Lesson test: The gradient

The gradient

1.[2p]
Find the directional derivative of $f (x, y) = x^{2} + x y + 2 y^{2}$ at $(1, 1)$ in the direction of $⟨ 3, 4 ⟩$ .

CorrectNot quite: 5.8

2.[2p]
Find the directional derivative of $f (x, y, z) = x y + y z + z x$ at $(1, 2, 3)$ in the direction of $⟨ 1, 2, 2 ⟩$ . Give it to two decimal places.

CorrectNot quite: 6.33

3.[2p]
What is the maximum rate of change of $f (x, y) = x^{2} y^{3}$ at the point $(1, 2)$ ?

CorrectNot quite: 20

4.[2p]
For $f (x, y) = x y$ at the point $(2, 1)$ , in which direction is the rate of change of $f$ zero?

$\frac{1}{\sqrt{5}} ⟨ 2, - 1 ⟩$

Correct

$\frac{1}{\sqrt{5}} ⟨ 1, 2 ⟩$

The answer is: $\frac{1}{\sqrt{5}}\langle 2, -1 \rangle$

$\frac{1}{\sqrt{5}} ⟨ 2, 1 ⟩$

The answer is: $\frac{1}{\sqrt{5}}\langle 2, -1 \rangle$

$\frac{1}{\sqrt{5}} ⟨ - 1, - 2 ⟩$

The answer is: $\frac{1}{\sqrt{5}}\langle 2, -1 \rangle$

5.[2p]
Match the angle $θ$ between a unit vector $𝐮$ and $\nabla f$ to the directional derivative $D_{𝐮} f$ .

$θ = 0$
$θ = π / 3$
$θ = π / 2$
$θ = π$

$| \nabla f |$
$0$
$- | \nabla f |$
$\frac{1}{2} | \nabla f |$

Show the answer

$θ = 0$ : $| \nabla f |$ $θ = π / 3$ : $\frac{1}{2} | \nabla f |$ $θ = π / 2$ : $0$ $θ = π$ : $- | \nabla f |$

6.[2p]
A hill has height $h (x, y) = 400 - 0.001 x^{2} - 0.001 y^{2}$ metres. On a map with contours every $10$ metres, roughly how far apart on the ground, in metres, are the contours near the point $(60, 80)$ ?

CorrectNot quite: 50

7.[2p]
The tangent plane to the ellipsoid $x^{2} + 2 y^{2} + 3 z^{2} = 21$ at $(1, 2, 2)$ can be written $x + a y + b z = d$ . What is $d$ ?

CorrectNot quite: 21

8.[3p]
Gradient descent is run on $f (x, y) = x^{2} + 4 y^{2}$ from $(2, 1)$ with step size $η = 0.1$ . What is the value of $f$ after two steps? Give it to four decimal places.

CorrectNot quite: 1.6448

9.[3p]
Which of these statements about a differentiable function $f$ of two variables are true?

Select all that apply

At a point where $\nabla f \neq 𝟎$ , the gradient is perpendicular to the level curve through that point

Correct

The largest directional derivative at a point equals $| \nabla f |$ there

Correct

If $f$ has a local minimum at a point, then $\nabla f = 𝟎$ there

Correct

If $\nabla f = 𝟎$ at a point, then $f$ has a local maximum or minimum there

The answer is: At a point where $\nabla f \ne \mathbf{0}$, the gradient is perpendicular to the level curve through that point, The largest directional derivative at a point equals $|\nabla f|$ there, If $f$ has a local minimum at a point, then $\nabla f = \mathbf{0}$ there

The gradient always points towards the highest point of the graph

1.[2p] Find the directional derivative of f(x,y)=x2+xy+2y2 at (1,1) in the direction of ⟨3,4⟩.

2.[2p] Find the directional derivative of f(x,y,z)=xy+yz+zx at (1,2,3) in the direction of ⟨1,2,2⟩. Give it to two decimal places.

3.[2p] What is the maximum rate of change of f(x,y)=x2y3 at the point (1,2)?

4.[2p] For f(x,y)=xy at the point (2,1), in which direction is the rate of change of f zero?

5.[2p] Match the angle θ between a unit vector 𝐮 and ∇f to the directional derivative D𝐮f.

6.[2p] A hill has height h(x,y)=400-0.001x2-0.001y2 metres. On a map with contours every 10 metres, roughly how far apart on the ground, in metres, are the contours near the point (60,80)?

7.[2p] The tangent plane to the ellipsoid x2+2y2+3z2=21 at (1,2,2) can be written x+ay+bz=d. What is d?

8.[3p] Gradient descent is run on f(x,y)=x2+4y2 from (2,1) with step size η=0.1. What is the value of f after two steps? Give it to four decimal places.

9.[3p] Which of these statements about a differentiable function f of two variables are true?

1.[2p]
Find the directional derivative of $f (x, y) = x^{2} + x y + 2 y^{2}$ at $(1, 1)$ in the direction of $⟨ 3, 4 ⟩$ .

2.[2p]
Find the directional derivative of $f (x, y, z) = x y + y z + z x$ at $(1, 2, 3)$ in the direction of $⟨ 1, 2, 2 ⟩$ . Give it to two decimal places.

3.[2p]
What is the maximum rate of change of $f (x, y) = x^{2} y^{3}$ at the point $(1, 2)$ ?

4.[2p]
For $f (x, y) = x y$ at the point $(2, 1)$ , in which direction is the rate of change of $f$ zero?

5.[2p]
Match the angle $θ$ between a unit vector $𝐮$ and $\nabla f$ to the directional derivative $D_{𝐮} f$ .

6.[2p]
A hill has height $h (x, y) = 400 - 0.001 x^{2} - 0.001 y^{2}$ metres. On a map with contours every $10$ metres, roughly how far apart on the ground, in metres, are the contours near the point $(60, 80)$ ?

7.[2p]
The tangent plane to the ellipsoid $x^{2} + 2 y^{2} + 3 z^{2} = 21$ at $(1, 2, 2)$ can be written $x + a y + b z = d$ . What is $d$ ?

8.[3p]
Gradient descent is run on $f (x, y) = x^{2} + 4 y^{2}$ from $(2, 1)$ with step size $η = 0.1$ . What is the value of $f$ after two steps? Give it to four decimal places.

9.[3p]
Which of these statements about a differentiable function $f$ of two variables are true?