# Finite Mathematics and Calculus with Applications

## Educators

TE
RE

### Problem 1

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

TE
Thomas E.

### Problem 2

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

TE
Thomas E.

### Problem 3

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

TE
Thomas E.

### Problem 4

Let $f(x, y)=\frac{\sqrt{9 x+5 y}}{\log x} .$ Find the following.
a. $f(10,2)$
b. $f(100,1)$
c. $f(1000,0)$
d. $f\left(\frac{1}{10}, 5\right)$

TE
Thomas E.

### Problem 5

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

TE
Thomas E.

### Problem 6

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

TE
Thomas E.

### Problem 7

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

TE
Thomas E.

### Problem 8

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

TE
Thomas E.

### Problem 9

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

TE
Thomas E.

### Problem 10

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

TE
Thomas E.

### Problem 11

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

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### Problem 12

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

TE
Thomas E.

### Problem 13

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 $z=4$.
$3 x+2 y+z=24$

RE
Rommel E.

### Problem 14

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 $z=4$.
$3 x+y+2 z=8$

TE
Thomas E.

### 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 $z=4$.
$y^{2}-x=-z$

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### 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 $z=4$.
$2 y-\frac{x^{2}}{3}=z$

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### Problem 17

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

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### Problem 18

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 $d ?$

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### Problem 19

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 20

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.

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### Problem 21

With its graph in a–f on the next page
$z=x^{2}+y^{2}$

TE
Thomas E.

### Problem 22

With its graph in a–f on the next page
$z^{2}-y^{2}-x^{2}=1$

TE
Thomas E.

### Problem 23

With its graph in a–f on the next page
$x^{2}-y^{2}=z$

TE
Thomas E.

### Problem 24

With its graph in a–f on the next page
$z=y^{2}-x^{2}$

TE
Thomas E.

### Problem 25

With its graph in a–f on the next page
$\frac{x^{2}}{16}+\frac{y^{2}}{25}+\frac{z^{2}}{4}=1$

TE
Thomas E.

### Problem 26

With its graph in a–f on the next page
$z=5\left(x^{2}+y^{2}\right)^{-1 / 2}$

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### Problem 27

Let $f(x, y)=4 x^{2}-2 y^{2},$ 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. $$\lim _{h \rightarrow 0} \frac{f(x, y+h)-f(x, y)}{h}$$

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### Problem 28

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. $$\lim _{h \rightarrow 0} \frac{f(x, y+h)-f(x, y)}{h}$$

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### Problem 29

Let $f(x, y)=x y e^{x^{2}+y^{2}} .$ Use a graphing calculator or spread- sheet 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. $$\lim _{h \rightarrow 0} \frac{f(1,1+h)-f(1,1)}{h}$$

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### Problem 30

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 31

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?

TE
Thomas E.

### Problem 32

The multiplier function
$$M=\frac{(1+i)^{n}(1-t)+t}{[1+(1-t) i]^{n}}$$
compares the growth of an Individual Retirement Account (IRA) with the growth of the same deposit in a regular savings account. The function $M$ depends on the three variables $n, i$ and $t,$ where $n$ represents the number of years an amount is left at interest, $i$ represents the interest rate in both types of accounts, and $t$ t represents the income tax rate. Values of $M>1$ indicate that the IRRA grows faster than the savings account. Let $M=f(n, i, t)$ and find the following.

Find the multiplier when funds are left for 25 years at 5$\%$ interest and the income tax rate is 33$\% .$ Which account grows faster?

TE
Thomas E.

### Problem 33

The multiplier function
$$M=\frac{(1+i)^{n}(1-t)+t}{[1+(1-t) i]^{n}}$$
compares the growth of an Individual Retirement Account (IRA) with the growth of the same deposit in a regular savings account. The function $M$ depends on the three variables $n, i$ and $t,$ where $n$ represents the number of years an amount is left at interest, $i$ represents the interest rate in both types of accounts, and $t$ t represents the income tax rate. Values of $M>1$ indicate that the IRRA grows faster than the savings account. Let $M=f(n, i, t)$ and find the following.

What is the multiplier when money is invested for 40 years at 6$\%$ interest and the income tax rate is 28$\% ?$ Which account grows faster?

TE
Thomas E.

### Problem 34

Find the level curve at a production of 500 for the production functions. Graph each level curve in the $xy$-plane.
In their original paper, $\mathrm{Cobb}$ and Douglas estimated the pro- duction 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.

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### Problem 35

Find the level curve at a production of 500 for the production functions. Graph each level curve in the $xy$-plane.
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

For the function in Exercise $34,$ what is the effect on z of doubling $x ?$ Of doubling $y ?$ Of doubling both?

TE
Thomas E.