1. 7
INVERSE FUNCTIONS

2. Exponential Functions
INVERSE FUNCTIONS
7.2
Exponential Functions
In this section, we will learn about:
Exponential functions and their derivatives.

3. The function f(x) = 2x is called an exponential function because
EXPONENTIAL FUNCTIONS
The function f(x) = 2x 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.

4. In general, an exponential function
EXPONENTIAL FUNCTIONS
In general, an exponential function
is a function of the form f(x) = ax
where a is a positive constant.
Let’s recall what this means.

5. If x = n, a positive integer, then:
EXPONENTIAL FUNCTIONS
If x = n, a positive integer,
then:

6. where n is a positive integer, then:
EXPONENTIAL FUNCTIONS
If x = 0, then a0 = 1, and if x = -n,
where n is a positive integer,
then:

7. If x is a rational number, x = p/q,
EXPONENTIAL FUNCTIONS
If x is a rational number, x = p/q,
where p and q are integers and q > 0,
then:

8. However, what is the meaning of ax if x is an irrational number?
EXPONENTIAL FUNCTIONS
However, what is the meaning of ax
if x is an irrational number?
For instance, what is meant by or ?

9. To help us answer that question,
EXPONENTIAL FUNCTIONS
To help us answer that question,
we first look at the graph of the function
y = 2x, where x is rational.
A representation of this graph is shown here.
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10. We want to enlarge the domain of y = 2x
EXPONENTIAL FUNCTIONS
We want to enlarge the domain of y = 2x
to include both rational and irrational
numbers.
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11. There are holes in the graph corresponding to irrational values of x.
EXPONENTIAL FUNCTIONS
There are holes in the graph
corresponding to irrational values of x.
We want to fill in the holes by defining f(x) = 2x, where , so that f is an increasing continuous function.
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12. In particular, since the irrational number satisfies , we must have
EXPONENTIAL FUNCTIONS
In particular, since the irrational number
satisfies , we must
have
We know what 21.7 and mean because 1.7 and 1.8 are rational numbers.
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13. Similarly, if we use better approximations
EXPONENTIAL FUNCTIONS
Similarly, if we use better approximations
for , we obtain better approximations
for :

14. It can be shown that there is exactly one
EXPONENTIAL FUNCTIONS
It can be shown that there is exactly one
number that is greater than all the numbers
21.7, 21.73, , , , …
and less than all the numbers
21.8, 21.74, , , , …

15. We define to be this number.
EXPONENTIAL FUNCTIONS
We define to be this number.
Using the preceding approximation process, we can compute it correct to six decimal places:
Similarly, we can define 2x (or ax, if a > 0) where x is any irrational number.

16. The figure shows how all the holes in
EXPONENTIAL FUNCTIONS
The figure shows how all the holes in
the earlier figure have been filled to
complete the graph of the function
f(x) = 2x,

17. In general, if is any positive number, we define r rational
EXPONENTIAL FUNCTIONS
Equation 1
In general, if is any positive number, we
define
r rational
This definition makes sense because any
irrational number can be approximated as
closely as we like by a rational number.

18. For instance, because has the decimal
EXPONENTIAL FUNCTIONS
For instance, because has the decimal
representation , Definition
1 says that is the limit of the sequence of
numbers
21.7, 21.73, , , , …

19. Similarly, 5π is the limit of the sequence of numbers
EXPONENTIAL FUNCTIONS
Similarly, 5π is the limit of the sequence of
numbers
53.1, 53.14, , , , ,
, . . .
It can be shown that Definition 1 uniquely specifies ax and makes the function f(x) = ax continuous.

20. The graphs of members of the family of
EXPONENTIAL FUNCTIONS
The graphs of members of the family of
functions y = ax are shown here for various
values of the base a.

21. Notice that all these graphs pass through
EXPONENTIAL FUNCTIONS
Notice that all these graphs pass through
the same point (0, 1) because a0 = 1 for a ≠ 0.

22. Notice also that as the base a gets larger,
EXPONENTIAL FUNCTIONS
Notice also that as the base a gets larger,
the exponential function grows more rapidly
(for x > 0).

23. The figure shows how the exponential
EXPONENTIAL FUNCTIONS
The figure shows how the exponential
function y = 2x compares with the power
function y = x2.

