Inverse Functions



9.1 Inverse Functions

Functions such as logarithms, exponential functions, and trigonometric functions are examples of transcendental functions. If a function is transcendental, it cannot be expressed as a polynomial or rational function. That is, it is not an algebraic function. In this chapter, we will begin by developing the concept of an inverse of a function and how it is linked to its original numerically, algebraically, and graphically. Later, we will take each type of elementary transcendental function—logarithmic, exponential, and trigonometric—individually and see the connection between them and their respective inverses, derivatives, and integrals.

Learning Objectives

A student will be able to:

• Understand the basic properties of the inverse of a function and how to find it.

• Understand how a function and its inverse are represented graphically.

• Know the conditions of invertability of a function.

One-to-One Functions

A function, as you know from your previous mathematics background, is a rule that assigns a single value in its range to each point in its domain. In other words, for each output number, there is one or more input numbers. However, a function never produces more than a single output for one input. A function is said to be a one-to-one function if each output is associated with only one single input. For example, [pic]assigns the output [pic]for both [pic]and [pic]and thus it is not a one-to-one function.

One-to-One Function

The function [pic]is one-to-one in a domain [pic]if [pic]whenever [pic]

There is an easy method to check if a function is one-to-one: draw a horizontal line across the graph. If the line intersects at only one point on the graph, then the function is one-to-one; otherwise, it is not. Notice in the figure below that the graph of [pic]is not one-to-one since the horizontal line intersects the graph more than once. But the function [pic]is a one-to-one function because the graph meets the horizontal line only once.

[pic]

[pic]

Example 1:

Determine whether the functions are one-to-one: (a) [pic](b) [pic]

Solution:

It is best to graph both functions and draw on each a horizontal line. As you can see from the graphs, [pic]is not one-to-one since the horizontal line intersects it at two points. The function [pic]however, is indeed one-to-one since only one point is intersected by the horizontal line.

[pic]

[pic]

The Inverse of a Function

We discussed above the condition for a one-to-one function: for each output, there is only one input. A one-to-one function can be reversed in such a way that the input of the function becomes the output and the output becomes an input. This reverse of the original function is called the inverse of the function. If [pic]is an inverse of a function [pic]then [pic]For example, the two functions [pic]and [pic]are inverses of each other since

[pic]

Thus [pic] and [pic]and [pic]are inverses of each other. Note: In general, [pic]

When is a function invertable?

It is interesting to note that if a function [pic]is always increasing or always decreasing over its domain, then a horizontal line will cut through this graph at one point only. Then [pic]in this case is a one-to-one function and thus has an inverse. So if we can find a way to prove that a function is constantly increasing or decreasing, then it is invertable or monotonic. From previous chapters, you have learned that if [pic]then [pic]must be increasing and if [pic]then [pic]must be decreasing.

To summarize, a function has an inverse if it is one-to-one in its domain or if its derivative is either [pic]or [pic]

Example 2:

Given the polynomial function [pic]show that it is invertable (has an inverse).

Solution:

Taking the derivative, we find that [pic]for all [pic]We conclude that [pic]is one-to-one and invertable. Keep in mind that it may not be easy to find the inverse of [pic](try it!), but we still know that it is indeed invertable.

How to find the inverse of a one-to-one function:

To find the inverse of a one-to-one function, simply solve for [pic]in terms of [pic]and then interchange [pic]and [pic]The resulting formula is the inverse [pic]

Example 3:

Find the inverse of [pic].

Solution:

From the discussion above, we can find the inverse by first solving for [pic]in [pic].

[pic]

Interchanging [pic],

[pic]

Replacing [pic]

[pic]

which is the inverse of the original function [pic].

Graphs of Inverse Functions

What is the relationship between the graphs of [pic]and [pic]? If the point [pic]is on the graph of [pic]then from the definition of the inverse, the point [pic]is on the graph of [pic]In other words, when we reverse the coordinates of a point on the graph of [pic]we automatically get a point on the graph of [pic]We conclude that [pic]and [pic]are reflections of one another about the line [pic]That is, each is a mirror image of the other about the line [pic]The figure below shows an example of [pic]and, when the domain is restricted, its inverse [pic]and how they are reflected about [pic].

[pic]

It is important to note that for the function [pic]to have an inverse, we must restrict its domain to [pic]since that is the domain in which the function is increasing.

Continuity and Differentiability of Inverse Functions

Since the graph of a one-to-one function and its inverse are reflections of one another about the line [pic]it would be safe to say that if the function [pic]has no breaks (no discontinuities) then [pic]will not have breaks either. This implies that if [pic]is continuous on the domain [pic]then its inverse [pic]is continuous on the range [pic]of [pic]For example, if [pic], then its domain is [pic]and its range is [pic]This means that [pic]is continuous for all [pic]The inverse of [pic]is [pic]where its domain is all [pic]and its range is [pic]We conclude that if [pic]is a function with domain [pic]and range [pic]and it is continuous and one-to-one on [pic]then its inverse [pic]is continuous and one-to-one on the range [pic]of [pic]

Suppose that [pic]has a domain [pic]and a range [pic]If [pic]is differentiable and one-to-one on [pic]then its inverse [pic]is differentiable at any value [pic]in [pic]for which [pic]and

[pic]

The formula above can be written in a form that is easier to remember:

[pic]

In addition, if [pic]on its domain is either [pic]or [pic]then [pic]has an inverse function [pic]and [pic]is differentiable at all values of [pic]in the range of [pic]In this case, [pic]is given by the formula above. The example below illustrate this important theorem.

Example 4:

In Example 3, we were given the polynomial function [pic]and we showed that it is invertable. Show that it is differentiable and find the derivative of its inverse.

Solution:

Since [pic]for all [pic][pic]is differentiable at all values of [pic]To find the derivative of [pic]if we let [pic]then

[pic]

So [pic] and [pic]

Since we are unable to solve for [pic] in terms of [pic] we leave the answer above in terms of [pic] Another way of solving the problem is to use Implicit Differentiation:

Since [pic] differentiating implicitly,

[pic]

Solving for [pic] we finally obtain [pic] which is the same result.

Review Questions

In problems #1-3, find the inverse function of [pic] and verify that [pic]

1. [pic]

2. [pic]

3. [pic]

In problems #4-6, use the horizontal line test to determine if the following functions have inverses.

4. [pic]

5. [pic]

6. [pic]

In problems #7-8, use the functions [pic] and [pic] to find the specified functions.

7. [pic]

8. [pic]

In problems #9-10, show that [pic] is monotonic (invertable) on the given interval (and therefore has an inverse).

9. [pic]

10. [pic]

Review Answers

1. [pic]

2. [pic]

3. [pic]

4. Function has an inverse.

5. Function does not have an inverse.

6. Function does not have an inverse.

7. [pic]

8. [pic]

9. [pic] on [pic]

10. [pic] which is negative on the interval in question, so [pic] is monotonically decreasing.

9.2 Exponential and Logarithmic Functions

Learning Objectives

A student will be able to:

• Understand and use the basic definitions of exponential and logarithmic functions and how they are related algebraically.

• Distinguish between an exponential and logarithmic functions graphically.

A Quick Algebraic Review of Exponential and Logarithmic Functions

Exponential Functions

Recall from algebra that an exponential function is a function that has a constant base and a variable exponent. A function of the form [pic]where [pic]is a constant and [pic]and [pic]is called an exponential function with base [pic]Some examples are [pic][pic]and [pic]All exponential functions are continuous and their graph is one of the two basic shapes, depending on whether [pic]or [pic]The graph below shows the two basic shapes:

[pic]

Logarithmic Functions

Recall from your previous courses in algebra that a logarithm is an exponent. If the base [pic]and [pic]then for any value of [pic]the logarithm to the base [pic]of the value of [pic]is denoted by

[pic]

This is equivalent to the exponential form

[pic]

For example, the following table shows the logarithmic forms in the first row and the corresponding exponential forms in the second row.

[pic]

Historically, logarithms with base of [pic] were very popular. They are called the common logarithms. Recently the base [pic]has been gaining popularity due to its considerable role in the field of computer science and the associated binary number system. However, the most widely used base in applications is the natural logarithm, which has an irrational base denoted by [pic]in honor of the famous mathematician Leonhard Euler. This irrational constant is [pic]Formally, it is defined as the limit of [pic]as [pic]approaches zero. That is,

[pic]

We denote the natural logarithm of [pic]by [pic]rather than [pic]So keep in mind, that [pic]is the power to which [pic]must be raised to produce [pic]That is, the following two expressions are equivalent:

[pic]

The table below shows this operation.

[pic]

A Comparison between Logarithmic Functions and Exponential Functions

Looking at the two graphs of exponential functions above, we notice that both pass the horizontal line test. This means that an exponential function is a one-to-one function and thus has an inverse. To find a formula for this inverse, we start with the exponential function [pic] Interchanging x and y, [pic]

Projecting the logarithm to the base [pic] on both sides,

[pic]

Thus [pic]is the inverse of [pic]

This implies that the graphs of [pic]and [pic]are reflections of one another about the line [pic]The figure below shows this relationship.

[pic]

Similarly, in the special case when the base [pic]the two equations above take the forms

[pic] and [pic]

The graph below shows this relationship:

[pic]

Before we move to the calculus of exponential and logarithmic functions, here is a summary of the two important relationships that we have just discussed:

• The function [pic]is equivalent to [pic]if [pic]and [pic]

• The function [pic]is equivalent to [pic]if [pic]and [pic]

You should also recall the following important properties about logarithms:

• [pic]

• [pic]

• [pic]

• To express a logarithm with base in terms of the natural logarithm: [pic]

• To express a logarithm with base [pic]in terms of another base [pic]: [pic]

Review Questions

Solve for x.

1. [pic]

2. [pic]

3. [pic]

4. [pic]

5. [pic]

6. [pic]

7. [pic]

8. [pic]

9. [pic]

10. [pic]

Review Answers

1. [pic]

2. [pic]

3. [pic]

4. [pic]

5. [pic]

6. [pic], [pic]

7. [pic]

8. [pic]

9. [pic]

10. [pic]

9.3 Differentiation and Integration of Logarithmic and Exponential Functions

Learning Objectives

A student will be able to:

• Understand and use the rules of differentiation of logarithmic and exponential functions.

• Understand and use the rules of integration of logarithmic and exponential functions.

In this section we will explore the derivatives of logarithmic and exponential functions. We will also see how the derivative of a one-to-one function is related to its inverse.

The Derivative of a Logarithmic Function

Our goal at this point to find an expression for the derivative of the logarithmic function [pic]Recall that the exponential number [pic]is defined as

[pic]

(where we have substituted [pic]for [pic]for convenience). From the definition of the derivative of [pic] that you already studied in Chapter 2,

[pic]

We want to apply this definition to get the derivative to our logarithmic function [pic]Using the definition of the derivative and the rules of logarithms from the Lesson on Exponential and Logarithmic Functions,

[pic]

At this stage, let [pic]the limit of [pic]then becomes [pic] Substituting, we get

[pic]

Inserting the limit,

[pic]

But by the definition [pic]

[pic]

From the box above, we can express [pic]in terms of natural logarithm by the using the formula [pic]Then

[pic]

Thus we conclude

[pic]

and in the special case where [pic]

[pic]

To generalize, if [pic]is a differentiable function of [pic]and if [pic]then the above two equations, after the Chain Rule is applied, will produce the generalized derivative rule for logarithmic functions.

