6.2

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6.2 Exponential Functions and

Their Derivatives

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Exponential Functions and Their Derivatives

The function f(x) = 2^x is called an exponential function because the variable, x, is the exponent. It should not be confused with the power function g(x) = x², in which the variable is the base.

In general, an exponential function is a function of the form

f(x) = b^x

where b is a positive constant. Let’s recall what this means.

If x = n, a positive integer, then

Exponential Functions and Their Derivatives

If x = 0, then b⁰ = 1, and if x = –n, where n is a positive integer, then

If x is a rational number, x = p /q, where p and q are integers and q > 0, then

But what is the meaning of b^x if x is an irrational number?

For instance, what is meant by or 5^π ?

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Exponential Functions and Their Derivatives

To help us answer this question we first look at the graph of the function y = 2^x, where x is rational. A representation of this graph is shown in Figure 1.

We want to enlarge the domain of y = 2^x to include both rational and irrational numbers.

There are holes in the graph in

Figure 1 corresponding to irrational values of x.

We want to fill in the holes by defining f(x) = 2^x, where x ∈ , so that f is an increasing function.

Figure 1

Representation of y = 2^x, x rational

Exponential Functions and Their Derivatives

In particular, since the irrational number satisfies

we must have

and we know what 2^1.7 and 2^1.8 mean because 1.7 and 1.8 are rational numbers.

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Exponential Functions and Their Derivatives

Similarly, if we use better approximations for we obtain better approximations for

Exponential Functions and Their Derivatives

It can be shown that there is exactly one number that is greater than all of the numbers

2^1.7, 2^1.73, 2^1.732, 2^1.7320, 2^1.73205, … and less than all of the numbers

2^1.8, 2^1.74, 2^1.733, 2^1.7321, 2^1.73206, … We define to be this number. Using the preceding approximation process we can compute it correct to six decimal places:

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Exponential Functions and Their Derivatives

Similarly, we can define 2^x (or b^x, if b > 0) where x is any irrational number.

Figure 2 shows how all the holes in Figure 1 have been filled to complete the graph of the function

f(x) = 2^x, x ∈

Figure 2

y = 2^x, x real

Figure 1

Representation of y = 2^x, x rational

Exponential Functions and Their Derivatives

In general, if b is any positive number, we define

This definition makes sense because any irrational number can be approximated as closely as we like by a rational

number.

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Exponential Functions and Their Derivatives

For instance, because has the decimal representation

= 1.7320508 . . . , Definition 1 says that is the limit of the sequence of numbers

2^1.7, 2^1.73, 2^1.732, 2^1.7320, 2^1.73205, 2^1.732050, 2^1.7320508, . . . Similarly, 5^π is the limit of the sequence of numbers

5^3.1, 5^3.14, 5^3.141, 5^3.1415, 5^3.14159, 5^3.141592, 5^3.1415926, . . . It can be shown that Definition 1 uniquely specifies b^x and makes the function f(x) = b^x continuous.

Exponential Functions and Their Derivatives

The graphs of members of the family of functions y = b^x are shown in Figure 3 for various values of the base b.

Member of the family of exponential functions

Figure 3

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Exponential Functions and Their Derivatives

Notice that all of these graphs pass through the same point (0, 1) because b⁰ = 1 for b ≠ 0. Notice also that as the base b gets larger, the exponential function grows more rapidly (for x > 0).

Figure 4 shows how the exponential function y = 2^x compares with the power function y = x².

Figure 4

Exponential Functions and Their Derivatives

The graphs intersect three times, but ultimately the

exponential curve y = 2^x grows far more rapidly than the parabola y = x². (See also Figure 5.)

Figure 5

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Exponential Functions and Their Derivatives

There are basically three kinds of exponential functions y = b^x.

If 0 < b < 1, the exponential function decreases; if b = 1, it is a constant; and if b > 1, it increases.

Exponential Functions and Their Derivatives

These three cases are illustrated in Figure 6.

Because (1/b)^x = 1/b^x = b^–x, the graph of y = (1/b)^x is just the reflection of the graph of y = b^x about the y-axis.

y = b^x, 0 < b < 1

Figure 6(a) Figure 6(b) Figure 6(c)

y = 1^x y = b^x, b > 1

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Exponential Functions and Their Derivatives

The properties of the exponential function are summarized in the following theorem.

Exponential Functions and Their Derivatives

The reason for the importance of the exponential function lies in properties 1–4, which are called the Laws of

Exponents.

If x and y are rational numbers, then these laws are well known from elementary algebra.

For arbitrary real numbers x and y these laws can be

deduced from the special case where the exponents are rational by using Equation 1.

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Exponential Functions and Their Derivatives

The following limits can be proved from the definition of a limit at infinity.

In particular, if b ≠ 1, then the x-axis is a horizontal

asymptote of the graph of the exponential function y = b^x.

Example 1

(a) Find .

(b) Sketch the graph of the function y = 2^–x – 1.

