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function f. Linear functions have the same rate of change no matter where one

starts.

For example, if x measures time in hours. if y = f(x) is the number of

kilometers travclcd in I hours, and f is linear, the slope off measures the number

of kilomctcrs traveled euctz hour. that is, the speed or velocity of the object under

study in kilometers per hour.

This view of the slope of a linear function as its rate of change plays a key

role in economic analysis. If C = I(y) is a linear cost function which gives the

total cost C of manufacturing y units of output, then the slope of F measures the

increase in the total manufacturing cost due to the production of one more unit.

In effect, it is the cust of making one more unit and is called the marginal cost.

It plays a central role in the hchavior of profit-maximizing firms. If u = U(x) is

LINEAR FVNCTIONS 21

12.21

a linear utility function which measures the utility u or satisfaction of having an

income ofx dollars, the slope of U measures the added utility from each additional

marginal utility of income. If y = G(z) is a linear

dollar of income. It is called the

function which measures the output y achieved by usingz units of labor input, then

its slope tells how much additional output can be obtained from hiring another unit

of labor. It is called the marginal product of labor. The rules which characterize

the utility-maximizing behavior of consumers and the profit-maximizing behavior

of firms all involve these marginal measwcs, since the decisions about whether

or not to consume another unit of some commodity or to produce another unit of

output are hascd not so much on the total amount consumed or produced to date,

but rather on how the next item consumed will affect total satisfaction or how the

next itenl produced will affect revenue, cost, and profit.

EXERCISES

2.7 Estimate tht: slqr of Ihe liner in Figure 2.7.

Figure

2.7

ONE-VARIABLE CALCULвЂќS: FOVNDATlONS I21

22

THE SLOPE OF NONLINEAR FUNCTIONS

2.3

We have just seen that the slope of a linear function as a measure of its marginal

effect is a key concept fbr liner functions in economic theory. However, nearly

all functions which arise in applications are nonlinear ones. How do we measure

the marginal effects of these nonlinear functions?

f(x)and thatcurrently

Supposethatwearcstudyingthcnonlinearfuncliony =

we are at the point (x,), f(,qI)) on the graph of f, as in Figure 2.8. We want tu

measure the mte of change off or the steepness of the graph off when x = x,,.

A natural solution to this problem is to draw the tangent line to the graph off at

x0, as pictured in Figure 2.8. Since the tangent line very closely approximates the

f(q)),

graph off around (q, it is a good proxy for the graph of / itself. Its slope.

which we know how to measure, should really he a good measure for the slope

of the nonlinear function at .Y,,. We note that for nonlinear functions, unlike linear

functions, the slope of the tangent line will vary from point to point.

We use the notion of the tangent lint approximation to a graph in our daily

lives. For example, contractws who plan to build a large mall or power plant and

farmers who want to subdivide large plots of land will generally assume that they

are working on a flut pluw, even though they know that they are working on

a rather round˜lurwr. In effect, they arc working with the tangent plane to the

earth and the computations that they make on it will hc exact to IO or 20 decimal

places-easily close enough for their purposes.

f at a

So, dcfinc the slope of a nonlinear function point (.a,, /(xi,)) on its

WC

graph as the slope of the tangent lint to the graph off at that point. We call the

at (x,,. lhe derivative of / iit TV,,.

,f ,f(,q,))

slope of the tangent lint to the graph uf

and we write it as

r2.31 THE SLOPE OF NONLINEAR FUNCTIONS 23

The latter notation comes from the fact that the slope is the change in f divided by

the change in x, or Af/Ax, where we follow the convention of writing a capital

Greek delta A to denote change.

Since the derivative is such an important concept, we need an analytic definition

that we can work with. The first step is to make precise the definition of the tangent

line to the graph off at a point. Try to formulate just such a definition. It is not

вЂњthe line which meets the graph off in just one point,вЂќ because point A in Figure

2.9 shows that we need to add more geometry to this first attempt at a definition.

We might expand our first attempt to вЂњthe line which meets the graph off at just

one point, hut does not cross the graph.вЂќ However, the x-axis in Figure 2.9 is the

true tangent line to the graph of y = x3 at (0, 0), and it does indeed cross the graph

of xвЂ™. So, we need to he yet more subtle.

Figure

Unfortunately, the only way to handle this problem is to use a limiting process.

First. recall that a line segment joining two points on a graph is called a secant

line. Now. back off a hit from the point (xi,, f(q,)) on the graph off to the point

(x,, + /I,, f(xo + h,)), where IJ, is some small number. Draw the secant line JZ, to

the graph joining these two points. as in Figure 2.10. Line 4, is an approximation

to the tangent line. By choosing the second point closer and closer to (x0, f(q)),

we will be drawing better and better approximations to the desired tangent line.

So. choose h: closer to zero than h, and draw the secant line JZ, of the graph of

j joining (xii, j(xll)) and (x0 + !I-. f(x,, + h?)). Continue in this way choosing a

sequence {h,,) of small numbers which converges monotonically to 0. For each n,

draw the secant line A?,, through the two dkrinct points on the graph (q, f (x0)) and

(q + h,,, f(q + h,,)). The secant lines {g,,} geometrically approach the tangent line

to the graph off at (Q, f(q)); and their slopes approach the slope of the tangent

line. Since <,, passes through the two points (xi,, f(q)) and (x,1 + h,,, f(xo + h,)),

if this limit exists. When this limit does exist, we say that the function ˜[ is

differentiable at ,r,, wjith derivative f '(xl,).

I2.41 COMPUTING DERIвЂќAT,вЂњES 2 5

2.4 COMPUTING DERIVATIVES

LetвЂ™s use formula (4) to compute the derivative of the simplest

Fxarnplr 2.4

nonlinear function, f(x) = IвЂ™, at the point x,, = 3. Since the graph of x2 is

fairly steep at the point (3, 9) as indicated in Figure 2. I I, we expect to find fвЂ™(3)

considcrahly larger than 1. For a sequence of h,,вЂ˜s converging to zero. choose

the sequence

[/I,,} = 0. I , 0 . 0 1 . 0.001.. .) ( 0 . I ) вЂњ , (5)

Table 2. I summarizes the computations we need to make

As /I,, - 0, the quotient in the Iat column ofTable 2. I approacha 6. Therefore.

the slope of the tangent line of the graph of j(.r) = .xвЂ™ af the point (3, 9) is 6; that

is. fвЂ™(3) = h.

Table

2.1

ONE˜VARlABLE cALCвЂќLUS: FOUNDATIONS [21

26

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