# Calculus with Applications

## Educators

AH
GL
+ 10 more educators

### Problem 1

Let $f(x, y)=2 x-4 y+7 .$ Find the following.
(a) $f(3,-1)$ (b) $f(-5,1)$ (c) $f(-5,-4)$ (d) $f(0,7)$

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### Problem 2

Let $f(x, y)=6 x-7 y+3 .$ Find the following.
(a) $f(4,-1)$ (b) $f(-5,1)$ $(c) \quad f(-5,-3)$ (d) $f(0,7)$

Sid W.
University of Louisville

### Problem 3

Let $f(x, y)=\sqrt{4 y^{2}+5 x^{2}} .$ Find the following.
(a) $f(5,-4) \quad$ (b) $f(-5,3)$ (c) $f(-1,-3)$ (d) $f(0,6)$

AH
Anh H.

### Problem 4

Let $f(x, y)=\sqrt{y^{2}+4 x^{2}} .$ Find the following.
(a) $f(1,-3)$ (b) $f(-3,5)$ (c) $f(-1,-2)$ (d) $f(0,8)$

Sid W.
University of Louisville

### Problem 5

Let $f(x, y)=e^{x}+\ln (x+y) .$ Find the following.
(a) $f(1,0)$ (b) $f(2,-1)$ (c) $f(0, e)$ (d) $f\left(0, e^{2}\right)$

GL
Gafari L.

### Problem 6

Let $f(x, y)=x e^{x+y} .$ Find the following.
(a) $f(1,0)$ (b) $f(2,-2)$ (c) $f(3,2)$ (d) $f(-1,4)$

Sid W.
University of Louisville

### Problem 7

Graph the first-octant portion of each plane.
$$x+y+z=9$$

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### Problem 8

Graph the first-octant portion of each plane.
$$x+y+z=15$$

Sid W.
University of Louisville

### Problem 9

Graph the first-octant portion of each plane.
$$2 x+3 y+4 z=12$$

Stuart H.

### Problem 10

Graph the first-octant portion of each plane.
$$4 x+2 y+3 z=24$$

Sid W.
University of Louisville

### Problem 11

Graph the first-octant portion of each plane.
$$x+y=4$$

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### Problem 12

Graph the first-octant portion of each plane.
$$y+z=5$$

Sid W.
University of Louisville

### Problem 13

Graph the first-octant portion of each plane.
$$x=5$$

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### Problem 14

Graph the first-octant portion of each plane.
$$z=4$$

Sid W.
University of Louisville

### Problem 15

Graph the level curves in the first quadrant of the $x y$ -plane for the following functions at heights of $z=0, z=2,$ and $z=4$ .
$$3 x+2 y+z=24$$

Sara S.

### Problem 16

Graph the level curves in the first quadrant of the $x y$ -plane for the following functions at heights of $z=0, z=2,$ and $z=4$ .
$$3 x+y+2 z=8$$

Sid W.
University of Louisville

### Problem 17

Graph the level curves in the first quadrant of the $x y$ -plane for the following functions at heights of $z=0, z=2,$ and $z=4$ .
$$y^{2}-x=-z$$

Neel F.

### Problem 18

Graph the level curves in the first quadrant of the $x y$ -plane for the following functions at heights of $z=0, z=2,$ and $z=4$ .
$$2 y-\frac{x^{2}}{3}=z$$

Sid W.
University of Louisville

### Problem 19

Discuss how a function of three variables in the form $w=f(x, y, z)$ might be graphed.

Yousef S.

### Problem 20

Suppose the graph of a plane $a x+b y+c z=d$ has a portion in the first octant. What can be said about $a, b, c,$ and $d ?$

Sid W.
University of Louisville

### Problem 21

In the chapter on Nonlinear Functions, the vertical line test was presented, which tells whether a graph is the graph of a function. Does this test apply to functions of two variables? Explain.

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### Problem 22

A graph that was not shown in this section is the hyperboloid of one sheet, described by the equation $x^{2}+y^{2}-z^{2}=1$ Describe it as completely as you can.