Derivatives of Logarithmic Functions

[pic]

Remark: Students often wonder why the constant [pic]is defined the way it is. The answer is in the derivative of [pic]With any other base the derivative of [pic]would be equal [pic]a more complicated expression than [pic]Thinking back to another unexpected unit, radians, the derivative of [pic]is the simple expression [pic]only if [pic]is in radians. In degrees, [pic], which is more cumbersome and harder to remember.

Example 1:

Find the derivative of [pic]

Solution:

Since [pic], for [pic]

[pic]

Example 2:

Find [pic].

Solution:

[pic]

Example 3:

Find [pic]

Solution:

Here we use the Chain Rule:

[pic]

Example 4:

Find the derivative of [pic]

Solution:

Here we use the Product Rule along with [pic]

[pic]

Example 5:

Find the derivative of [pic]

Solution:

We use the Quotient Rule and the natural logarithm rule:

[pic]

Integrals Involving Natural Logarithmic Function

In the last section, we have learned that the derivative of [pic]is [pic]. The antiderivative is

[pic]

If the argument of the natural logarithm is [pic]then [pic]thus

[pic]

Example 6:

Evaluate [pic]

Solution:

In general, whenever you encounter an integral with an integrand as a rational function, it might be possible that it can be integrated with the rule of natural logarithm. To do so, determine the derivative of the denominator. If it is the numerator itself, then the integration is simply the [pic]of the absolute value of the denominator. Let’s test this technique.

[pic]

Notice that the derivative of the denominator is [pic], which is equal to the numerator. Thus the solution is simply the natural logarithm of the absolute value of the denominator:

[pic]

The formal way of solving such integrals is to use [pic]substitution by letting [pic]equal the denominator. Here, let [pic]and [pic]Substituting,

[pic]

Remark: The integral must use the absolute value symbol because although [pic]may have negative values, the domain of [pic]is restricted to [pic]

Example 7:

Evaluate [pic]

Solution:

As you can see here, the derivative of the denominator is [pic]Our numerator is [pic]However, when we multiply the numerator by 2 we get the derivative of the denominator. Hence

[pic]

Again, we could have used [pic]substitution.

Example 8:

Evaluate [pic].

Solution:

To solve, we rewrite the integrand as

[pic]

Looking at the denominator, its derivative is [pic]. So we need to insert a minus sign in the numerator:

[pic]

Derivatives of Exponential Functions

We have discussed above that the exponential function is simply the inverse function of the logarithmic function. To obtain a derivative formula for the exponential function with base [pic]we rewrite [pic]as

[pic]

Differentiating implicitly,

[pic]

Solving for [pic]and replacing [pic]with [pic]

[pic]

Thus the derivative of an exponential function is

[pic]

In the special case where the base is [pic]since [pic]the derivative rule becomes

[pic]

To generalize, if [pic]is a differentiable function of [pic]with the use of the Chain Rule the above derivatives take the general form

[pic]

And if [pic]

[pic]

Derivatives of Exponential Functions

[pic]

[pic]

Example 9:

Find the derivative of [pic].

Solution:

Applying the rule for differentiating an exponential function,

[pic]

Example 10:

Find the derivative of [pic].

Solution:

Since

[pic]

Example 11:

Find [pic]if

[pic]

where [pic]and [pic]are constants and [pic]

Solution:

We apply the exponential derivative and the Chain Rule:

[pic]

Integrals Involving Exponential Functions

Associated with the exponential derivatives in the box above are the two corresponding integration formulas:

[pic]

The following examples illustrate how they can be used.

Example 12: Evaluate [pic].

Solution:

[pic]

Example 13:

[pic]

Solution:

[pic]

In the next chapter, we will learn how to integrate more complicated integrals, such as [pic], with the use of [pic]substitution and integration by parts along with the logarithmic and exponential integration formulas.

Multimedia Links

For a video presentation of the derivatives of exponential and logarithmic functions (4.4), see Math Video Tutorials by James Sousa, The Derivatives of Exponential and Logarithmic Functions (8:26)[pic].

Review Questions

1. Find [pic] of [pic]

2. Find [pic] of [pic]

3. Find [pic] of [pic]

4. Find [pic] of [pic]

5. Find [pic] of [pic]

6. Find [pic] of [pic]

7. Integrate [pic]

8. Integrate [pic]

9. Integrate [pic]

10. Integrate [pic]

11. Evaluate [pic]

12. Evaluate [pic]

Review Answers

1. [pic]

2. [pic]

3. [pic]

4. [pic]

5. [pic]

6. [pic]

7. [pic]

8. [pic]

9. [pic]

10. [pic]

11. [pic]

12. [pic]

Practice on Differentiating Exponential Functions

Find the derivative of the function.

1. [pic] 2. [pic] 3. [pic] 4. [pic]

5. [pic] 6. [pic] 7. [pic]

8. [pic] 9. [pic]

10. Find [pic].

a. [pic] b. [pic]

Answers:

1. [pic] 2. [pic] 3. [pic] 4. [pic]

5. [pic] 6. [pic] 7.[pic]

8. [pic] 9. [pic] or [pic]

10. a. [pic] b. [pic]

More Practice on Differentiating Exponential Functions

Find the derivative of the function.

1. [pic] 2. [pic] 3. [pic] 4. [pic]

5. [pic] 6. [pic] 7. [pic]

8. [pic] 9. [pic] 10. Find [pic] if [pic].

11. Find the slope of the line tangent to [pic] at [pic]

Answers:

1. [pic] 2. [pic] 3. [pic] 4. [pic]

5. [pic] 6. [pic] 7. [pic]

8.[pic] 9. [pic] 10. [pic] 11. 5

Derivatives of Exponential Functions HW

Find the derivative of each function.

