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Applied Calculus For Business, Economics, and Finance

Warren B. Gordon, Walter O. Wang, April Allen Materowski

Chapter 1

Functions and their Applications

Educators


Problem 1

Determine the equation of the line whose slope and $y$ -intercept are given.
$$2,(0,4).$$

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Problem 2

Determine the equation of the line whose slope and $y$ -intercept are given.
$$-3,(0,0).$$

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Problem 3

Determine the equation of the line whose slope and $y$ -intercept are given.
$$0,(0,-2).$$

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Problem 4

Determine the equation of the line whose slope and $y$ -intercept are given.
$$1 / 2,(0,1 / 4).$$

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Problem 5

Determine the equation of the line whose slope and $y$ -intercept are given.
$$1 / 4,(0,3).$$

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Problem 6

Find the slope, $x$ -intercept and $y$ -intercept of the given line.
$$3 x-4 y=12.$$

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Problem 7

Find the slope, $x$ -intercept and $y$ -intercept of the given line.
$$2 x+4 y-9=0.$$

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Problem 8

Find the slope, $x$ -intercept and $y$ -intercept of the given line.
$$4 y=5 x+12.$$

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Problem 9

Find the slope, $x$ -intercept and $y$ -intercept of the given line.
$$1.3 x+4.7 y+11.2=0.$$

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Problem 10

Find the slope, $x$ -intercept and $y$ -intercept of the given line.
$$16-4 y=35.$$

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Problem 11

Give an equation of the line with the given slope and passing through the given point.
$$-3,(2,5)$$

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Problem 12

Give an equation of the line with the given slope and passing through the given point.
$$1 / 2,(-2,0)$$

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Problem 13

Give an equation of the line with the given slope and passing through the given point.
$$1 / 6,(0,0)$$

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Problem 14

Give an equation of the line with the given slope and passing through the given point.
$$14,(-1,0.3)$$

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Problem 15

Give an equation of the line with the given slope and passing through the given point.
$$4.1, (3, 0)$$

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Problem 16

Find equation for the given line.A vertical line passing through $(-1,8).$

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Problem 17

Find equation for the given line.A horizontal line passing through $(2,-6).$

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Problem 18

Find equation for the given line.A line with intercepts $(0,4)$ and$ ( -2,0 ).$

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Problem 19

Find equation for the given line.A vertical line passing through 4$ (12,2).$

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Problem 20

Find equation for the given line.A horizontal line passing through $(-3,8).$

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Problem 21

Plot the lines found in Exercise $1$.

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Problem 22

Plot the lines found in Exercise $2$.

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Problem 23

Plot the lines found in Exercise $3$.

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Problem 24

Plot the lines found in Exercise $4$.

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Problem 25

Plot the lines found in Exercise $5$.

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Problem 26

Plot the lines found in Exercise $6$.

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Problem 27

Plot the lines found in Exercise $7$.

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Problem 28

Plot the lines found in Exercise $8$.

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Problem 29

Plot the lines found in Exercise $9$.

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Problem 30

Plot the lines found in Exercise $10$.

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Problem 31

Plot the lines found in Exercise $11$.

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Problem 32

Plot the lines found in Exercise $12$.

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Problem 33

Plot the lines found in Exercise $13$.

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Problem 34

Plot the lines found in Exercise $14$.

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Problem 35

Plot the lines found in Exercise $15$.

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Problem 36

Plot the lines found in Exercise $16$.

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Problem 37

Plot the lines found in Exercise $17$.

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Problem 38

Plot the lines found in Exercise $18$.

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Problem 39

Plot the lines found in Exercise $19$.

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Problem 40

Plot the lines found in Exercise $20$.

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Problem 41

(a) Find the slope of the line whose equation is $2 x-5 y=6 .$ Find the $x$ and $y$ intercepts of the line. Plot the line. (b) Find the equation of the line parallel to the line given in (a) and passing through $(-1,7) .$ Plot the line on the same set of axes.

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Problem 42

(a) Find the slope of the line whose equation is $3 x+7 y+42=0$. Find the $x$ and $y$ -intercepts of the line. Plot the line. (b) Find the equation of the line parallel to the line given in (a) and four units above it. Plot the line on the same set of axes.

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Problem 43

(a) Find an equation for the horizontal line passing through the $y$ -intercept of the line in $41(\text { a). }(b)$ Find an equation for the vertical line passing through the $x$ -intercept of the line in $41(b).$

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Problem 44

(a) Find an equation for the horizontal line passing through the $y$ -intercept of the line in $42(a) .$ (b) Find an equation for the vertical line passing through the $x$ -intercept of the line in $42(b).$

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Problem 45

Find the equations of two lines parallel to the line $y=-3,$ and 4 units from it.

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Problem 46

Find the equations of two lines parallel to $x=2$, and 6 units from it.

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Problem 47

Find the slope of the line passing through each pair of points.
$$(1,-2) \text { and }(1,-1.4).$$

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Problem 48

Find the slope of the line passing through each pair of points.
$$(2,-9) \text { and }(12,5).$$

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Problem 49

Find the slope of the line passing through each pair of points.$$(-1 / 4,2 / 5) \text { and }(0,0).$$

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Problem 50

Find the slope of the line passing through each pair of points.$$(1,4) \text { and }(2,4).$$

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Problem 51

Find the slope of the line passing through each pair of points.$$(12,16) \text { and }(12,-73).$$

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Problem 52

Find the slope of the line passing through each pair of points.
$$(0,3) \text { and }(-6,0).$$

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Problem 53

Find the slope of the line passing through each pair of points.
$$(1 / 2,-2) \text { and }(1 / 4,-1 / 4).$$

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Problem 54

Find the slope of the line passing through each pair of points.
$$(0,-9) \text { and }(1 / 2,3).$$

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Problem 55

Find the slope of the line passing through each pair of points.
$$(-1 / 3,2 / 3) \text { and }(0,0).$$

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Problem 56

Find the slope of the line passing through each pair of points.
$$(1,-5) \text { and }(2,-5).$$

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Problem 57

Find the slope of the line passing through each pair of points.
$$(1 / 2,16) \text { and }(1 / 2,-73).$$

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Problem 58

Find the slope of the line passing through each pair of points.
$$(0,4) \text { and }(-7,0).$$

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Problem 59

Find the equations of the lines and plot the lines from Exercise $47$.

