Museos y enseñanza

The very definition of the derivative assumes that arbitrarily small increments in the in- dependent variable are possible. In practical problems it is impossible to implement, or even measure, arbitrarily small changes in the variable. For example, economic quantities that vary with time, such as a nation’s domestic product or the number of its people who are employed, are usually measured at intervals of days, weeks, or years. Further, the cost functions of the type we discussed in Example 2 are often defined only for integer values of x. In all these cases, the variables only take discrete values. The graphs of such func- tions, therefore, will only consist of discrete points. For functions of this type in which time and numbers both change discretely, the concept of the derivative is not defined. To remedy this, the actual function is usually replaced by a “smooth” function that is a “good approximation” to the original function.

1928 1929 30 000 20 000 10 000 Unemployment t _{1928 1929} 30 000 20 000 10 000 Unemployment t

Figure 1 Unemployment in Norway

(1928–1929)

Figure 2 A smooth curve approximating

As an illustration, Fig. 1 graphs observations of the number of registered unemployed in Norway for each month of the years 1928–1929. This was a period in which Norway was still largely an agricultural economy, in which unemployment rose during the autumn and fell during the spring. In Fig. 2 we show the graph of a “smooth” function that approximates the points plotted in Fig. 1.

P R O B L E M S F O R S E C T I O N 6 . 4

1. Let C(x)_{= x}2_{+ 3x + 100 be the cost function of a firm. Show that the average per unit rate} of change when x is changed from 100 to 100_{+ h is}

C(100_{+ h) − C(100)}

h = 203 + h (h̸= 0)

What is the marginal cost C′_{(100)? Use (6.2.6) to find C}′_{(x)and, in particular, C}′_(100).

2. If the cost function of a firm is C(x)_{= kx + I, give economic interpretations of the parameters} kand I .

3. If the total saving of a country is a function S(Y ) of the national product Y , then S′_{(Y )is called} the marginal propensity to save (MPS). Find the MPS for the following functions:

(a) S(Y )_{= a + bY} (b) S(Y )_{= 100 + 0.1Y + 0.0002Y}2

4. If the tax a family pays is a function of its income y given by T (y), then T′_{(y)is called the}

marginal tax rate. Characterize the following tax function by determining its marginal rate: T (y)= ty (tis a constant number in (0, 1))

5. Let x(t) denote the number of barrels of oil left in a well at time t, where time is measured in minutes. What is the interpretation of_{˙x(0) = −3?}

6. The total cost of producing x units of a commodity is C(x)_{= x}3_{− 90x}2_{+ 7500x, x ≥ 0.} (a) Compute the marginal cost function C′_{(x). (Use the result in Problem 6.2.9.)}

(b) For which value of x is the marginal cost the least?

7. (a) The profit function is π(Q)_{= 24Q − Q}2_{− 5. Find the marginal profit, and find the value} Q∗of Q that maximizes profits.

(b) The revenue function is R(Q)_{= 500Q −}1

3Q3. Find the marginal revenue.

(c) Find marginal cost when C(Q)_{= −Q}3_{+ 214.2Q}2_{− 7900Q + 320 700. (This particular} cost function is mentioned in Example 4.7.1.)

8. Refer to the definition given in Example 3. Compute the marginal cost in the following two cases:

(a) C(x)_{= a}1x2+ b1x+ c1 (b) C(x)_{= a}1x3+ b1

S E C T I O N 6 . 5 / A D A S H O F L I M I T S 169

6.5 A Dash of Limits

In Section 6.2 we defined the derivative of a function based on the concept of a limit. The same concept has many other uses in mathematics, as well as in economic analysis, so now we should take a closer look. Here we give a preliminary definition and formulate some important rules for limits. In Section 7.9, we discuss the limit concept more closely.

E X A M P L E 1 Consider the function F defined by the formula F (x)=e

x _{− 1}

x

where e≈ 2.7 is the base for the natural exponential function. (See Section 4.9.) Note that if x= 0, then e0 _{= 1, and the fraction collapses to the absurd expression “0/0”. Thus, the}

function F is not defined for x = 0, but one can still ask what happens to F (x) when x is close to 0. Using a calculator (except when x= 0), we find the values shown in Table 1.

Table 1 Values of F (x)_{= (e}x_{− 1)/x when x is close to 0}

x −1 −0.1 −0.001 −0.0001 0.0 0.0001 0.001 0.1 1

F (x) 0.632 0.956 0.999 1.000 ∗ 1.000 1.001 1.052 1.718

∗_{not defined}

From the table it appears that as x gets closer and closer to 0, so the fraction F (x) gets closer and closer to 1. It therefore seems reasonable to assume that F (x) tends to 1 in the limit as x tends to 0. We write2

lim x→0 ex_{− 1} x = 1 or ex_{− 1} x → 1 as x → 0

Figure 1 shows a portion of the graph of F . The function F is defined for all x, except at x = 0, and limx→0F (x) = 1. (A small circle is used to indicate that the corresponding

point (0, 1) is not in the graph of F .)

#2 #1 1 2 3 2 1 y x ex _{# 1} x F(x) ! Figure 1

Suppose, in general, that a function f is defined for all x near a, but not necessarily at x = a. Then we say that f (x) has the number A as its limit as x tends to a, if f (x) tends to A as x tends to (but is not equal to) a. We write

lim

x→af (x)= A or f (x)→ A as x → a

It is possible, however, that the value of f (x) does not tend to any fixed number as x tends to a. Then we say that limx→af (x)does not exist, or that f (x) does not have a limit as x

tends to a.

E X A M P L E 2 Examine the limit lim

h→0

√

h+ 1 − 1

h using a calculator.

Solution: By choosing numbers h close to 0, we find the following table:

Table 2 Values of F (h)_{= (}√h+ 1 − 1)/h when h is close to 0

h −0.5 −0.2 −0.1 −0.01 0.0 0.01 0.1 0.2 0.5

F (h) 0.586 0.528 0.513 0.501 ∗ 0.499 0.488 0.477 0.449 ∗_{not defined}

This suggests that lim

h→0

√

h+ 1 − 1

h = 0.5.

The limits we claim to have found in Examples 1 and 2 are both based on a rather shaky numerical procedure. For instance, in Example 2, can we really be certain that our guess is correct? Could it be that if we chose h values even closer to 0, the fraction would tend to a limit other than 0.5, or maybe not have any limit at all? Further numerical computations will support our belief that the initial guess is correct, but we can never make a table that has all the values of h close to 0, so numerical computation alone can never establish with certainty what the limit is. This illustrates the need to have a rigorous procedure for finding limits, based on a precise mathematical definition of the limit concept. This definition is given in Section 7.8, but here we merely give a preliminary definition which will convey the right idea.

Writing limx→af (x) = A means that we can make f (x) as close to A as we

want for all x sufficiently close to (but not equal to) a. (1)

We emphasize:

(a) The number limx→af (x)depends on the value of f (x) for x-values close to a, but not

on how f behaves at the precise value of x = a. When finding limx→af (x), we are

simply not interested in the value f (a), or even in whether f is defined at a.

(b) When we compute limx→af (x), we must take into consideration x-values on both

S E C T I O N 6 . 5 / A D A S H O F L I M I T S 171