Q: Suppose $$f(x, y) is $$continuous at $$(a, b). $$Which of the following must be true?

$$\displaystyle \lim_{(x, y) \to (a, b)} f(x, y) = f(a, b)$$

Question 1

Which of the following functions has level curves that are circles centered at the origin?

Accepted Answer

$$f(x, y) = e$$^{x^2 + y^2}$$

Question 2

Consider the function $$F(x, y, z) = x^2$ + y^2$ + z^2$. What is the geometric description of the level surface F(x$$, y, z) = c$$ for c$$ \u003e 0$$?

Accepted Answer

A sphere of radius $$\sqrt{c}$$

Question 3

Evaluate the following limit:

$$\displaystyle \lim_{(x, y) 	o (1, 1)} x^2$ y + y^5$

Accepted Answer

$$2$$

Question 4

Which of the following statements about the limit laws for functions of two variables is FALSE?

Accepted Answer

The limit of a function always exists if the limits along the $$x- $$and $$y-$$axes exist and are equal.

Question 5

Evaluate the following limit using limit laws:

$$\displaystyle \lim_{(x, y) 	o (2, 3)} \frac{2 + x - y}{\sqrt{x^4$ + y^2$}}$$

Accepted Answer

$$\frac{1}{5}$$

Question 6

Suppose $$f(x, y) = \frac{\sin(x^2$ + y^2$)}{x^2$ + y^2$}. $$What is $$\displaystyle \lim_{(x, y) 	o (0, 0)} f(x, y)$$?

Accepted Answer

$$1$$

Question 7

Compute the following limit:

$$\displaystyle \lim_{(x, y) 	o (0, 1)} \frac{xy - x^3$ y - x + x^3$}{x y^2$ - x y}$$

Accepted Answer

$$1$$

Question 8

Which of the following is a correct application of the two-path test for nonexistence of a limit?

Accepted Answer

If the limit along two different paths yields different values, the limit does not exist.

Question 9

Evaluate $$\displaystyle \lim_{(x, y) 	o (0, 0)} \frac{x}{\sqrt{x^2$ + y^2$}} or $$determine if it exists.

Accepted Answer

The limit does not exist

Question 10

For the function $$G(x, y, z) = x^2$ + y^2$ - z^2$, what is the level surface G(x$$, y, z) = 0$$?

Accepted Answer

A cone

Question 11

Which of the following is TRUE about polynomial functions of two variables?

Accepted Answer

They are continuous everywhere in their domain.

Question 12

Evaluate $$\displaystyle \lim_{(x, y) 	o (0, 0)} \frac{xy}{x^2$ + y^2$} or $$determine if it exists.

Accepted Answer

The limit does not exist

Question 13

Suppose $$f(x, y) is $$continuous at $$(a, b). $$Which of the following must be true?

Accepted Answer

$$\displaystyle \lim_{(x, y) 	o (a, b)} f(x, y) = f(a, b)$$

Question 14

A chemical reaction's rate depends on temperature $$T$$ and pressure $$P$$ via $$R(T, P) = T^2$ + P^2$. What do the level curves R(T$$, P) = c$$ represent in the T$-P$ plane?

Accepted Answer

Circles centered at the origin

Question 15

Let $$f(x, y) = x^2$ + y^2$. What is $$\displaystyle \lim_{(x, y) 	o (0, 0)} f(x, y)$$?

Accepted Answer

$$0$$

Question 16

Which of the following is a necessary condition for $$\displaystyle \lim_{(x, y) 	o (a, b)} f(x, y) = L to $$exist?

Accepted Answer

The limit along every path to $$(a, b)$$ must be $$L.$$

Question 17

Suppose $$f(x, y) = x^2$ y + y^5$ models the cost (in dollars) of producing a chemical at location (x$$, y)$$. What is the instantaneous cost at (1$$, 1)$$?

Accepted Answer

$$2$$

Question 18

Which of the following is the correct definition of the limit of a function of two variables $$f(x, y) as$$ $$(x, y) 	o (a, b)$$?

Accepted Answer

For every $$\epsilon \u003e 0$$, there exists $$\delta \u003e 0$$ such that $$|f(x, y) - L| \u003c \epsilon$$ whenever $$0 \u003c \sqrt{(x-a)^2$ + (y-b)^2$} \u003c \delta.$$

Calc 10: Limits and Continuity of Multivariate Functions

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