1.) [pic]

2.) [pic]

3.) [pic]

4.) [pic]

5.) [pic]

6.) [pic]

7.) [pic]

8.) [pic]

9.) Find the 2nd derivative of [pic]

10.) Find [pic] implicitly: [pic]

Answers!

1.) [pic]

2.) [pic]

3.) [pic]

4.) [pic]

5.) [pic]

6.) [pic]

7.) [pic]

8.) [pic]

9.) [pic]

10.) [pic]

Practice with Integration involving Exponential Function

1. [pic] 2. [pic] 3. [pic] 4. [pic]

5. [pic] 6. [pic] 7. [pic] 8. [pic]

9. Because of an insufficient oxygen supply, the trout population in a lake is dying. The population’s rate of change can be modeled by [pic], where [pic]is the time in days. When [pic], the population is 2500.

a. Write an equation that models the population [pic]in terms of the time [pic].

b. What is the population after 15 days?

c. According to this model, how long will it take for the entire trout population to die?

Answers:

1. [pic] 2. [pic] 3. [pic] 4. [pic]

5. [pic] 6. [pic] 7. [pic] 8. [pic]

9. a. [pic] b. approx. 1180 fish c. infinitely long

More Practice with Integration involving Exponential Function

1. [pic] 2. [pic] 3. [pic] 4. [pic]

5. [pic] 6. [pic] 7. [pic] 8. [pic]

9. [pic] 10. [pic] 11. [pic] 12. [pic]

13. The marginal price for the demand of a product can be modeled by [pic], where [pic] is the quantity demanded. When the demand is 600 units, the price is $30.

a. Find the demand function, [pic].

b. Use your calculator to graph the demand function. Does the price increase or decrease as the demand increases?

c. Find the quantity demanded when the price is $22.

Answers:

1. [pic] 2. [pic] 3. [pic] 4. [pic]

5. [pic] 6. [pic] 7. [pic] 8. [pic]

9. [pic] 10. [pic] 11. [pic] 12. [pic]

13. a. [pic] b. The price increases as the demand increases c. 387

Integrals of Exponential Functions HW

Integrate! Give both exact and approximate answers for definite integrals.

1.) [pic]

2.) [pic]

3.) [pic]

4.) [pic]

5.) [pic]

6.) [pic]

7.) [pic]

8.) [pic]

9.) [pic]

10.) [pic]

Answers!

|1.) [pic] |7.) [pic] |

|2.) [pic] |8.) [pic] |

|3.) [pic] |9.) [pic] |

|4.) [pic] |10.) [pic] |

|5.) [pic] | |

|6.) [pic] | |

Review Problems on Differentiation & Integration involving Exponential Functions

Find the derivative of each function.

1.) [pic]

2.) [pic]

3.) [pic]

4.) [pic]

5.) [pic]

6.) [pic]

7.) [pic]

8.) Find [pic] implicitly: [pic]

Integrate.

9.) [pic]

10.) [pic]

11.) [pic]

12.) [pic]

13.) [pic]

14.) [pic]

15.) [pic]

16.) [pic]

Answers

|1.) [pic] |5.) [pic] |9.) [pic] |13.) [pic] |

|2.) [pic] |6.) [pic] |10.) [pic] |14.) [pic] |

|3.) [pic] |7.) [pic] |11.) [pic] |15.) [pic] |

|4.) [pic] |8.) [pic] |12.) [pic] |16.) [pic] |

Practice with Differentiating the Natural Log Function

Find the derivative.

1. [pic] 2. [pic] 3. [pic] 4. [pic]

5. [pic] 6. [pic] 7. [pic] 8. [pic]

9. [pic] 10. [pic] 11. [pic]

12. [pic] 13. [pic] 14. [pic]

15. Write the equation of the line tangent to the graph of[pic] at [pic].

Answers:

1. [pic] 2. [pic] 3. [pic] 4. [pic]

5. [pic] 6. [pic] 7. [pic] 8. [pic]

9. [pic] 10. [pic] 11. [pic] 12. [pic]

13. [pic] 14. [pic] 15. [pic]

More Practice with Differentiating the Natural Log Function

Find the derivative.

1. [pic] 2. [pic] 3. [pic] 4. [pic]

5. [pic] 6. [pic] 7. [pic] 8. [pic]

9. [pic] 10. [pic] 11. [pic]

12. [pic] 13. [pic] 14. [pic]

15. Write the equation of the line tangent to the graph of[pic] at [pic].

Answers:

1. [pic] 2. [pic] 3. [pic] 4. [pic]

5. [pic] 6. [pic] 7. [pic] 8. [pic]

9. [pic] 10. [pic] 11. [pic] 12. [pic]

13. [pic] 14. [pic] 15. [pic] or [pic]

Derivatives of Logarithmic Functions HW

Find the derivative of each function.

1.) [pic]

2.) [pic]

3.) [pic]

4.) [pic]

5.) [pic]

6.) [pic]

7.) [pic]

8.) [pic]

Answers!

1.) [pic]

2.) [pic]

3.) [pic]

4.) [pic]

5.) [pic]

6.) [pic]

7.) [pic]

8.) [pic]

Practice with Integration involving the Natural Log Function

1. [pic] 2. [pic] 3. [pic]

4. [pic] 5. [pic] 6. [pic]

7. [pic] 8. [pic] 9. [pic]

10. [pic] 11. [pic] 12. [pic]

13. [pic] 14. [pic] 15. [pic]

16. [pic] 17. [pic]

18. A population of bacteria is growing at the rate of [pic], where [pic]is the time in days. When [pic], the population is 1000.

a. Write the equation that models the population [pic]in terms of the time [pic].

b. What is the population after 3 days?

c. After how many days will the population be 12,000?

