The Logarithmic Derivative

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7 SE Transcendental

Functions - TI

Chapter Outline

7.1 INVERSES OFFUNCTIONS

www.ck12.org Chapter 7. SE Transcendental Functions - TI

7.1 Inverses of Functions

This activity is intended to supplement Calculus, Chapter 6, Lesson 1.

Exploring the Data

What happens when the domain and range of a function trade places? This special “trade” creates inverses of functions, which will be explored in this activity.

To the right is wind tunnel data obtained from www.doe.state.la.us.

Wind tunnel experiments are used to test the wind friction or resistance of an automobile at different speeds. The given data shows wind speed in miles per hour versus resistance in pounds.

Record this data inL1 andL2 by pressingSTATand selectEdit. Make sure to clear the lists before you input any data.

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7.1:

Speed (MPH) Resistance (lbs.)

10 6.4

21 9.2

34 17.0

40 22.4

45 30.2

55 59.2

1. Construct a scatter plot and graph your data on the grid below. Recall that to construct a scatter plot, press 2^ndY=, select the scatter plot option, and make sure thatL1 andL2 are selected for Xlist and Ylist. PressZOOMand select theZoomStatoption.

2. Now graph a second scatter plot on your calculator (Plot 2) and graph the new data on the grid above. For this graph, you will switch the domain and range. This means that in the StatPlot menu,L1 andL2 trade places.

7.1. Inverses of Functions www.ck12.org

Exploring the Problem

3. After graphing the original wind tunnel data and then graphing the data with the domain and range switched, what did you notice about the graphs of the two sets of points?

4. Find the midpoint between the first points from the first and second scatter plots and the midpoint of the last points from the first and second scatter plots.

5. Find the equation of the line that connects these two midpoints. Graph this line on the grid above and on your calculator. What does this line represent?

Developing the Pattern Further

Clear all data inL1 andL2. Also, clear all plots; Plot1, Plot2, andY1.

6. Graphy=2x+3 in a standard viewing window. Use thezerocommand (2^nd TRACE) to find thex−intercept.

UseTRACEto find they−intercept.

7. Switch the locations of thex−andy−coordinates of these points to(y,x)and input these asL1 andL2. Graph these two points usingPlot1. Find the equation of the line connecting these two points and graph it inY2.

8. Switch x andy in the equation y=2x+3 and solve fory. How does your result compare to your answer to Question 7?

9. List 3 ways in which the inverse of a function may be obtained.

10. The inverse of a function is always a function.2Agree 2Disagree

Explain your answer choice and provide at least one example to illustrate why you chose it.

11. Find the inverse of the following functions if an inverse exists. If an inverse does not exist, write “does not exist.”

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7.2:

Function Inverse

f(x) =6x−2 f⁻¹(x) =

f(x) =¹₂x−³₄ f⁻¹(x) =

|f(x) =|x| f⁻¹(x) =

7.2. The Logarithmic Derivative www.ck12.org

7.2 The Logarithmic Derivative

This activity is intended to supplement Calculus, Chapter 6, Lesson 3.

Problem 1 –The Derivative of

If(x,y)is a point ony= f(x)andy=g(x)is the inverse of f(x), then(y,x)is a point ong(x). We know thate⁰=1 and ln(1) =0, so(0,1)is a point ony=e^xand(1,0)is a point ony=ln(x). We could do this for several points and keep getting the same inverse results.

Thus, ify=e^x, then x=e^y will be equivalent toy=ln(x)because they are inverses of one another. Now we can take the implicit derivative with respect toxofx=e^y.

x=e^y→1=dy

dx·e^y→1=dy

dx·x→ 1 x =dy

dx Use thelimitcommand to test this formula. Be careful with your parentheses.

• Find limh→0ln(2+h)−ln(2)

h .

• Do the same with limh→0ln(3+h)−ln(3)

h .

• What is limh→0ln(x+h)−ln(x)

h ?

Use thederivativecommand to find the derivative of the logarithmic function f(x) =ln(x).

Problem 2 –The Derivative of

What happens if our logarithm has a base other thane? We need to know how to take the derivative of the function y=log_a(x).

First we want to comparey1=ln(x)andy2=log₂(x).

To enter log₂(x), use the alpha keys to spell outlog.

Within the parentheses, enter the expression, then the base.

• Graph both functions (ln(x)and log₂(x)) on the same set of axes. Sketch your graph to the right. What do you notice?

www.ck12.org Chapter 7. SE Transcendental Functions - TI

• Do the same steps withy1=ln(x)andy3=log₄(x). What do you notice?

• Simplify the following ratios.

ln(x) log₂(x)

ln(x) log₄(x)

ln(x) log_a(x) Sometimes the ratio_log^ln(x)

a(x)is written as ln(x) =ln(a)·log_a(x). We can rewrite this ratio as log_a(x) =^ln(x)_ln(a) and call it an identity.

• Graph the following functions on the same set of axes: y1=ln(x),y2=ln(2)·log₂(x),y3=ln(3)·log₃(x).

What was the result?

What happens when we take the derivative ofy=log_a(x). Use thederivative command to find the derivatives of the functions below.

f(x) =log₂(x) g(x) =log₃(x) h(x) =log_a(x)

• Do you notice a pattern?

What does log₂(e)equal? If we use the formula from earlier in this class, we get log₂(e) =^ln(e)_ln(2)=_ln(2)¹ . Therefore, the general result isy=log_a(x)→ ^dy_dx=_(xln(a))¹ .

Problem 3 –Derivative of Exponential and Logarithmic Functions Using the Chain Rule

Now we want to take the derivative of more complicated functions:

Recall:y=a^u→ ^dy_dx=a^{u du}_dx whereudepends onx.

7.2. The Logarithmic Derivative www.ck12.org

• Suppose thaty=log_a(u), whereudepends onx. Using the chain rule, take the derivative of this function.

Find the derivative of the following functions with the chain rule.

Identifyu(x)andafor each function before you find the derivative.

• f(x) =5^(x2) u(x) = a= f⁰(x) =

• g(x) =e^(x3+2) u(x) = a= g⁰(x) =

• h(x) =log₃(x⁴+7) u(x) = a= h⁰(x) =

• j(x) =ln p x⁶+2

u(x) = a=

j⁰(x) =

www.ck12.org Chapter 8. SE Integration Techniques - TI

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8 SE Integration Techniques - TI

Chapter Outline

8.1 INTEGRATION BY SUBSTITUTION