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If only one endpoint is included, the interval is called half-open (or half-closed)

and written as (a, b] or [a, b). There are also five kinds of infinite intervals:

(a, =) = (x E RвЂ™ : x > a},

[u, x) = {x E RвЂ™ : x 2 a},

(-x, a) = (x E RвЂ™ : x < a},

(-=, a] = (x E RвЂ™ : x 5 a),

(-2, +x) = RвЂ™.

EXERCISES

For each of the following functions, plot enough points to sketch a complete graph.

2.1

Then answer the following questions:

ui) Whcrc is the function increasing and where is it decreasing?

h) Find the local and glahal maxima and minima of these functions:

i) y = 3x 2: ii) y = -2x; iii) y = 2 + 1;

ii,) y = 1 + x: v) y = x3 x: vi) y = 1x1.

In economic models. it is natural to assume that total cost functions are increasing

2.2

functions of output. since more output requires more input, which must he paid for.

Name two more types of functions which arise in economics models and are naturally

L2.21 LINCAR FUNCT,вЂњNS 1 7

Figure

2.5

Figure

Computing the slope of line l? three ways.

2.6

This use of two arbitrary points of a line to compute its slope leads to the following

most general definition of the slope of a line.

Let (x0, yo) and (XI, yj) be arbitrary points on a line e. The ratio

Definition

m YI - Yu

XI XII

is called the slope of line 2. The analysis in Figure 2.6 shows that the slope of X

is independent of the two points chosen on 2. The same analysis shows that two

lines are parallel if and only if they have the same slope.

Example 2.2 The slope of the line joining the points (4,6) and (0,7) is

I

This line slopes downward at an angle just less than the horizontal. The slope

of the line joining (4, 0) and (0, 1) is also I /4; so these two lines are parallel.

1 2 . 2 1 LINEAKFVNCTIONS 19

The Equation of a Line

We next find the equation which the points on a given line must satisfy. First,

suppose that the line 4 has slope m and that the line intercepts the y-axis at the

point (0. h). This point (0, h) is called the y-intercept of P. Let (I, y) denote an

arbitrary point on the line. Using (1, y) and (0, h) to compute the slope of the lint,

we conclude that

or y - h = mx; y = mx + b.

that is,

The following theorem summarizes this simple calculation.

The orem 2 .1 The line whose slope is no and whose y-intercept is the point

(0, h) has the equation y = mr + h.

Polynomials of Degree One Have Linear Graphs

b. Its graph is

Now,. consider the general polynomial of degree one fix) = mx +

b. Given any two

the locus of all points (I, y) which satisfy the equation y = vzx +

points (.r,, J;,) and (x2, yz) on this graph, the slope of the line connecting them is

Since the slope of this locus is )?I everywhere. this Incus describes a straight line.

One checks directly that its y-intercept is h. So, polynomials of degree one do

indeed have straight lines as their graphs. and it is natural to call such functions

linear functions.

In applications. wmctimcs need to construct the formula of the linear

WC

function from given analytic data. For cxamplc. by Thcorcm 2. I, the lint with

b) has equation y = nz.r + h. What is the equation of

slops ,n and x-intercept (0,

the lint with slope wz which passes through a ˜more general point, say (xc,, ye)? As

in the proof˜˜f Thcorcm 2. I USC the given point (Q, y,,) and ii gcncric point on the

lint (TV, y) to compute the slope of the line:

It follows that the equation of the given line is y = !n(x- -˜ x1,) + y,,, or

,I = 171x + (!, mx,,). (3

20 ONE˜вЂњARlABLECALCVLUS:FOVI\вЂ˜DATtO˜S 121

If, instead, we are given two points on the line, say (nil, yO) and (xl, y,), we can

use these two points to compute the slope m of the line:

We can then substitute this value form in (3).

Example 2 . 3 Let x denote the temperature in degrees Centigrade and let y denote

the temperature in degrees Fahrenheit. We know that x andy are linearly related,

that O0 Centigrade OI 32вЂ™ Fahrenheit is the freezing temperature of water and

that 100вЂќ Centigrade or 212вЂ™ Fahrenheit is the boiling temperature of water. To

find the equation which relates degrees Fahrenheit to degrees Centigrade, we

find the equation of the line through the points (0, 32) and (100,212). The slope

of this line is

212-32 180 9

100-O 100 JвЂ™

This means that an increase of lo Centigrade corresponds to an increase of

Y/SвЂќ Fahrenheit. Use the slope 9/j and the point (932) to express the linear

relationship:

v-32 Y

вЂњ1 J = вЂњx+ 72

5 -вЂ™

x-0 s

Interpreting the Slope of a Linear Function

The slope of the graph of a linear function is a key concept. We will simply call

it the slope of the linear function. Recall that the slope of a line measures how

much y changes as one moves along the line increasing x by one unit. Therefore,

the slope of a linear function f measurer how much f(.r) increases for each unit

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