Q: Let $$f(x, y) = x^2$ + 3y^2$. What is the gradient $$\nabla f$$ at the point (1$$, -2)$$?

$$\nabla f = \langle 2, -12 \rangle$$

Question 1

Consider the surface $$z = x^2$ + y^2$. Which of the following best describes the level curves for z = k$, where k$ is a constant?

Accepted Answer

Circles centered at the origin with radius $$\sqrt{k}$$

Question 2

Evaluate the limit $$\displaystyle \lim_{(x, y) 	o (0, 0)} \frac{x^2$ y}{x^4$ + y^2$}$$, or show that it does not exist.

Accepted Answer

The limit does not exist because approaching along $$y = x^2$ and y$$ = $$-x^2$$ gives different values.

Question 3

Given $$f(x, y) = x^2$ y + e$$^{xy}$$, compute f$$_{xy}$$.

Accepted Answer

$$f_{xy} = 2x + x e$$^{xy}$$

Question 4

Let $$f(x, y) = x^2$ + 3y^2$. What is the gradient $$
abla f$$ at the point (1$$, -2)$$?

Accepted Answer

$$
abla f = \langle 2, -12 angle$$

Question 5

Suppose the temperature at a point $$(x, y) in a $$metal plate is given by $$T(x, y) = 100 - x^2$ - y^2$. In which direction from the point (3$$, 4)$$ does the temperature increase most rapidly?

Accepted Answer

In the direction of the vector $$\langle -3, -4 \rangle$$

Question 6

Given $$f(x, y) = x^2$ - y^2$, classify the critical point at (0$$, 0)$$ using the second derivative test.

Accepted Answer

Saddle point

Question 7

Let $$f(x, y) = x^2$ y + y^3$. Find all critical points.

Accepted Answer

Only $$(0, 0)$$

Question 8

Suppose $$z = f(x, y) = x^2$ + y^2$. What is the directional derivative at (1$$, 2)$$ in the direction of the unit vector $$\vec{u} = \frac{1}{\sqrt{5}} \langle 2, 1 \rangle$$?

Accepted Answer

$$2 \sqrt{5}$$

Question 9

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

Accepted Answer

It is perpendicular to the level curve of $$f at$$ $$(a, b).$$

Question 10

Let $$f(x, y) = x^2$ + y^2$ - 4x - 6y + 13. $$Find the minimum value of $$f(x, y)$$ and the point where it occurs.

Accepted Answer

Minimum at $$(2, 3)$$ with value $$0$$

Question 11

Given $$f(x, y) = x^2$ y + y^3$, compute f$$_{yy}$$.

Accepted Answer

$$6y$$

Question 12

Evaluate $$\displaystyle \lim_{(x, y) 	o (1, 2)} \frac{x^2$ y - 2x + y}{x + y}.$$

Accepted Answer

$$3$$

Question 13

Suppose $$f(x, y) = \ln(x^2$ + y^2$). $$Compute $$f_x.$$

Accepted Answer

$$\frac{2x}{x^2$ + y^2$}$$

Question 14

A chemical reaction's rate at $$(x, y) is $$given by $$R(x, y) = x e$$^{y}$$. What is the rate of change of R$ at (1$$, 0)$$ in the direction of the vector $$\langle 3, 4 \rangle$$?

Accepted Answer

$$5$$

Question 15

Let $$f(x, y) = x^3$ + y^3$ - 3xy. $$Find all critical points.

Accepted Answer

$$(0, 0)$$ and $$(1, 1)$$ and $$(-1, -1)$$ and $$(0, 0)$$

Question 16

Suppose $$f(x, y) = x^2$ + y^2$. What is the value of the discriminant D$ at (0$$, 0)$$?

Accepted Answer

$$D = 4$$

Question 17

Given $$f(x, y) = x^2$ y + y^3$, use the chain rule to compute $$\frac{dz}{dt}$$ at t = 0$ if x$$ = $$t^2$$, $$y = \sin t.$$

Accepted Answer

$$0$$

Question 18

Let $$f(x, y) = x^2$ y + y^3$. Compute f$$_{xx}$$.

Accepted Answer

$$2y$$

Question 19

Suppose $$f(x, y) = x^2$ - 4xy + 4y^2$. Classify the critical point at (0$$, 0)$$ using the second derivative test.

Accepted Answer

Local minimum

Calculus III Study Guide: Multivariable Functions, Partial Derivatives, and Optimization

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