Sid W.
University of Louisville

### Problem 23

Match each equation in Exercises $23-28$ with its graph in $(a)-(f)$ below and on the next page.
$$z=x^{2}+y^{2}$$

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### Problem 24

Match each equation in Exercises $23-28$ with its graph in $(a)-(f)$ below and on the next page.
$$z^{2}-y^{2}-x^{2}=1$$

Sid W.
University of Louisville

### Problem 25

Match each equation in Exercises $23-28$ with its graph in $(a)-(f)$ below and on the next page.
$$x^{2}-y^{2}=z$$

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### Problem 26

Match each equation in Exercises $23-28$ with its graph in $(a)-(f)$ below and on the next page.
$$z=y^{2}-x^{2}$$

Sid W.
University of Louisville

### Problem 27

Match each equation in Exercises $23-28$ with its graph in $(a)-(f)$ below and on the next page.
$$\frac{x^{2}}{16}+\frac{y^{2}}{25}+\frac{z^{2}}{4}=1$$

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### Problem 28

Match each equation in Exercises $23-28$ with its graph in $(a)-(f)$ below and on the next page.
$$z=5\left(x^{2}+y^{2}\right)^{-1 / 2}$$

Sid W.
University of Louisville

### Problem 29

Let $f(x, y)=4 x^{2}-2 y^{2},$ and find the following.
(a) $\frac{f(x+h, y)-f(x, y)}{1}$ (b) $\frac{f(x, y+h)-f(x, y)}{h}$ (c) $\lim _{h \rightarrow 0} \frac{f(x+h, y)-f(x, y)}{h}$ (d) $\lim _{h \rightarrow 0} \frac{f(x, y+h)-f(x, y)}{h}$

Jonathon B.

### Problem 30

Let $f(x, y)=5 x^{3}+3 y^{2},$ and find the following.
(a) $\frac{f(x+h, y)-f(x, y)}{h}$ (b) $\frac{f(x, y+h)-f(x, y)}{h}$ (c) $\lim _{h \rightarrow 0} \frac{f(x+h, y)-f(x, y)}{h}$ (d) $\lim _{h \rightarrow 0} \frac{f(x, y+h)-f(x, y)}{h}$

Sid W.
University of Louisville

### Problem 31

Let $f(x, y)=x y e^{x^{2}+y^{2}}$ . Use a graphing calculator or spreadsheet to find each of the following and give a geometric interpretation of the results. (Hint: First factor $e^{2}$ from the limit and then evaluate the quotient at smaller and smaller values of $h . )$
(a) $\lim _{h \rightarrow 0} \frac{f(1+h, 1)-f(1,1)}{h}$ (b) $\lim _{h \rightarrow 0} \frac{f(1,1+h)-f(1,1)}{h}$

Ethan S.

### Problem 32

The following table provides values of the function $f(x, y) .$ However, because of potential errors in measurement, the functional values may be slightly inaccurate. Using the statistical package included with a graphing calculator or spreadsheet and critical thinking skills, find the function $f(x, y)=a+b x+c y$ that best estimates the table where $a, b,$ and $c$ are integers. (Hint: Do a linear regression on each column with the value of $y$ fixed and then use these four regression equations to determine the coefficient $c . )$

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### Problem 33

Production Production of a digital camera is given by
$$P(x, y)=100\left(\frac{3}{5} x^{-2 / 5}+\frac{2}{5} y^{-2 / 5}\right)^{-5},$$
where $x$ is the amount of labor in work-hours and $y$ is the amount of capital. Find the following.
(a) What is the production when 32 work-hours and 1 unit of capital are provided?
(b) Find the production when 1 work-hour and 32 units of capital are provided.
(c) If 32 work-hours and 243 units of capital are used, what is the production output?

FT
Felipe T.

### Problem 34

In their original paper, Cobb and Douglas estimated the production function for the United States to be $z=1.01 x^{3 / 4} y^{1 / 4}$ , where $x$ represents the amount of labor and $y$ the amount of capital. Source: American Economic Review.

Sid W.
University of Louisville

### Problem 35

A study of the connection between immigration and the fiscal problems associated with the aging of the baby boom generation considered a production function of the form $z=x^{0.6} y^{0.4}$ where $x$ represents the amount of labor and $y$ the amount of capital. Source: Journal of Political Economy.

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### Problem 36

Production For the function in Exercise $35,$ what is the effect on $z$ of halving $x ?$ Of halving $y$ ? Of halving both?

Sid W.
University of Louisville

(a) 3/4 (b) 2/5

Carson M.