Solution:

(a)

= 0 – 1

= –1

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Example 1 – Solution

(b) We write as in part (a). The graph of is shown in Figure 3, so we shift it down one unit to obtain the graph of shown in Figure 7.

Part (a) shows that the line y = –1 is a horizontal asymptote.

cont’d

Figure 7

Member of the family of exponential functions

Figure 3

Applications of Exponential

Functions

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Applications of Exponential Functions

Table 1 shows data for the population of the world in the 20th century and Figure 8 shows the corresponding scatter plot.

Table 1 Figure 8

Scatter plot for world population growth

Applications of Exponential Functions

The pattern of the data points in Figure 8 suggests

exponential growth, so we use a graphing calculator with exponential regression capability to apply the method of least squares and obtain the exponential model

P = (1436.53)  (1.01395)^t where t = 0 corresponds to the year 1900.

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Applications of Exponential Functions

Figure 9 shows the graph of this exponential function together with the original data points.

Figure 9

Exponential model for population growth

Applications of Exponential Functions

We see that the exponential curve fits the data reasonably well.

The period of relatively slow population growth is explained by the two world wars and the Great Depression of the

1930s.

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Derivatives of Exponential

Functions

Derivatives of Exponential Functions

Let’s try to compute the derivative of the exponential function f(x) = b^x using the definition of a derivative:

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Derivatives of Exponential Functions

The factor b^x doesn’t depend on h, so we can take it in front of the limit:

Notice that the limit is the value of the derivative of f at 0, that is,

Derivatives of Exponential Functions

Therefore we have shown that if the exponential function f(x) = b^x is differentiable at 0, then it is differentiable

everywhere and

f′(x) = f′(0)b^x

This equation says that the rate of change of any

exponential function is proportional to the function itself.

(The slope is proportional to the height.)

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Derivatives of Exponential Functions

Numerical evidence for the existence of f′(0) is given in the table below for the cases b = 2 and b = 3. (Values are

stated correct to four decimal places.)

Derivatives of Exponential Functions

It appears that the limits exist and for b = 2,

for b = 3,

In fact, it can be proved that these limits exist and, correct to six decimal places, the values are

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Derivatives of Exponential Functions

Thus, from Equation 4, we have

Of all possible choices for the base b in Equation 4, the simplest differentiation formula occurs when f′(0) = 1.

Derivatives of Exponential Functions

In view of the estimates of f′(0) for b = 2 and b = 3, it

seems reasonable that there is a number b between 2 and 3 for which f′(0) = 1. It is traditional to denote this value by the letter e.

Thus we have the following definition.

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Derivatives of Exponential Functions

Geometrically, this means that of all the possible

exponential functions y = b^x, the function f(x) = e^x is the one whose tangent line at (0, 1) has a slope f′(0) that is exactly 1. (See Figures 12 and 13.)

Figure 13 Figure 12

Derivatives of Exponential Functions

We call the function f(x) = e^x the natural exponential function.

If we put b = e and, therefore, f′(0) = 1 in Equation 4, it becomes the following important differentiation formula.

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Derivatives of Exponential Functions

Thus the exponential function f(x) = e^x has the property that it is its own derivative.

The geometrical significance of this fact is that the slope of a tangent line to the curve y = e^x is equal to the y-coordinate of the point (see Figure 13).

Figure 13

Example 2

Differentiate the function y = e^{tan x}.

Solution:

To use the Chain Rule, we let u = tan x. Then we have y = e^u, so

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Derivatives of Exponential Functions

In general if we combine Formula 8 with the Chain Rule, as in Example 2, we get

Derivatives of Exponential Functions

We have seen that e is a number that lies somewhere between 2 and 3, but we can use Equation 4 to estimate the numerical value of e more accurately. Let e = 2^c. Then e^x = 2^cx.

If f(x) = 2^x, then from Equation 4 we have f′(x) = k2^x, where the value of k is f′(0) ≈ 0.693147.

Thus, by the Chain Rule,

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Derivatives of Exponential Functions

Putting x = 0, we have 1 = ck , so c = 1/k and e = 2^1/k ≈ 21/0.693147 ≈ 2.71828

It can be shown that the approximate value to 20 decimal places is

e ≈ 2.71828182845904523536

The decimal expansion of e is nonrepeating because e is an irrational number.

Exponential Graphs

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Exponential Graphs

The exponential function f(x) = e^x is one of the most frequently occurring functions in calculus and its

applications, so it is important to be familiar with its graph (Figure 14) and properties.

Figure 14

The natural exponential function

Exponential Graphs

We summarize these properties as follows, using the fact that this function is just a special case of the exponential functions considered in Theorem 2 but with base b = e > 1.

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Example 6

Find

Solution:

We divide numerator and denominator by e^2x:

= 1

Example 6 – Solution

We have used the fact that as and so

= 0

cont’d

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Integration

Integration

Because the exponential function y = e^x has a simple derivative, its integral is also simple:

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Example 8

Evaluate

Solution:

We substitute u = x³. Then du = 3x² dx , so x² dx = du and