direction of $(9 ∕ 4) i + 3 j$	d)	$15 ∕ 4$ , in the direction of $(- 9 ∕ 4) i + 3 j$
e)	$9 ∕ 4$ , in the direction of $(9 ∕ 4) i + 3 j$

a)	$12 i - 92 j$	b)	$12 i + 92 j$
c)	$- 12 i - 92 j$	d)	$- 12 i + 92 j$

The University of Sydney - School of Mathematics and Statistics

Quiz 9: Directional derivatives and the gradient

Question 1

Question 2

Question 3

Question 4

Question 5

Question 6

Question 7

Question 8

Question 9

Question 10

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