24. The graphs intersect three times, but
EXPONENTIAL FUNCTIONS
The graphs intersect three times, but
ultimately the exponential curve y = 2x
grows far more rapidly
than the parabola y = x2.

25. You can see that there are basically three
EXPONENTIAL FUNCTIONS
You can see that there are basically three
kinds of exponential functions y = ax.
If 0 < a < 1, the exponential function decreases.
If a = 1, it is a constant.
If a > 1, it increases.

26. EXPONENTIAL FUNCTIONS
Those three cases are
illustrated here.

27. Notice also that, since (1/a)x = 1/ax = a-x,
EXPONENTIAL FUNCTIONS
Notice also that, since (1/a)x = 1/ax = a-x,
the graph of y = (1/a)x is just the reflection
of the graph of y = ax about the y-axis.

28. PROPERTIES OF EXPONENTIAL FUNCTIONS
The properties of the exponential function are summarized in the following theorem.

29. If 0 < a < 1, f(x) = ax is a decreasing function.
PROPERTIES OF EXP. FUNCTIONS
Theorem 2
If a > 0 and a ≠ 1, f(x) = ax is a continuous function with domain and range (0, ∞).
In particular, ax > 0 for all x.
If 0 < a < 1, f(x) = ax is a decreasing function.
If a > 1, f is an increasing function.

30. PROPERTIES OF EXP. FUNCTIONS
Theorem 2
If a, b > 0 and x, y , then

31. LAWS OF EXPONENTS
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, 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.

32. The following limits can be read from the accompanying graphs.
EXPONENTIAL FUNCTIONS
The following limits can be read from the accompanying graphs.
Alternatively, they can be proved from the definition of a limit at infinity.

33. If a > 1, then If 0 < a < 1, then EXPONENTIAL FUNCTIONS
Equation 3
If a > 1, then
If 0 < a < 1, then
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34. EXPONENTIAL FUNCTIONS
In particular, if a ≠ 1, the x-axis is a horizontal asymptote of the graph of the exponential function y = ax.
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35. Sketch the graph of the function y = 2–x – 1.
EXPONENTIAL FUNCTIONS
Example 1
Find:
Sketch the graph of the function y = 2–x – 1.

36. EXPONENTIAL FUNCTIONS
Example 1 a

37. We write as in part a. The graph of is shown. EXPONENTIAL FUNCTIONS
Example 1 b
We write as in part a.
The graph of is shown.
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38. EXPONENTIAL FUNCTIONS
Example 1 b
So, we shift that graph down one unit to obtain the graph of shown here.
For a review of shifting graphs, see Section 1.3.
Part (a) shows that the line y = -1 is a horizontal asymptote.
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39. APPLICATIONS OF EXPONENTIAL FUNCTIONS
The exponential function occurs very frequently in mathematical models of nature and society.
Here, we indicate briefly how it arises in the description of population growth.
In Section 7.5, we will pursue these and other applications in greater detail.

40. APPLICATIONS OF EXPONENTIAL FUNCTIONS
In Section 3.7, we considered a bacteria population that doubles every hour.
We saw that, if the initial population is n0, the population after t hours is given by the function f(t) = n02t

41. APPLICATIONS OF EXPONENTIAL FUNCTIONS
This population function is a constant multiple of the exponential function y = 2t.
Hence, it exhibits the rapid growth we observed in these earlier figures.
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42. APPLICATIONS OF EXPONENTIAL FUNCTIONS
Under ideal conditions (unlimited space and nutrition and freedom from disease), this exponential growth is typical of what actually occurs in nature.

43. APPLICATIONS OF EXPONENTIAL FUNCTIONS
What about the
human population?

44. The table shows data for the population of
APPLICATIONS: HUMAN POPULATION
The table shows data for the population of
the world in the 20th century.
The figure shows the corresponding scatter
plot.

45. The pattern of the data points suggests exponential growth.
APPLICATIONS: HUMAN POPULATION
The pattern of the data points
suggests exponential growth.

46. So, we use a graphing calculator with
APPLICATIONS: HUMAN POPULATION
So, we use a graphing calculator with
exponential regression capability to apply
the method of least squares and obtain the
exponential model

47. The figure shows the graph of this exponential function together with
APPLICATIONS: HUMAN POPULATION
The figure shows the graph of this
exponential function together with
the original data points.

48. We see that the exponential curve fits the data reasonably well.
APPLICATIONS: HUMAN POPULATION
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.