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Problem 60

Find the equations of the lines and plot the lines from Exercise $48$.

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Problem 61

Find the equations of the lines and plot the lines from Exercise $49$.

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Problem 62

Find the equations of the lines and plot the lines from Exercise $50$.

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Problem 63

Find the equations of the lines and plot the lines from Exercise $51$.

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Problem 64

Find the equations of the lines and plot the lines from Exercise $52$.

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Problem 65

Find the equations of the lines and plot the lines from Exercise $53$.

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Problem 66

Find the equations of the lines and plot the lines from Exercise $54$.

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Problem 67

Find the equations of the lines and plot the lines from Exercise $55$.

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Problem 68

Find the equations of the lines and plot the lines from Exercise $56$.

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Problem 69

Find the equations of the lines and plot the lines from Exercise $57$.

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Problem 70

Find the equations of the lines and plot the lines from Exercise $58$.

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Problem 71

Determine an equation for the line parallel to $y=3 x-7$ and passing through the point $(1,-5)$.

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Problem 72

Determine an equation for the line (a) parallel (b) perpendicular to $2 x-5 y=9$ and passing through the point $(-2,-4)$.

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Problem 73

Determine an equation for the line (a) parallel (b) perpendicular to $3 x+7 y=11$ and passing through the point $(1,-3)$.

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Problem 74

(a) Plot the line $4 x+6 y+12=0$. Find the area of the triangle formed by the line, the $x$ -axis and the $y$ -axis. (b) Repeat for $A x+B y+C=0$ for $A$
$B, C$ positive.

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Problem 75

Find an equation of a line whose $y$ -intercept is 4 and such that the area of the triangle formed by the line and the two axes is 20 square units.( Two possible answers.)

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Problem 76

The area of a triangle formed by a line and the two axes is 40 and the slope of the line is $-5 .$ Find an equation for the line. (Two possible answers.)

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Problem 77

(a) Find the length of the portion of the line $5 x+12 y=84$ that is cut off by the two axes. (b) Repeat for $A x+B y+C=0$ for $A, B, C$ positive.

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Problem 78

(a) Plot the points (-1,-7),(4,2) and $(8,4) .$ (b) Do they lie on the same line? (c) How can you tell without plotting?

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Problem 79

In 1990 the Massachusetts Non-Resident State Income Tax calls for a tax of $5 \%$ on earned income and $10 \%$ on unearned income. Suppose a person has total income of $\$ 40,000 dollars of which amount $x$ is earned. Find her $\operatorname{tax}, t,$ as a function of $x$.

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Problem 80

When the price for a color television is $\$ 240,$ the average monthly sales for this item at a department store is $450 .$ For each $\$ 10$ increase in price, the average monthly sales fall by 20 units. What is the average monthly sales if the price is $\$ 400$ per color television?

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Problem 81

When the price is $\$ 50$ per radio, a producer will supply 100 radios each month for sale. For each $\$ 2$ increase in price the producer will supply an additional 6 radios. How many radios are supplied if their per unit price is $\$ 72 ?$

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Problem 82

Plot each of the following lines on the same set of axes. (a) $y=2 x$
(b) $y=2(x-3)$ (c) $y=2(x+3)$ (d) How are these lines related?

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Problem 83

Plot each of the following lines on the same set of axes. (a) $y=2 x$
(b) $y-4=2 x(\text { c) } y+4=2 x$ (d) How are these lines related?

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Problem 84

Plot each of the following lines on the same set of axes. (a) $y=2 x$
(b) $y-4=2(x-3)$ (c) $y+4=2(x-3)$ (d) $y-4=2(x+3)$
(e) $y+4=2(x+3)$ (f) How are these lines related?

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Problem 85

In general, how are the lines $y=m x+b$ and $y-k=m(x-h)+b \mathrm{re}-$ lated? ( $m, b, h,$ and $k$ are constants.)

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Problem 86

Show if $a b \neq 0,$ then the line with intercepts $(a, 0)$ and $(0, b)$ has the equa$\operatorname{tin} \frac{x}{a}+\frac{y}{b}=1$.

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Problem 87

Using the previous exercise, determine the equation of the line with intercepts (a) (3,0),(0,6) (b) (2,0),(0,-4) (c) $(1 / 2,0),(0,2 / 3)$

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Problem 88

Find the point on the line $y=2 x+3$ that is equidistant from the points $(-5,6)$ and $(0,0)$.

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Problem 89

Given the two parallel lines $y=m x+b$ and $y=m x+B,$ determine the perpendicular distance between these two lines.

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Problem 90

Let $A\left(x_{1}, y_{1}\right)$ and $B\left(x_{2}, y_{2}\right)$ be in any two points in the plane. (a) Plot these points, (b) Obtain the right triangle formed by drawing a horizontal line from $A$ and a vertical line through $B$. What are the coordinates of the point at which these two lines intersect? (c) Using the theorem of Pythagoras, derive the distance formula.

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