Inverse Function Theorem Exercise

The purpose of this exercise is to use an applet and a graphing calculator to understand what the Inverse Function Theorem says and visualize why it is true.  Click here to display the applet in a separate window.      Smaller applet

1.

Position the point x as close to π/3  1.047 as possible.  The applet shows a red line tangent to y = sin(x) at the point .  What is the slope of the tangent line?  What is an equation for the tangent, in the form ?  View an answer

You may be able to position x more accurately if you enlarge the image by moving the unit point (1, 0) further away from the origin then repositioning the origin to view important items.

2.

The inverse of f(x) = sin(x) is f ¹(x) = sin ¹x, and the graph of the inverse is the reflection about the line y = x.  What are the coordinates of the point (y, x) corresponding to (x, y)?  What is the slope of the blue line tangent to sin ¹x at (y, x)?  How does the Inverse Function Theorem express this relationship?  View an answer

3.

Drag point x as close as possible to .  What is the slope of sin(x) here?  If we let g(x) = sin^-1(x), which of the following statements is implied by the inverse function theorem?

        View an answer

The next two questions will explore the relationship between the lines tangent to sin(x) and sin^-1x at corresponding points.

4.

Move x as close as possible to −π/6  −0.524.  Write an equation in the form  for the line tangent to y = sin(x) at this point.  Find the inverse of this linear function in the usual way, by solving for x in terms of y and then exchanging the variables x and y.  View an answer

5.

Now find the equation in the form  for the line tangent to y = sin ¹x at the point .  This should be the same as the inverse of the linear function from question 4.  Explain in your own words why the line tangent to sin(x) at point (x, y) and line tangent to sin ¹x at (y, x) are inverses, and why their slopes are reciprocals. Answer

Recall that if f(x) = e^x then f  ^-1(x) = ln x.  The next set of questions will use the graphing calculator to explore what the Inverse Function Theorem says about these functions.

6.

Sketch the graph of f(x) = e^x.  What is the derivative of this function at x = 2?  Write the equation for the line tangent to the curve at (2, f(2) ) and sketch its graph. View an answer

7.

Let g(x) = ln x.  Then g(x) = f ¹(x).  If we know from the previous question that , which of the following statements about g(x) is implied by the Inverse Function Theorem?

            
View Answer

8.

Find an equation in the form  for the line tangent to  at  and sketch its graph.  View an answer

9.

How can you convince yourself that the tangent to  at  and the tangent to  at  are reflections about the line y = x?  Are the slopes of these tangent lines reciprocals, as stated by the Inverse Function Theorem? View an answer

10.

Again letting g(x) = ln x while f(x) = e^x, use your graphing calculator to confirm that   What does the Inverse Function Theorem say about ?  Why? View an answer

1.

The slope of the tangent line at x = π/3 is  and the equation for the tangent is  or y = 0.5x + 0.342

2.

Corresponding to the point  on y = sin(x) will be its reflection .  The slope of sin ¹x at  is 2.

The Inverse Function Theorem says that the slope of sin ¹x at  is .

3.

The slope of sin(x) at  is 0.707.  The Inverse Function Theorem says that then the slope of sin^-1(x) at  is .  So (D) is correct.

4.

The slope of sin(x) at  is 0.866, so the tangent is y = 0.866x  0.0466.  Solving this for x in terms of y gives x = 1.155y + 0.0538.  Thus the inverse of the linear function is f ¹(x) = 1.155x + 0.0538.

5.

Answers to this will vary.

6.

The derivative of e^x at (2, e²)  (2, 7.389) is 7.389.  The tangent line at this point is y  7.389 = 7.389(x  2) or y = 7.389x  7.389.

7.

If  then  so (D) is implied.

8.

 so the tangent at  is  or  

9.

A way to obtain an equation for the inverse of  is to exchange the variables x and y, then solve for y.

10.