49. DERIVATIVES OF EXPONENTIAL FUNCTIONS
Let’s try to compute the derivative of the exponential function f(x) = ax using the definition of a derivative:

50. The factor ax doesn’t depend on h.
DERIVATIVES OF EXP. FUNCTIONS
The factor ax 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,

51. DERIVATIVES OF EXP. FUNCTIONS
Equation 4
Thus, we have shown that, if the exponential function f(x) = ax is differentiable at 0, then it is differentiable everywhere and f’(x) = f’(0)ax
The equation states that the rate of change of any exponential function is proportional to the function itself.
The slope is proportional to the height.

52. DERIVATIVES OF EXP. FUNCTIONS
Numerical evidence for the existence of f’(0) is given in the table for the cases a = 2 and a = 3.
Values are stated correct to four decimal places.

53. It appears that the limits exist and for a = 2,
DERIVATIVES OF EXP. FUNCTIONS
It appears that the limits exist and for a = 2,
for a = 3,

54. DERIVATIVES OF EXP. FUNCTIONS
Equation 5
In fact, it can be proved that these limits exist and, correct to six decimal places, the values are:

55. Thus, from Equation 4, we have:
DERIVATIVES OF EXP. FUNCTIONS
Equation 6
Thus, from Equation 4, we have:

56. DERIVATIVES OF EXP. FUNCTIONS
Of all possible choices for the base a in Equation 4, the simplest differentiation formula occurs when f’(0) = 1.
In view of the estimates of f’(0) for a = 2 and a = 3, it seems reasonable that there is a number a between 2 and 3 for which f’(0) = 1.

57. It is traditional to denote this value by the letter e.
Thus, we have the following definition.

58. e is the number such that:
DEFINITION OF THE NUMBER e
Definition 7
e is the number such that:

59. DERIVATIVES OF EXP. FUNCTIONS
Geometrically, this means that, of all the possible exponential functions y = ax, the function f(x) = ex is the one whose tangent line at (0, 1) has a slope f’(0) that is exactly 1.

60. We call the function f(x) = ex the natural exponential function.

61. NATURAL EXPONENTIAL FUNCTION
If we put a = e and, therefore, f’(0) = 1 in Equation 4, it becomes the following important differentiation formula.

62. The derivative of the natural exponential function is:
NATURAL EXP. FUNCTION
Formula 8
The derivative of the natural exponential function is:

63. NATURAL EXP. FUNCTION
Thus, the exponential function f(x) = ex 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 = ex is equal to the y-coordinate of the point.

64. Differentiate the function y = e tan x
NATURAL EXP. FUNCTION
Example 2
Differentiate the function y = e tan x
To use the Chain Rule, we let u = tan x.
Then, we have y = eu.
Thus,

65. NATURAL EXP. FUNCTION
Formula 9
In general, if we combine Formula 8 with the Chain Rule, as in Example 2, we get:

66. NATURAL EXP. FUNCTION
Example 3
Find y’ if y = e-4x sin 5x.
Using Formula 9 and the Product Rule, we have:

67. We have seen that e is a number that lies somewhere between 2 and 3.
NATURAL EXP. FUNCTION
We have seen that e is a number that lies somewhere between 2 and 3.
However, we can use Equation 4 to estimate the numerical value of e more accurately.

68. NATURAL EXP. FUNCTION
Let e = 2c.
Then, ex = 2cx.
If f(x) = 2x, then from Equation 4, we have f’(x) = k2x, where the value of k is: f’(0) ≈
Hence, by the Chain Rule:

69. NATURAL EXP. FUNCTION
Putting x = 0, we have 1 = ck.
So, c = 1/k and e = 21/k ≈ 21/ ≈
It can be shown that the approximate value to 20 decimal places is: e ≈
The decimal expansion of e is non-repeating because e is an irrational number.

70. NATURAL EXP. FUNCTION
Example 4
In Example 6 in Section 3.7, we considered a population of bacteria cells in a homogeneous nutrient medium.
We showed that, if the population doubles every hour, then the population after t hours is: n = n02t where n0 is the initial population.

