Q: Let $$f(x, y) = \dfrac{x^2$ - y}{xy}. $$What is the domain of $$f$$?

All $$(x, y)$$ such that $$x \neq 0$$ and $$y \neq 0$$

Question 1

Let $$f(x, y) = \dfrac{x^2$ - y}{xy}. $$What is the domain of $$f$$?

Accepted Answer

All $$(x, y)$$ such that $$x 
eq 0$$ and $$y 
eq 0$$

Question 2

Which of the following statements is TRUE about the gradient $$
abla f(x, y) of a $$differentiable function $$f$$?

Accepted Answer

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

Question 3

Evaluate $$\displaystyle \lim_{(x, y) 	o (0, 0)} \dfrac{5xy}{3x^2$ - 4y^2$}$$, or determine if it does not exist.

Accepted Answer

The limit does not exist.

Question 4

Let $$f(x, y, z) = 5x \cos(xy) - e^z \ln(z^3$). $$What is $$\dfrac{\$$partial^2$$ f}{\partial x \partial y}$$?

Accepted Answer

$$-5y \sin(xy) - 5x \sin(xy)$$

Question 5

Suppose $$f(x, y) = x^2$y - e$$^{xy}$$. Find the gradient $$
abla f$$ at the point (1$$, 2)$$.

Accepted Answer

$$\langle 4 - 2e^2$, 1 - e^2$ \rangle$$

Question 6

Given $$f(x, y) = x^2$y - e$$^{xy}$$ at (1$$, 2)$$, what is the rate of change in the direction of $$\langle 5, -1 \rangle$$?

Accepted Answer

$$\dfrac{21 - 11e^2$}{\sqrt{26}}$$

Question 7

Which of the following sets is both open and closed in $$\mathbb{R}^2$$?

Accepted Answer

The empty set $$\emptyset$$

Question 8

Let $$f(x, y) = x^3$ + y^3$ + 3x^2$ - 3y^2$ - 8. $$Find and classify the critical point(s).

Accepted Answer

A local maximum at $$(-2, 0)$$ and a local minimum at $$(0, 2)$$

Question 9

Which of the following best describes the tangent plane to the surface $$xyz - xz + 3xy = 5 at $$the point $$(1, 2, -1)$$?

Accepted Answer

$$-3x + 2y - 2z = -9$$

Question 10

Let $$f(x, y) = g(u(x, y), v(x, y)). $$Given $$g_u(3, 4) = 10$$, $$g_v(3, 4) = -2$$, $$u_x(0, 1) = 4$$, $$v_x(0, 1) = -8$$, $$u(0, 1) = 3$$, $$v(0, 1) = 4$$, what is $$f_x(0, 1)$$?

Accepted Answer

$$10 \cdot 4 + (-2) \cdot (-8) = 56$$

Question 11

Which of the following is NOT a correct interpretation of the gradient $$
abla f at a $$point?

Accepted Answer

It is always tangent to the level curves of $$f at $$that point.

Question 12

Suppose you want to maximize the product $$xyz$$ for positive $$x, y, z$$ subject to $$x + y + z^2$ = 16. $$Which method should you use?

Accepted Answer

Lagrange multipliers

Question 13

Which of the following is the correct equation for the tangent plane to $$f(x, y) = x^2$y - e$$^{xy}$$ at (1$$, 2)$$?

Accepted Answer

$$z = 2 - e^2$ + (4 - 2e^2$)(x - 1) + (1 - e^2$)(y - 2)$$

Question 14

Which of the following is a necessary condition for a function $$f(x, y) to $$have a local extremum at $$(a, b)$$?

Accepted Answer

$$
abla f(a, b) = \vec{0}$$

Question 15

Let $$f(x, y) = x^2$ + y^2$. Which of the following are the level curves of f$?

Accepted Answer

Circles centered at the origin

Question 16

Suppose $$f(x, y) is $$differentiable and $$
abla f(a, b) = \vec{0}. $$Which of the following must be true?

Accepted Answer

$$(a, b) is a $$critical point of $$f$$

Question 17

Which of the following is the correct application of the multivariable chain rule for $$f(x, y) = g(u(x, y), v(x, y))$$?

Accepted Answer

$$f_x = g_u u_x + g_v v_x$$

Question 18

Which of the following is NOT a valid notation for the partial derivative of $$f$$ with respect to $$x$$?

Accepted Answer

$$f'_{y}$$

Multivariable Calculus: Functions, Limits, Derivatives, and Optimization

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