Answers:

1. [pic] 2. [pic] 3. [pic] 4. [pic]

5. [pic] 6. [pic] 7. [pic]

8. [pic] 9. [pic] 10. [pic]

11. [pic] 12. [pic] 13. [pic]

14. [pic] 15. [pic] 16. [pic]

17. [pic] 18. a. [pic] b. [pic] c. [pic]

More Practice with Integration involving the Natural Log Function

1. [pic] 2. [pic] 3. [pic]

4. [pic] 5. [pic] 6. [pic]

7. [pic] 8. [pic] 9. [pic]

10. [pic] 11. [pic] 12. [pic]

13. [pic] 14. [pic] 15. [pic]

16. [pic] 17. [pic]

18. From 1986 through 1992, the number of automatic teller machine (ATM) transactions [pic](in millions) in the United States changed at the rate of [pic], where [pic]corresponds to 1986. In 1992, there were 600 million transactions..

a. Write the model that gives the total number of ATM transactions per year.

b. Use the model to find the number of ATM transactions in 1987.

Answers:

1. [pic] 2. [pic] 3. [pic] 4. [pic]

5. [pic] 6. [pic] 7. [pic]

8. [pic] 9. [pic] 10. [pic] 11. [pic]

12. [pic] 13. [pic] 14. [pic] 15. [pic]

16. [pic] 17. [pic]

18. a. [pic] b. [pic]million transactions

Practice with Integrating the Other Trig Functions

1. [pic] 2. [pic] 3. [pic]

4. [pic] 5. [pic] 6. [pic]

7. [pic] 8. [pic] 9. [pic]

10. [pic] 11. [pic] 12. [pic]

13. [pic] 14. [pic] 15. [pic]

Answers:

1. [pic] 2. [pic] 3. [pic]

4. [pic] 5. [pic] 6. [pic]

7. [pic] 8. [pic] 9. [pic]

10. [pic] 11. [pic] 12. [pic]

13. [pic] 14. [pic] 15. [pic]

Mo’ Trig Integrals HW

Integrate! Don’t forget that you can check by differentiating.

|1.) [pic] |7.) [pic] |

| | |

| | |

| | |

| |8.) [pic] |

|2.) [pic] | |

| | |

| | |

| |9.) [pic] |

| | |

|3.) [pic] | |

| | |

| | |

| |10.) [pic] |

| | |

|4.) [pic] | |

| | |

| | |

| |11.) [pic] |

| | |

|5.) [pic] | |

| | |

| | |

| | |

| |12.) [pic] |

|6.) [pic] | |

| |(HINT: [pic]) |

Answers!

|1.) [pic] |7.) [pic] |

|2.) [pic] |8.) [pic] |

|3.) [pic] |9.) [pic] |

|4.) [pic] |10.) [pic] |

|5.) [pic] |11.) [pic] |

|6.) [pic] |12.) [pic] |

Derivatives with Other Bases HW

|Find the derivative of each function. |Find [pic] implicitly: |

| | |

|1.) [pic] |12.) [pic] |

| | |

|2.) [pic] |13.) [pic] |

| | |

|3.) [pic] |14.) [pic] |

| | |

|4.) [pic] |15.) [pic] |

| | |

|5.) [pic] | |

| | |

|6.) [pic] | |

| | |

|7.) [pic] | |

| | |

|8.) [pic] | |

| | |

|9.) [pic] | |

| | |

|10.) [pic] | |

| | |

|11.) [pic] | |

Answers!

|1.) [pic] |7.) [pic] |12.) [pic] |

| | | |

|2.) [pic] |8.) [pic] |13.) [pic] |

| | | |

|3.) [pic] |9.) [pic] |14.) [pic] |

| | | |

|4.) [pic] |10.) [pic] |15.) [pic] |

| | | |

|5.) [pic] |11.) [pic] | |

| | | |

|6.) [pic] | | |

Integrals with Other Bases HW

Integrate!

1.) [pic]

2.) [pic]

3.) [pic]

4.) [pic]

5.) [pic]

6.) [pic]

7.) [pic]

Answers!

1.) [pic]

2.) [pic]

3.) [pic]

4.) [pic]

5.) [pic]

6.) [pic]

7.) [pic]

9.4 Exponential Growth and Decay

Learning Objectives

A student will be able to:

• Apply the laws of exponential and logarithmic functions to a variety of applications.

• Model situations of growth and decay in a variety of problems.

When the rate of change in a substance or population is proportional to the amount present at any time t, we say that this substance or population is going through either a decay or a growth, depending on the sign of the constant of proportionality.

This kind of growth is called exponential growth and is characterized by rapid growth or decay. For example, a population of bacteria may increase exponentially with time because the rate of change of its population is proportional to its population at a given instant of time (more bacteria make more bacteria and fewer bacteria make fewer bacteria). The decomposition of a radioactive substance is another example in which the rate of decay is proportional to the amount of the substance at a given time instant. In the business world, the interest added to an investment each day, month, or year is proportional to the amount present, so this is also an example of exponential growth.

Mathematically, the relationship between amount [pic]and time [pic]is a differential equation:

[pic]

Separating variables,

[pic]

and integrating both sides,

[pic]

gives us

[pic]

So the solution to the equation [pic]has the form [pic]The box below summarizes the details of this function.

The Law of Exponential Growth and Decay

The function [pic]is a model for exponential growth or decay, depending on the value of [pic]

• If [pic]: The function represents exponential growth (increase).

• If [pic]: The function represents exponential decay (decrease).

Where [pic]is the time, [pic]is the initial population at [pic]and [pic]is the population after time [pic]

Applications of Growth and Decay

Radioactive Decay

In physics, radioactive decay is a process in which an unstable atomic nucleus loses energy by emitting radiation in the form of electromagnetic radiation (like gamma rays) or particles (such as beta and alpha particles). During this process, the nucleus will continue to decay, in a chain of decays, until a new stable nucleus is reached (called an isotope). Physicists measure the rate of decay by the time it takes a sample to lose half of its nuclei due to radioactive decay. Initially, as the nuclei begins to decay, the rate starts very fast and furious, but it slows down over time as more and more of the available nuclei have decayed. The figure below shows a typical radioactive decay of a nucleus. As you can see, the graph has the shape of an exponential function with [pic]

[pic]

The equation that is used for radioactive decay is [pic]We want to find an expression for the half-life of an isotope. Since half-life is defined as the time it takes for a sample to lose half of its nuclei, then if we starting with an initial mass [pic](measured in grams), then after some time [pic][pic]will become half the amount that we started with, [pic]Substituting this into the exponential decay model,

[pic]

Canceling [pic]from both sides,

[pic]

Solving for [pic]which is the half-life, by taking the natural logarithm on both sides,

[pic]

Solving for [pic]and denoting it with new notation [pic]for half-life (a standard notation in physics),

[pic]

This is a famous expression in physics for measuring the half-life of a substance if the decay constant [pic]is known. It can also be used to compute [pic]if the half-life [pic]is known.

Example 1:

A radioactive sample contains [pic]of nobelium. If you know that the half-life of nobelium is [pic], how much will remain after [pic]?

Solution:

Before we compute the mass of nobelium after [pic], we need to first know its decay rate [pic]Using the half-life formula,

[pic]

So the decay rate is [pic]The common unit for the decay rate is the Becquerel [pic]: [pic]is equivalent to [pic]decay per sec. Since we found [pic], we are now ready to calculate the mass after [pic]. We use the radioactive decay formula. Remember, [pic]represents the initial mass, [pic], and [pic]. Thus

[pic]

So after [pic], the mass of the isotope is approximately [pic].

Population Growth

The same formula [pic]can be used for population growth, except that [pic]since it is an increasing function.

Example 2:

A certain population of bacteria increases continuously at a rate that is proportional to its present number. The initial population of the bacterial culture is [pic]and jumped to [pic]bacteria in [pic].

1. How many will be there in [pic]?

2. How long will it take the population to double?

Solution:

From reading the first sentence in the problem, we learn that the bacteria is increasing exponentially. Therefore, the exponential growth formula is the correct model to use.

1. Just like we did in the previous example, we need to first find [pic]the growth rate. Notice that [pic]and [pic]Substituting and solving for [pic]

[pic]

Dividing both sides by [pic]and then projecting the natural logarithm on both sides,

[pic]

Now that we have found [pic]we want to know how many will be there after [pic]. Substituting,

[pic]

2. We are looking for the time required for the population to double. This means that we are looking for the time at which [pic]Substituting,

[pic]

Solving for [pic]requires taking the natural logarithm of both sides:

[pic]

Solving for [pic]

[pic]

This tells us that after about [pic](around [pic]) the population of the bacteria will double in number.

Compound Interest

Investors and bankers depend on compound interest to increase their investment. Traditionally, banks added interest after certain periods of time, such as a month or a year, and the phrase was “the interest is being compounded monthly or yearly.” With the advent of computers, the compounding could be done daily or even more often. Our exponential model represents continuous, or instantaneous, compounding, and it is a good model of current banking practices. Our model states that

[pic]

where [pic]is the initial investment (present value) and [pic]is the future value of the investment after time [pic]at an interest rate of [pic]The interest rate [pic]is usually given in percentage per year. The rate must be converted to a decimal number, and [pic]must be expressed in years. The example below illustrates this model.

Example 3:

An investor invests an amount of [pic]and discovers that its value has doubled in 5 years. What is the annual interest rate that this investment is earning?

Solution:

We use the exponential growth model for continuously compounded interest,

[pic]

Thus

[pic]

The investment has grown at a rate of [pic]per year.

Example 4:

Going back to the previous example, how long will it take the invested money to triple?

Solution:

[pic]

Other Exponential Models and Examples

Not all exponential growths and decays are modeled in the natural base [pic]or by [pic]Actually, in everyday life most are constructed from empirical data and regression techniques. For example, in the business world the demand function for a product may be described by the formula

[pic]

where [pic]is the price per unit and [pic]is the number of units produced. So if the business is interested in basing the price of its unit on the number that it is projecting to sell, this formula becomes very helpful. If a motorcycle factory is projecting to sell [pic]in one month, what price should the factory set on each motorcycle?

[pic]

Thus the factory’s base price for each motorcycle should be set at [pic]

As another example, let’s say a medical researcher is studying the spread of the flu virus through a certain campus during the winter months. Let’s assume that the model for the spread is described by

[pic]

where [pic]represents the total number of infected students and [pic]is the time, measured in days. Suppose the researcher is interested in the number of students who will be infected in the next week ([pic] days). Substituting [pic]into the model,

[pic]

According to the model, [pic]students will become infected with the flu virus. Assume further that the researcher wants to know how long it will take until [pic]students become infected with the flu virus. Solving for [pic]

[pic]

Cross-multiplying,

[pic]

Projecting ln on both sides,

[pic]

Substituting for [pic],

[pic]

So the flu virus will spread to [pic]students in [pic]days.

Other applications are introduced in the exercises.

Multimedia Links

For a video presentation of exponential growth involving bacteria (some calculus in part c) (14.0), see Khan Academy, Exponential Growth and Decay (16:00)[pic].

For a video presentation of exponential decay (14.0), see Just Math Tutoring, Exponential Decay, Finding Half-Life (6:08)[pic].

For additional problems on exponential growth and decay (14.0), see Khan Academy, Word Problem Solving, Exponential Growth and Decay (7:21)[pic].

Review Questions

1. In 1990, the population of the USA was 249 million. Assume that the annual growth rate is 1.8%.

a. According to this model, what was the population in the year 2000?

b. According to this model, in which year the population will reach 1 billion?

2. Prove that if a quantity A is exponentially growing and if A1 is the value at t1 and A2 at time t2, then the growth rate will be given by [pic]

3. Newton’s Law of Cooling states that the rate of cooling is proportional to the difference in temperature between the object and the surroundings. The law is expressed by the formula

[pic]

where T0 is the initial temperature of the object at t = 0, Tr is the room temperature (i.e., temperature of the surroundings), and k is a constant that is unique for the measuring instrument (the thermometer) called the time constant. Suppose a liter of juice at 23°C is placed in the refrigerator to cool. If the temperature of the refrigerator is kept at 11°C and k = 0.417, what is the temperature of the juice after 3 minutes?

4. Referring back to problem 3, if it takes an object 320 seconds to cool from 40°C above room temperature to 22°C above room temperature, how long will it take to cool another 10°C?

5. Polonium-210 is a radioactive isotope with a half-life of 140 days. If a sample has a mass of 10 grams, how much will remain after 10 weeks?

6. In the physics of acoustics, there is a relationship between the subjective sensation of loudness and the physically measured intensity of sound. This relationship is called the sound level (. It is specified on a logarithmic scale and measured with units of decibels (dB). The sound level ( of any sound is defined in terms of its intensity I (in the SI-MKS unit system, it is measured in watts per meter squared, W / m²) as [pic] For example, the average decibel level of a busy street traffic is 70 dB, normal conversation at a dinner table is 55 dB, the sound of leaves rustling is 10 dB, the siren of a fire truck at 30 meters is 100 dB, and a loud rock concert is 120 dB. The sound level 120 dB is considered the threshold of pain for the human ear and 0 dB is the threshold of hearing (the minimum sound that can be heard by humans.)

a. If at a heavy metal rock concert a dB meter registered 130 dB, what is the intensity I of this sound level?

b. What is the sound level (in dB) of a sound whose intensity is 2.0 x 10-6 W / m²?

7. Referring to problem #6, a single mosquito 10 meters away from a person makes a sound that is barely heard by the person (threshold 0 dB). What will be the sound level of 1000 mosquitoes at the same distance?

8. Referring back to problem #6, a noisy machine at a factory produces a sound level of 90 dB. If an identical machine is placed beside it, what is the combined sound level of the two machines?

Review Answers

1.

a. [pic]

b. [pic]

2. Hint: use[pic]

3. [pic]

4. [pic], about [pic]

5. [pic]

6.

a. [pic]

b. [pic]

7. [pic]

8. [pic]

9.5 Derivatives and Integrals Involving Inverse Trigonometric Functions

Learning Objectives

A student will be able to:

• Learn the basic properties inverse trigonometric functions.

• Learn how to use the derivative formula to use them to find derivatives of inverse trigonometric functions.

• Learn to solve certain integrals involving inverse trigonometric functions.

A Quick Algebraic Review of Inverse Trigonometric Functions

You already know what a trigonometric function is, but what is an inverse trigonometric function? If we ask what is [pic]equal to, the answer is [pic]That is simple enough. But what if we ask what angle has a sine of [pic]? That is an inverse trigonometric function. So we say [pic]but [pic]The “[pic]” is the notation for the inverse of the sine function. For every one of the six trigonometric functions there is an associated inverse function. They are denoted by

[pic]

Alternatively, you may see the following notations for the above inverses, respectively,

[pic]

Since all trigonometric functions are periodic functions, they do not pass the horizontal line test. Therefore they are not one-to-one functions. The table below provides a brief summary of their definitions and basic properties. We will restrict our study to the first four functions; the remaining two, [pic]and [pic]are of lesser importance (in most applications) and will be left for the exercises.

|Inverse Function |Domain |Range |Basic Properties |

|[pic] |[pic] |[pic] |[pic] |

|[pic] |[pic] |[pic] |[pic] |

|[pic] |all [pic] |[pic] |[pic] |

|[pic] |[pic] |[pic] |[pic] |

The range is based on limiting the domain of the original function so that it is a one-to-one function.

Example 1:

What is the exact value of [pic]?

Solution:

This is equivalent to [pic]. Thus [pic]. You can easily confirm this result by using your scientific calculator.

Example 2:

Most calculators do not provide a way to calculate the inverse of the secant function, [pic]A practical trick however is to use the identity

[pic]

(Recall that [pic])

For practice, use your calculator to find [pic]

Solution:

Since

[pic]

[pic]

Here are two other identities that you may need to enter into your calculator:

[pic]

The Derivative Formulas of the Inverse Trigonometric Functions

If [pic]is a differentiable function of [pic]then the generalized derivative formulas for the inverse trigonometric functions are (we introduce them here without a proof):

[pic]

Example 3:

Differentiate [pic]

Solution:

Let [pic]so

[pic]

Example 4:

Differentiate [pic]

Solution:

Let [pic]so

[pic]

Example 5:

Find [pic]if [pic]

Solution:

Let [pic]

[pic]

The Integration Formulas of the Inverse Trigonometric Functions

The derivative formulas in the box above yield the following integrations formulas for inverse trigonometric functions:

[pic]

Example 6:

Evaluate [pic]

Solution:

Before we integrate, we use [pic]substitution. Let [pic](the square root of [pic]). Then [pic]Substituting,

[pic]

Example 7:

Evaluate [pic]

Solution:

We use [pic]substitution. Let [pic]so [pic]Substituting,

[pic]

Example 8:

Evaluate the definite integral [pic].

Solution:

Substituting [pic][pic]

To change the limits,

[pic]

Thus our integral becomes

[pic]

Multimedia Links

For a video presentation of the derivatives of inverse trigonometric functions (4.4), see Math Video Tutorials by James Sousa, The Derivatives of Inverse Trigonometric Functions (8:55)[pic].

For three presentations of integration involving inverse trigonometric functions (18.0), see Math Video Tutorials by James Sousa, Integration Involving Inverse Trigonometric Functions, Part 1 (7:39)[pic]; Math Video Tutorials by James Sousa, Integration Involving Inverse Trigonometric Functions, Part 2 (6:39)[pic]; This last video includes an example showing completing the square (19.0), Math Video Tutorials by James Sousa, Integration Involving Inverse Trigonometric Functions, Part 3 (6:18)[pic].

Review Questions

1. Find [pic] of [pic]

2. Find [pic] of [pic]

3. Find [pic] of [pic]

4. Find [pic] of [pic]

5. Find [pic] of [pic]

6. Evaluate [pic]

7. Evaluate [pic]

8. Evaluate [pic]

9. Evaluate [pic]

10. Given the points A(2, 1) and B(5, 4), find a point Q in the interval [2, 5] on the x-axis that maximizes angle [pic].

Review Answers

1. [pic]

2. [pic]

3. [pic]

4. [pic]

5. [pic]

6. [pic]

7. [pic]

8. [pic]

9. [pic]

10. [pic]

9.6 L’Hôspital’s Rule

Learning Objectives

A student will be able to:

• Learn how to find the limit of indeterminate form [pic] by L’Hospital’s rule.

If the two functions [pic]and [pic]are both equal to zero at [pic]then the limit

[pic]

cannot be found by directly substituting [pic]The reason is because when we substitute [pic]the substitution will produce [pic]known as an indeterminate form, which is a meaningless expression. To work around this problem, we use L’Hospital’s rule, which enables us to evaluate limits of indeterminate forms.

L’Hospital’s Rule

If [pic], and [pic]and [pic]exist, where [pic], then

[pic]

The essence of L’Hospital’s rule is to be able to replace one limit problem with a simpler one. In each of the examples below, we will employ the following three-step process:

1. Check that [pic]is an indeterminate form [pic]To do so, directly substitute [pic]into [pic]and [pic]If you get [pic]then you can use L’Hospital’s rule. Otherwise, it cannot be used.

2. Differentiate [pic]and [pic]separately.

3. Find [pic]If the limit is finite, then it is equal to the original limit [pic].

Example 1:

Find [pic]

Solution:

When [pic]is substituted, you will get [pic]

Therefore L’Hospital’s rule applies:

[pic]

Example 2:

Find [pic]

Solution:

We can see that the limit is [pic]when [pic]is substituted.

Using L’Hospital’s rule,

[pic]

Example 3:

Use L’Hospital’s rule to evaluate [pic].

Solution:

[pic]

Example 4:

Evaluate [pic]

Solution:

[pic]

Example 5:

Evaluate [pic].

Solution:

We can use L’Hospital’s rule since the limit produces the [pic]once [pic]is substituted. Hence

[pic]

A broader application of L’Hospital’s rule is when [pic]is substituted into the derivatives of the numerator and the denominator but both still equal zero. In this case, a second differentiation is necessary.

Example 6:

Evaluate [pic]

Solution:

[pic]

As you can see, if we apply the limit at this stage the limit is still indeterminate. So we apply L’Hospital’s rule again:

[pic]

Review Questions

Find the limits.

1. [pic]

2. [pic]

3. [pic]

4. [pic]

5. [pic]

6. Assume k is a nonzero constant and x > 0.

a. Show that [pic]

b. Use L’Hospital’s rule to find [pic]

7. Cauchy’s Mean Value Theorem states that if the functions f and g are continuous on the interval (a, b) and [pic], then there exists a number c such that [pic] Find all possible values of c in the interval (a, b) that satisfy this property for [pic] on the interval [pic].

Review Answers

1. [pic]

2. [pic]

3. [pic]

4. [pic]

5. [pic]

6. [pic]

7. [pic]

Texas Instruments Resources

In the CK-12 Texas Instruments Calculus FlexBook, there are graphing calculator activities designed to supplement the objectives for some of the lessons in this chapter. See .

Practice with L’Hopital’s Rule

1. [pic] 2. [pic] 3. [pic] 4. [pic]

5. [pic] 6. [pic] 7. [pic] 8. [pic]

9. [pic] 10. [pic] 11. [pic] 12. [pic]

13. [pic] 14. [pic] 15. [pic] 16. [pic]

17. [pic] 18. [pic] 19. [pic] 20. [pic]

21. [pic] 22. [pic] 23. [pic] 24. [pic]

25. [pic]

In general, [pic], [pic] .

26. [pic] 27. [pic] 28. [pic]

29. [pic] 30. [pic] 31. [pic]

Answers:

1. 0 2. -1 3. NL 4. NL 5. NL 6. [pic] 7. 1 8. 0 9. [pic] 10. NL 11. 0 12. [pic]13. 0 14. NL 15. [pic] 16. 0 17. 1 18. 2 19. [pic] 20. 0 21. [pic] 22. 4 23. [pic] 24. [pic] 25. [pic]

26. [pic] 27. [pic] 28. [pic] 29. [pic] 30. [pic] 31. [pic]

L’Hôpital’s Rule HW

Evaluate the following limits.

1.) [pic]

2.) [pic]

3.) [pic]

4.) [pic]

5.) [pic]

6.) [pic]

7.) [pic]

8.) [pic]

9.) [pic]

10.) [pic]

Answers!

1.) [pic] 2.) 1 3.) 0 4.) 0 5.) DNE

6.) [pic] 7.) 2 8.) ( 9.) 4 10.) [pic]

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