71. Now, we can use Equations 4 and 5 to compute the growth rate:
NATURAL EXP. FUNCTION
Example 4
Now, we can use Equations 4 and 5 to compute the growth rate:
For instance, if the initial population is n0 = 1000 cells, then the growth rate after two hours is:

72. Find the absolute maximum value of the function f(x) = xe-x.
NATURAL EXP. FUNCTION
Example 5
Find the absolute maximum value of the function f(x) = xe-x.
We differentiate to find any critical numbers: f ’(x) = xe-x (-1) + e-x(1) = e-x(1-x)
As exponential functions are always positive, we see f ’(x) > 0 when 1 - x > 0, that is, when x < 1.
Similarly, f ’(x) < 0 when x > 1.

73. NATURAL EXP. FUNCTION
Example 5
By the First Derivative Test for Absolute Extreme Values, f has an absolute maximum value when x = 1.
The value is:

74. EXPONENTIAL GRAPHS
The exponential function f(x) = ex is one of the most frequently occurring functions in calculus and its applications.
So, it is important to be familiar with its graph and properties.
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75. We summarize these properties as follows.
EXPONENTIAL GRAPHS
We summarize these properties as follows.
We use the fact that this function is just a special case of the exponential functions considered in Theorem 2 but with base a = e > 1.

76. PROPERTIES OF f(x) = ex
Definition 10
The exponential function f(x) = ex is an increasing continuous function with domain and range (0, ∞).
Thus, ex > 0 for all x.
Also,
So, the x-axis is a horizontal asymptote of f(x) = ex.

77. Find: We divide numerator and denominator by e2x:
EXPONENTIAL GRAPHS
Example 6
Find:
We divide numerator and denominator by e2x:
We have used the fact that t = -2x → -∞ as x → ∞ and so

78. EXPONENTIAL GRAPHS
Example 7
Use the first and second derivatives of f(x) = e1/x, together with asymptotes, to sketch its graph.
Notice that the domain of f is {x | x ≠ 0}.
Hence, we check for vertical asymptotes by computing the left and right limits as x → 0.

79. As x→ 0+, we know that t = 1/x → ∞.
EXPONENTIAL GRAPHS
Example 7
As x→ 0+, we know that t = 1/x → ∞.
So,
This shows that x = 0 is a vertical asymptote.

80. As x → 0-, we know that t = 1/x →-∞. So,
EXPONENTIAL GRAPHS
Example 7
As x → 0-, we know that t = 1/x →-∞.
So,
As x→ ∞, we have 1/x→ 0.
This shows that y = 1 is a horizontal asymptote.

81. Now, let’s compute the derivative.
EXPONENTIAL GRAPHS
Example 7
Now, let’s compute the derivative.
The Chain Rule gives:
Since e1/x > 0 and x2 > 0 for all x ≠ 0, we have f ‘(x) < 0 for all x ≠ 0.
Thus, f is decreasing on (-∞, 0) and on (0, ∞).
There is no critical number, so the function has no maximum or minimum.

82. The second derivative is:
EXPONENTIAL GRAPHS
Example 7
The second derivative is:

83. EXPONENTIAL GRAPHS
Example 7
Since e1/x > 0 and x4 > 0, we have f”(x) > 0 when x > -½(x ≠ 0) and f”(x) < 0 when x < -½.
So, the curve is concave downward on (-∞, -½) and concave upward on (-½, 0) and on (0, ∞).
The inflection point is (-½, e-2).

84. EXPONENTIAL GRAPHS
Example 7
To sketch the graph of f, we first draw the horizontal asymptote y = 1 as a dashed line in a preliminary sketch.
We also draw the parts of the curve near the asymptotes.
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85. EXPONENTIAL GRAPHS
Example 7
These parts reflect the information concerning limits and the fact that f is decreasing on both (-∞, 0) and (0, ∞).
Notice that we have indicated that f(x) → 0 as x → 0- even though f(0) does not exist.
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86. EXPONENTIAL GRAPHS
Example 7
We finish the sketch by incorporating the information concerning concavity and the inflection point.
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87. We check our work with a graphing device.
EXPONENTIAL GRAPHS
Example 7
We check our work with a graphing device.
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88. INTEGRATION
Formula 11
As the exponential function y = ex has a simple derivative, its integral is also simple:

89. Evaluate: We substitute u = x3. Then, du = 3x2 dx. So, and INTEGRATION
Example 8
Evaluate:
We substitute u = x3.
Then, du = 3x2 dx.
So, and

90. Find the area under the curve y = e-3x from 0 to 1.
INTEGRATION
Example 9
Find the area under the curve y = e-3x from 0 to 1.
The area is: