BY DIFFERENTIALS

approximation of a function. Since this material is the essence of what calculus is

about, it is included in this chapter alongside the fundamental concepts of calculus.

Recall that for a linear function f(x) = mx + b, the derivative f™(x) = m gives

the slope of the graph off and measures the rate of change or marginal change of

f: the increase in the value off for every unit increase in the value of J.

Let™s carry over this marginal analysis to nonlinear functions. After all, this

was one of the main reasons for defining the derivative of such an f In formulating

the analytic definition of the derivative off, we used the fact that the slope of the

tangent line to the graph at (q, f(q))1s well approximated by the slope of the

secant line through (xu, f(q)) and a nearby point (1” + h, f(xn + h)) on the graph.

In symbols,

fh + h) - fh) = f'cxoj

(11)

h

for h small, where = means “is well approximated by” or “is close in value to.”

If WC set h = 1 in (1 I), then (11) becomes

f(xo + 1) - fh) = f™ko); (12)

in words, the derivative off at xc, is a good approximation to the marginal change

of /™ at x,,. Of course, the less curved the graph of f at xc,, the better is the

approximation in (12).

Emmpte 2. I I Consider the production function F(x) = f&. Suppose that the

firm is currently using 100 units of labor input A-, so that its output is 5 units.

The derivative of the production function F at I = 100;

= ;lOO˜ I” = h = 0.025,

F™(l00)

is a good rnc˜asurc of the uddirional output that can be achievedby hiring one

more unit oflahor. the marginal product of labur. The actual increase in output

is F( ](!I) F( ltltl) = 0.02494 ., pretty close to 0.025.

Even though it is not eructl,: the increase in )˜ = F(x) due to a one unit increase

in x. economists still USC F™(x) as the marginal change in F because it is easier

10 work with the single term P(x) than with the difference b-(x + I) - F(x) and

hccause using the simple turn F™(x) avoids the question of what unit to use to

measure ii one unit increase in x.

What if the change in the amount of input ,x is not exactly one unit? Return to

(I I) and substitute Ax, the exact change in .x, for /I. Multiplying (I I) out yields:

sy = f(x,, f Ax) f(x,,) = f yx,+x, (13)

or f(x,, + ill) = f(.X(,) + f™(x&r, (14)

36 CALCULUS : 121

ONE-VARIABLE FOUNDATIONS

where we write Ay for the exact change in y = f(x) when x changes by Ax. Once

again, the less curved the graph and/or the smaller the change Ax in x, the better

the approximation in (13) and (14).

Example2.12 Consider again a firm withproductionfunctiony = i$. Suppose

it cuts its labor forcen from 900 to 896 units. Let™s estimate the change in output

Ay and the new output y at x = 896. We substitute

F(X) = 1,v2 X” = 900, and Ax = -4

z ™

into (13) and (14) and compute that

F™@) = ax- I/? and F™(YO0) =4 &=&

By (13), output will decrease by approximately

I 1 1

F™(x,)Ax = * 4 = 30 units.

By (14), the new output will be approximatel)

E(900) + P(900)(-4) = 1s - & = 14; = 14.Y666.

i The actual new output is Q896) = 14.9663 ˜: once again the approximation

1 derivatives is a good one.

by

From a mathematical point of view, we can consider (14) as an eflective

way of approximating f(x) for x close to some xg where f(q) and f ˜(xc,) are

easily computed. For example, in Example 2.12, we computed &/%. using our

familiarity with i,/% = 15.

Exumple 2.13 Let™s use (14) to estimate the cube root of 1001.5. We know that

the cube root of 1000 is IO. Choose f(x) = xl”, x,1 = 1000, and Ax = + 1.5.

) Then,

f™(m) = ;x-˜˜/˜ and f™(l000) = ;(IOOO)-˜I™ = A.

I

Therefore

f(lOOl.5) = f(l000) + f™(lOO0)˜ 1.5 = 10 + g = 10.005.

I

close to the true value 10.004YYH˜ of &iiiii?

L2.71 APPROXIMATION 8˜ DIFFERENTIALS 37

Equations (13) and (14) are merely analytic representations of the geometric

fact that the tangent line P to the graph of y = f(x) at (˜0, I) is a good

approximation to the graph itself for x near x0. As Figure 2.15 indicates, the

left-hand sides of (13) and (14) pertain to movement along the graph off, while

the right-hand sides pertain to movement along the tangent line 4, because the

equation of the tangent line, the line through the point (x0, f(x”)) with slope f ˜(x0),

is

= f(4 + f™h)(x- -xc,) = f@o) + f™kMx

Y

Continue to write Ay for the actual change in f as x changes by Ax, that is, for

the change along the graph off, as in Figure 2.15. Write dy for the change in y

along the tangent line JZ as x changes by Ax. Then, (13) can be written as

Ay = dy = f ˜(xo)Ax.

We usually write dx instead of Ax when we are working with changes along the

tangent line, even though Ax is equal to dx. The increments dy and dx along the

tangent line 1 are called differentials. We sometimes write the differential df in

place of the differential dy. The equation of differentials

f

df = ˜(x,,)dx or dy = f™(x,j)dx

for the variation along the tangent line to the graph off gives added weight to the

notation g for the derivative f™(x).

Figure

Comparing dy and Ay, 2.1s

EXERCISES

2.22 Suppose that the total cost of manufacturing x units of a certain commodity is

C(x) = 2x™ + 6.x + 12. Use differentials to approximate the cost of producing the

21s˜ unit. Compare this estimate with the cost of actually producing the 21st unit.

A manufacturer™s total cost is C(x) = 0. Ix? 0.25˜™ + 300x + 100 dollars, where

2.23

x is the level of production. Estimate the effect on the total cost of an increase in the

level of production from 6 to 6.1 units.

2.24 It is estimated that f years from now, the population of a certain town will he

F(r) = 40 - [S/(t + 2)]. Use differentials to estimate the amount by which the