Q: A meteorologist models the temperature at a point $$(x, y, z) in a $$room as $$T(x, y, z) = 2x - y^2$ + 3z. At $$the point $$(1, -2, 0)$$, in which direction should a thermometer be moved to increase the temperature most rapidly?

In the direction of $$\nabla T = 2\mathbf{i} - 4\mathbf{j} + 3\mathbf{k}$$

Q: A hiker is climbing a hill whose height at point $$(x, y) is $$given by $$h(x, y) = 100 - x^2$ - y^2$. At the point (3$$, 4)$$, in which direction should the hiker walk to descend most rapidly?

In the direction of $$-\nabla h = 6\mathbf{i} + 8\mathbf{j}$$

Question 1

Given the function $$f(x, y) = 10x - 8y^2$, compute the gradient $$
abla f$$ at the point (8$$, -5)$$. Which of the following is correct?

Accepted Answer

$$160\,\mathbf{i} + 40\,\mathbf{j}$$

Question 2

Let $$f(x, y, z) = x + y^2$ - 2z + z \ln x^2$. Find $$
abla f$$ at the point (1$$, 5, 4)$$. Which of the following is correct?

Accepted Answer

$$\mathbf{i} + 10\mathbf{j} + (\ln 1 + 1 - 2)\mathbf{k}$$

Question 3

Find the directional derivative of $$f(x, y) = -4x + 8y^2$ at (-6$$, -2)$$ in the direction of $$\mathbf{A} = 3\mathbf{i} - 4\mathbf{j}$$. Which of the following is correct?

Accepted Answer

$$\dfrac{112}{5}$$

Question 4

Given $$f(x, y, z) = xy + yz + zx at$$ $$P_0 = (-3$$, 3, 2)$$, find the derivative in the direction of $$\mathbf{A} = 2\mathbf{i} + 6\mathbf{j} - 9\mathbf{k}$$. Which of the following is correct?

Accepted Answer

$$\dfrac{12}{\sqrt{121}}$$

Question 5

For $$f(x, y) = x + xy + y^2$ at P_0$ = (-4, -2)$$, in which direction does the function increase most rapidly?

Accepted Answer

$$\mathbf{u} = \dfrac{1}{\sqrt{13}}(-1\mathbf{i} + 3\mathbf{j})$$

Question 6

For $$f(x, y) = x + xy + y^2$ at P_0$ = (-4, -2)$$, what is the maximum rate of increase of the function?

Accepted Answer

Question 7

Assuming all necessary derivatives exist, how are $$D_{\mathbf{i}}f$$, $$D_{\mathbf{j}}f$$, and $$D_{\mathbf{k}}f$$ related to $$f_x$$, $$f_y$$, and $$f_z$$?

Accepted Answer

$$D_{\mathbf{i}}f$$, $$D_{\mathbf{j}}f$$, and $$D_{\mathbf{k}}f$$ are equal to $$f_x$$, $$f_y$$, and $$f_z$$ respectively, because when $$\mathbf{u} = \mathbf{i}$$, the directional derivative at $$P_0 is$$ $$f_x$$ evaluated at that point, etc.

Question 8

Given $$f(x, y) = 4y - 10x at$$ $$(1, 9)$$, what is the gradient $$
abla f at $$this point?

Accepted Answer

$$-10\mathbf{i} + 4\mathbf{j}$$

Question 9

Suppose you are given the curve $$x^2$$ + $$y^2 = 4$ and the point (1$$, \sqrt{3})$$. What is the equation of the tangent line to the curve at this point?

Accepted Answer

$$2x - 2y = -8/3$$

Question 10

Find a parametric equation for the line perpendicular to the surface $$3x^2$$ + $$4y^2$$ + $$z^2 = 43$ at the point (1$$, 3, 2)$$. Which is correct?

Accepted Answer

$$\mathbf{r}(t) = (1 + 6t)\mathbf{i} + (3 + 24t)\mathbf{j} + (2 + 4t)\mathbf{k}$$

Question 11

A meteorologist models the temperature at a point $$(x, y, z) in a $$room as $$T(x, y, z) = 2x - y^2$ + 3z. At $$the point $$(1, -2, 0)$$, in which direction should a thermometer be moved to increase the temperature most rapidly?

Accepted Answer

In the direction of $$
abla T = 2\mathbf{i} - 4\mathbf{j} + 3\mathbf{k}$$

Question 12

Suppose $$f(x, y) = x^2$ + y^2$ represents the concentration of a pollutant at point (x$$, y)$$ in a lake. At the point (2$$, 1)$$, what is the rate of change of concentration in the direction toward the origin?

Accepted Answer

$$-2\sqrt{5}$$

Question 13

A hiker is climbing a hill whose height at point $$(x, y) is $$given by $$h(x, y) = 100 - x^2$ - y^2$. At the point (3$$, 4)$$, in which direction should the hiker walk to descend most rapidly?

Accepted Answer

In the direction of $$-
abla h = 6\mathbf{i} + 8\mathbf{j}$$

Question 14

Let $$f(x, y, z) = xyz. At $$the point $$(1, 2, 3)$$, what is the directional derivative in the direction of the vector $$\mathbf{v} = 2\mathbf{i} - 2\mathbf{j} + \mathbf{k}$$?

Accepted Answer

$$\dfrac{2}{3}$$

Question 15

Suppose $$f(x, y) = e$$^{xy}$$. At the point (0$$, 1)$$, what is the gradient $$
abla f$$?

Accepted Answer

$$\mathbf{j}$$

Question 16

A chemical reaction's rate at point $$(x, y, z) is $$modeled by $$R(x, y, z) = x^2$y + yz^2$. At (1$$, 2, 3)$$, what is the gradient $$
abla R$$?

Accepted Answer

$$4\mathbf{i} + 11\mathbf{j} + 12\mathbf{k}$$

Question 17

Let $$f(x, y) = x^2$ - y^2$. At the point (1$$, 2)$$, what is the directional derivative in the direction of $$\mathbf{u} = \dfrac{1}{\sqrt{2}}(\mathbf{i} + \mathbf{j})$$?

Accepted Answer

$$-2\sqrt{2}$$

Multivariable Calculus: Directional Derivatives and The Gradient Study Guide

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