1. 7
INVERSE FUNCTIONS

2. The Natural Exponential Function
INVERSE FUNCTIONS
7.3*
The Natural Exponential Function
In this section, we will learn about:
The natural exponential function and its properties.

3. THE NATURAL EXPONENTIAL FUNCTION
Since ln is an increasing function, it is one-to-one and, therefore, has an inverse function.
We denote this by: exp

4. Thus, according to the definition of an inverse function,
NATURAL EXP. FUNCTION
Notation 1
Thus, according to the definition of an inverse function,

5. The cancellation equations are:
NATURAL EXP. FUNCTION
Equations 2
The cancellation equations are:
In particular, we have: exp(0) = 1 since ln 1 = exp(0) = e since ln e = 1

6. NATURAL EXP. FUNCTION
We obtain the graph of y = exp x by reflecting the graph of y = ln x about the line y = x.
The domain of exp is the range of ln, that is, (-∞, ∞).
The range of exp is the domain of ln, that is, (0, ∞).
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7. NATURAL EXP. FUNCTION
If r is any rational number, the third law of logarithms gives: ln(er ) = r ln e = r
Therefore, by Notation 1, exp(r) = er
Thus, exp(x) = ex whenever x is a rational number.

8. NATURAL EXP. FUNCTION
This leads us to define ex, even for irrational values of x, by the equation ex = exp(x)
In other words, for the reasons given, we define ex to be the inverse of the function ln x.

9. In this equation, Notation 1 becomes:
NATURAL EXP. FUNCTION
Notation 3
In this equation, Notation 1 becomes:

10. The cancellation Equations 2 become:
NATURAL EXP. FUNCTION
Equations 4 & 5
The cancellation Equations 2 become:

11. NATURAL EXP. FUNCTION
E. g. 1—Solution 1
Find x if ln x = 5.
From Notation 3, we see that: ln x = means e5 = x
Therefore, x = e5.

12. Start with the equation ln x = 5
NATURAL EXP. FUNCTION
E. g. 1—Solution 2
Start with the equation ln x = 5
Then, apply the exponential function to both sides of the equation: e ln x = e5
However, Equation 4 says that: e ln x = x
Therefore, x = e5.

13. Solve the equation e 5-3x = 10.
NATURAL EXP. FUNCTION
Example 2
Solve the equation e 5-3x = 10.
We take natural logarithms of both sides of the equation and use Equation 5:

14. NATURAL EXP. FUNCTION
Example 2
Since the natural logarithm is found on scientific calculators, we can approximate the solution to four decimal places: x ≈

15. NATURAL EXP. FUNCTION
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.
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16. It is also important to be familiar with its properties.
NATURAL EXP. FUNCTION
It is also important to be familiar with its properties.
These follow from the fact that it is the inverse of the natural logarithmic function.

17. PROPERTIES
Properties 6
The exponential function f(x) = ex is an increasing continuous function with domain and range (0, ∞).
Thus, e x > 0 for all x

18. Also, So, the x-axis is a horizontal asymptote of f(x) = ex.
PROPERTIES
Properties 6
Also,
So, the x-axis is a horizontal asymptote of f(x) = ex.

19. Find: We divide numerator and denominator by e2x:
PROPERTIES
Example 3
Find:
We divide numerator and denominator by e2x:
We have used the fact that t = -2x → -∞ as x → ∞ and so

20. PROPERTIES
Now, we verify that f(x) = ex has the properties expected of an exponential function.

21. If x and y are real numbers and r is rational, then
LAWS OF EXPONENTS
Laws 7
If x and y are real numbers and r is rational, then
1. e x+y = e xe y
2. e x-y = e x/e y
3. (e x)r = e rx

22. Using the first law of logarithms and Equation 5, we have:
LAW 1 OF EXPONENTS
Proof
Using the first law of logarithms and Equation 5, we have:
Since ln is a one-to-one function, it follows that: e xe y = e x+y

23. Laws 2 and 3 are proved similarly.
LAWS 2 AND 3 OF EXPONENTS
Laws 2 and 3 are proved similarly.
See Exercises 95 and 96.
As we will see in Section 7.4*, Law 3 actually holds when r is any real number.

24. DIFFERENTIATION
Formula 8
The natural exponential function has the remarkable property that it is its own derivative.

25. To find its derivative, we use the inverse function method.
DIFFERENTIATION
Formula 8—Proof
The function y = ex is differentiable because it is the inverse function of y = ln x.
We know this is differentiable with nonzero derivative.
To find its derivative, we use the inverse function method.

26. DIFFERENTIATION
Formula 8—Proof
Let y = ex.
Then, ln y = x and, differentiating this latter equation implicitly with respect to x, we get:

27. The geometric interpretation of Formula 8 is:
DIFFERENTIATION
The geometric interpretation of Formula 8 is:
The slope of a tangent line to the curve y = ex at any point is equal to the y-coordinate of the point.
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28. DIFFERENTIATION
This property implies that the exponential curve y = ex grows very rapidly.
See Exercise 100.
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29. Differentiate the function y = e tan x.
DIFFERENTIATION
Example 4
Differentiate the function y = e tan x.
To use the Chain Rule, we let u = tan x.
Then, we have y = e u.
Hence,

30. DIFFERENTIATION
Formula 9
In general, if we combine Formula 8 with the Chain Rule, as in Example 4, we get:

31. DIFFERENTIATION
Example 5
Find y’ if y = e-4x sin 5x.
Using Formula 9 and the Product Rule, we have:

32. Find the absolute maximum value of the function f(x) = xe -x.
DIFFERENTIATION
Example 6
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)

33. Similarly, f ’(x) < 0 when x > 1.
DIFFERENTIATION
Example 6
Since exponential functions are always positive, we see that f ’(x) > 0 when 1 – x > 0, that is, when x < 1.
Similarly, f ’(x) < 0 when x > 1.

34. DIFFERENTIATION
Example 6
By the First Derivative Test for Absolute Extreme Values, f has an absolute maximum value when x = 1.
The absolute maximum value is:

35. DIFFERENTIATION
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.

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

37. As x → 0-, we know that t = 1/x →-∞. So,
DIFFERENTIATION
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.

38. Now, let’s compute the derivative.
DIFFERENTIATION
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.

39. The second derivative is:
DIFFERENTIATION
Example 7
The second derivative is:

40. DIFFERENTIATION
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).

41. DIFFERENTIATION
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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42. DIFFERENTIATION
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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43. DIFFERENTIATION
Example 7
We finish the sketch by incorporating the information concerning concavity and the inflection point.
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44. We check our work with a graphing device.
DIFFERENTIATION
Example 7
We check our work with a graphing device.
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45. INTEGRATION
Formula 10
As the exponential function y = ex has a simple derivative, its integral is also simple:

46. 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

47. 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: