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Derivatives of Exponential Functions and the Number e

One of the most important constants in mathematics is the number e, whose value is approximately 2.71828. In this section we introduce e and show that it has the following remarkable properties.

(1)    The function y = ex is equal to its own derivative.

(2)    e is the limit 08_exp-log_functions-88.gif

Either property can be used as the definition of e. Because of property 1, it is convenient in the calculus to use exponential and logarithmic functions with the base e instead of 10. However, it is not at all easy to prove that such a number e exists. Before going into further detail we shall discuss these properties intuitively.

A function which equals its own derivative may be described as follows. Imagine a point moving on the (x, y) plane starting at (0, 1). The point is equipped with a little man and a steering wheel which controls the direction of motion of the point. The man always steers directly away from the point (x - 1,0), so that

08_exp-log_functions-89.gif

Then the point will trace out a curve y = f(x) which equals its own derivative, as in Figure 8.3.1.

08_exp-log_functions-90.gif

Figure 8.3.1

Another intuitive description is based on the example of the population growth function y = at. If the birth rate minus the death rate is equal to one, then the derivative of at is at, and a is the constant e. Imagine a country with one million people (one unit of population) at time f = 0 which has an annual birth rate of one million births per million people, and zero death rate. Then after one year the population will be approximately e million, or 2,718,282. (This high a growth rate is not recommended.)

The limit e = limx→∞ (1 + 1/x)x is suggested intuitively by the notion of continuously compounded interest. Suppose a bank gives interest at the annual rate of 100%, and we deposit one dollar in an account at time t = 0. If the interest is compounded annually, then after t = 1 year our account will have 2 dollars. If the interest is compounded quarterly (four times per year), then our account will grow to 1 + dollars at time t = , (1 + )2 dollars at time t = j, and so on. After one year our account will have (1 + )4 ~ 2.44 dollars. Similarly, if our account is compounded daily then after one year it will have (1 + 1/365)365 dollars, and if it is compounded n times per year it will have (1 + 1/n)n dollars after one year.

Table 8.3.1 shows the value of (1 + 1/n)n for various values of n. (The last few values can be found with some hand calculators.)

Table 8.3.1

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This table strongly suggests that the limit e = limx→∞ (1 + 1/x)x exists. A proof will be given later. Thus for H positive infinite,

08_exp-log_functions-92.gif

If the interest is compounded H times per year, then in t years each dollar will grow to

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Thus if the 100% interest is continuously compounded, each dollar in the account grows to et dollars in t years. At the interest rate r, each dollar in a continuously compounded account will grow to ert dollars in t years. For more information, see Section 8.4. We now turn to a detailed discussion of e.

LEMMA

The limit 08_exp-log_functions-94.gif exists.

We shall save the proof of this lemma for the end of the section.

DEFINITION

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As we have indicated before, e has the approximate value

e ~ 2.71828.

The function y = ex is called the exponential function and is sometimes written y = exp x.

THEOREM 1

e is the unique real number such that

08_exp-log_functions-96.gif

PROOF

Our plan is to show that whenever t and t + Δt are finite and differ by a nonzero infinitesimal Δt,

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We may assume that t is the smaller of the two numbers, so that Δt is positive. By the rules of exponents,

(1)

08_exp-log_functions-98.gif

Let 08_exp-log_functions-99.gif

(2) Then

08_exp-log_functions-100.gif

Since ex is continuous and e0 = 1, we see from Equation 2 that b Δt is positive infinitesimal. Thus H = 1/b Δt is positive infinite. From Equation 2,

08_exp-log_functions-101.gif

Taking standard parts,

08_exp-log_functions-102.gif

Therefore st(b) = 1, and by Equation 1,

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We conclude that for real x,

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It remains to prove that e is the only real number with this property. Let a be any positive real number different from e, a ≠ e. We may then differentiate ax by the Chain Rule.

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Since a ≠ e, loge a ≠ 1, so

(d(ax))/dx ≠ ax.

Since ex is its own derivative, it is also its own antiderivative. We thus have a new differentiation formula and a new integration formula which should be memorized.

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We are now ready to plot the graph of the exponential curve y = ex. Here is a short table. It gives both the value y and the slope y', because y = y' = ex.

x

-2

-1

0

1

2

ex

1/e2 ~ 0.14

1/e ~ 0.37

1

e ~ 2.7

e2 ~ 7.39

The number ex is always positive, and y, y', and y" all equal ex. From this we can draw three conclusions.

y = ex > 0 the curve lies above the x-axis,

y' = ex > 0 increasing,

y" = ex > 0 concave upward.

If H is positive infinite, then by Rule (vii),

eH ≥ 1 + H(e - 1).

So

eH is infinite, e-H = 1/eH is infinitesimal.

Therefore,

limx→∞ ex = ∞, limx→-∞ ex = 0.

We use this information to draw the curve in Figure 8.3.2.

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Figure 8.3.2


Last Update: 2010-11-25