Q: A temperature function in a metal plate is given by $$T(x, y) = 10x - 8y^2$. At the point (8$$, -5)$$, in which direction does the temperature increase most rapidly?

In the direction of $$\nabla T at$$ $$(8, -5)$$

Q: Suppose $$f(x, y, z) is $$differentiable. Which of the following statements is true about the gradient $$\nabla f at a $$point $$P$$?

$$\nabla f$$ points in the direction of the greatest rate of increase of $$f at$$ $$P.$$

Question 1

Let $$f(x, y) = 10x - 8y^2$. What is the gradient $$
abla f$$ at the point (8$$, -5)$$?

Accepted Answer

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

Question 2

Given $$f(x, y, z) = x^2$ + y^3$ - 2z^2$ + z^2$ \ln x$$, find $$
abla f at $$the point $$(1, 5, 4). $$Which of the following is correct?

Accepted Answer

$$2\,\mathbf{i} + 75\,\mathbf{j} - 16\,\mathbf{k}$$

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 is correct?

Accepted Answer

$$\dfrac{208}{5}$$

Question 4

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

Accepted Answer

$$\dfrac{25}{\sqrt{121}}$$

Question 5

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

Accepted Answer

$$\mathbf{u} = \dfrac{(-10, -6)}{\sqrt{136}}$$

Question 6

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

Accepted Answer

$$\sqrt{136}$$

Question 7

Which of the following best describes the relationship between the directional derivatives $$D_{\mathbf{i}}f$$, $$D_{\mathbf{j}}f$$, $$D_{\mathbf{k}}f$$ and the partial derivatives $$f_x$$, $$f_y$$, $$f_z$$?

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, and similarly for $$\mathbf{j}$$ and $$\mathbf{k}.$$

Question 8

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

Accepted Answer

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

Question 9

Suppose you are standing on a hill whose height at point $$(x, y) is $$given by $$f(x, y) = x^2$ + xy + y^2$. At the point (-4$$, -2)$$, in which direction should you walk to descend most rapidly?

Accepted Answer

In the direction of $$\mathbf{u} = -\dfrac{(-10, -6)}{\sqrt{136}}$$

Question 10

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

Accepted Answer

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

Question 11

Let $$f(x, y) = x^2$ + y^2$ and consider the curve x + y = 4$. What is the equation of the tangent line to the curve at the point (1$$, 3)$$?

Accepted Answer

$$2x - 2y = -8$$

Question 12

A temperature function in a metal plate is given by $$T(x, y) = 10x - 8y^2$. At the point (8$$, -5)$$, in which direction does the temperature increase most rapidly?

Accepted Answer

In the direction of $$
abla T at$$ $$(8, -5)$$

Question 13

Suppose $$f(x, y, z) is $$differentiable. Which of the following statements is true about the gradient $$
abla f at a $$point $$P$$?

Accepted Answer

$$
abla f$$ points in the direction of the greatest rate of increase of $$f at$$ $$P.$$

Question 14

If $$f(x, y) = x^2$ + xy + y^2$, what is the value of the directional derivative at (-4$$, -2)$$ in the direction of the unit vector $$\mathbf{u} = \dfrac{(-10, -6)}{\sqrt{136}}$$?

Accepted Answer

$$\sqrt{136}$$

Question 15

A biologist models the population density of a species as $$P(x, y) = 4y - 10x. At $$the point $$(1, 9)$$, in which direction should the biologist move to find the greatest increase in population density?

Accepted Answer

In the direction of $$-10\,\mathbf{i} + 4\,\mathbf{j}$$

Question 16

Given $$f(x, y, z) = x^2$ + y^3$ - 2z^2$ + z^2$ \ln x$$, what is $$\dfrac{\partial f}{\partial x} at$$ $$(1, 5, 4)$$?

Accepted Answer

$$2$$

Question 17

A weather balloon is rising in the atmosphere, where the temperature at position $$(x, y, z) is$$ $$T(x, y, z) = xy + yz + zx. At $$the point $$(-3, 3, 2)$$, what is the rate of change of temperature in the direction of the vector $$\mathbf{A} = 2\mathbf{i} + 6\mathbf{j} - 9\mathbf{k}$$?

Accepted Answer

$$\dfrac{25}{\sqrt{121}}$$

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