Q: Let $$f(x, y) = x^2$ + y^2$. At the point (1$$, 1)$$, in which direction does the function increase most rapidly?

In the direction of $$\nabla f(1, 1) = (2, 2)$$

Q: For a function $$f(x, y)$$, the directional derivative in the direction of a unit vector $$\u00e2$$ can be written as:

$$D_{\u00e2}f(x$$, y) = \nabla f(x, y) \cdot \u00e2$$

Q: A meteorologist models the temperature at a point $$(x, y) on a $$flat plain as $$T(x, y) = 10 - x^2$ - y^2$. At the point (1$$, 2)$$, in which direction should a weather balloon move to experience the fastest increase in temperature?

In the direction of $$\nabla T(1, 2) = (2, 4)$$

Question 1

Given $$f(x, y) = x^2$ + y^2$, what is the directional derivative of f$ at the point (1$$, 2)$$ in the direction of the unit vector $$ = \left( \frac{3}{5}, \frac{4}{5} \right)$$?

Accepted Answer

$$10$$

Question 2

Which of the following statements about the gradient vector $$
abla f(x_0$, y_0$) is $$TRUE?

Accepted Answer

It is orthogonal to the level curve of $$f at$$ $$(x_0$, y_0$).$$

Question 3

Let $$f(x, y) = 3x - 2y^2$. What is the maximum value of the directional derivative of f$ at the point (1$$, 1)$$?

Accepted Answer

Question 4

Suppose $$f(x, y) = x^2$y. $$Compute the gradient $$
abla f(x, y).$$

Accepted Answer

$$\left(2xy, x^2$\right)$$

Question 5

For a differentiable function $$f(x, y)$$, the directional derivative in the direction of a unit vector $$\u00e2 = (a, b) is $$given by:

Accepted Answer

$$D_{\u00e2}f(x$$, y) = f_x(x, y)a + f_y(x, y)b$$

Question 6

If $$f(x, y) = e$$^{xy}$$, what is the directional derivative of f$ at (0$$, 1)$$ in the direction of the vector $$\vec{v} = (1, 2)$$?

Accepted Answer

$$0$$

Question 7

Which of the following is a property of the gradient vector $$
abla f at a $$point $$(x_0$, y_0$)$$?

Accepted Answer

It points in the direction of the fastest increase of $$f.$$

Question 8

Let $$f(x, y) = x^2$ + y^2$. At the point (1$$, 1)$$, in which direction does the function increase most rapidly?

Accepted Answer

In the direction of $$
abla f(1, 1) = (2, 2)$$

Question 9

Suppose $$f(x, y) = x^2$y - y^3$. Find the directional derivative at (1$$, 2)$$ in the direction of the unit vector $$\u00e2 = \left( \frac{3}{5}, \frac{4}{5} \right)$$.

Accepted Answer

$$-2$$

Question 10

For a function $$f(x, y, z)$$, the directional derivative at $$(x_0$, y_0$, z_0$) in $$the direction of a unit vector $$\u00e2 = (a, b, c) is$$:

Accepted Answer

$$D_{\u00e2}f$$($$x_0$$, $$y_0$$, $$z_0) = f_x$$($$x_0$$, $$y_0$$, $$z_0)a + f_y$$($$x_0$$, $$y_0$$, $$z_0)b + f_z$$($$x_0$$, $$y_0$$, $$z_0)c$

Question 11

If $$f(x, y) = x^2$y + y^3$, what is the value of the directional derivative at (1$$, 1)$$ in the direction of the unit vector $$\u00e2 = \left( \frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}} \right)$$?

Accepted Answer

$$2\sqrt{2}$$

Question 12

Which of the following best describes the geometric significance of the gradient vector $$
abla f(x_0$, y_0$)$$?

Accepted Answer

It points in the direction of the fastest increase of $$f$$ and is orthogonal to the level curve at $$(x_0$, y_0$).$$

Question 13

Let $$f(x, y) = x^2$ + y^2$. What is the directional derivative at (1$$, 2)$$ in the direction of the vector $$\vec{v} = (2, 1)$$?

Accepted Answer

$$6$$

Question 14

If $$f(x, y) = x^2$y$$, what is the value of the directional derivative at $$(2, 1) in $$the direction of the unit vector $$\u00e2 = (0, 1)$$?

Accepted Answer

$$4$$

Question 15

Suppose $$f(x, y) = x^2$ + y^2$. At the point (1$$, 1)$$, what is the minimum value of the directional derivative and in which direction does it occur?

Accepted Answer

Minimum value is $$-2\sqrt{2} in $$the direction opposite to $$
abla f(1, 1)$$

Question 16

If $$f(x, y) = x^2$y - y^3$, what is the gradient $$
abla f(x, y)$$?

Accepted Answer

$$\left(2xy, x^2$ - 3y^2$\right)$$

Question 17

For a function $$f(x, y)$$, the directional derivative in the direction of a unit vector $$\u00e2$$ can be written as:

Accepted Answer

$$D_{\u00e2}f(x$$, y) = 
abla f(x, y) \cdot \u00e2$$

Question 18

If $$f(x, y) = x^2$y + y^3$, what is the value of the directional derivative at (1$$, 1)$$ in the direction of the unit vector $$\u00e2 = \left( \frac{1}{\sqrt{2}}, -\frac{1}{\sqrt{2}} \right)$$?

Accepted Answer

$$-2\sqrt{2}$$

Question 19

A meteorologist models the temperature at a point $$(x, y) on a $$flat plain as $$T(x, y) = 10 - x^2$ - y^2$. At the point (1$$, 2)$$, in which direction should a weather balloon move to experience the fastest increase in temperature?

Accepted Answer

In the direction of $$
abla T(1, 2) = (2, 4)$$

Section 4.6 - Directional Derivatives and the Gradient Vector

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