1. HWQ 1/12/15
Evaluate the definite integral:
No calculator please.

2. Section 5.3 20143 Derivatives of Inverse Functions
Objective: To find the derivative of the inverse of a function.

3. Inverse Functions - Review
Inverse functions are basically functions that “cancel” when we perform a composition.
Formally, we say that the function g (x) is the inverse function of f (x) if:
and

4. Inverse Functions
The graph of the inverse of f (x), f -1(x), is the reflection of f (x) across the line y = x.
The inverse function exists (without any constraints) if it satisfies one condition:
A function has an inverse iff it is one-to-one
For every x, there is only one y and for every y, there is only one x.
If f is strictly monotonic for its entire domain, then it is one-to-one and therefore has an inverse function.

5. Inverse Functions If an inverse is a function, then following is true:
If f is continuous on its domain, then f -1is continuous on its domain.
If f is increasing on its domain, then f -1is increasing on its domain.
If f is decreasing on its domain, then f -1is decreasing on its domain.
If f is differentiable on an interval and its derivative is not zero, then f -1is differentiable for all x where the derivative of f is not zero.
If g is the inverse of f, then the derivative of the inverse is given by:

6. Example 1 – Verifying Inverse Functions
Show that the functions are inverse functions of each other.
and
Solution:
Because the domains and ranges of both f and g consist of all real numbers, you can conclude that both composite functions exist for all x.
The compositions f (g(x)) and g(f(x)) are given by:

7. Example 1 – Solution
cont'd
Because f (g(x)) = x and g(f (x)) = x, you can conclude that f and g are inverse functions of each other (see Figure 5.11).
Figure 5.11

8. Inverse Functions
In Figure 5.11, the graphs of f and g = f –1 appear to be mirror images of each other with respect to the line y = x. The graph of f –1 is a reflection of the graph of f in the line y = x.

9. Inverse Functions
The idea of a reflection of the graph of f in the line y = x is generalized in the following theorem.
Figure 5.12

10. Inverse Functions

11. Finding an Inverse Function

12. Derivative of an Inverse Function
If (x,y) is a point on the graph of a function, the derivative of the function at x will be the reciprocal of the derivative of its inverse at y.
Graphs of inverse functions have reciprocal slopes at inverse points.

13. Derivative of an Inverse Function
If (x,y) is a point on the graph of a function, the derivative of the function at x will be the reciprocal of the derivative of its inverse at y.
Graphs of inverse functions have reciprocal slopes at inverse points.

14. The Derivative of the Inverse
Let’s try to get a better understanding of what this formula is actually stating:
Let this function be f (x)
f (x)
The inverse function will be reflected across the line y = x
g (x)
Let this function be f -1 (x) = g (x)

15. The Derivative of the Inverse
Let’s try to get a better understanding of what this formula is actually stating:
Let’s say we wish to find the slope of the tangent line of g (x) at x = 5
f (x)
g (x)
The coordinate will be (5, g (5))
The slope of the tangent line will be some constant; we’ll call it g ‘(x)
(5, g (5)) 5

16. The Derivative of the Inverse
Let’s try to get a better understanding of what this formula is actually stating:
Let’s say we wish to find the slope of the tangent line of g (x) at x = 5
f (x)
This coordinate will be the reverse or the inverse (g (5), 5)
g (x)
We know that if there exists a point on g (x), then there is an inverse point on f (x)
(g (5), 5)
(5, g (5)) 5

17. The Derivative of the Inverse
Let’s try to get a better understanding of what this formula is actually stating:
According to this formula, the reciprocal of the slope of the tangent line of f(x) at x = g(5) is the SAME as the slope of the tangent line of g(x) at x = 5
In other words, if we want to find the slope of the tangent line of g(x) for some x, all we have to do is find the reciprocal of the derivative of the f(x) when f(x) = (the given x of the inverse)
f (x)
g (x)
(g (5), 5)
(5, g (5)) 5

18. The Derivative of the Inverse
Let’s try to get a better understanding of what this formula is actually stating:
f (x)
If we let the coordinate be more generic such as (a, b), then we could say:
g (x)
(b, a)
(a,b)
Graphs of inverse functions
have reciprocal slopes at inverse points. a

19. Examples What are your options for answering this question?
1) Find the inverse, then find its derivative at x=8.
2) Find where f(x) = 8, find the derivative there,
and reciprocate.
3) Use the formula.

20. Examples
First, let’s try the first option. Find the inverse by switching x and y and solving for y:
Find the derivative of f -1:

21. Examples
Now, let’s try the second option. Find where f(x) = 8, find the derivative there, and reciprocate.
This is usually the easiest method.

22. Examples
Now, let’s try to find the derivative of the inverse by using the formula:
We still need to find g (x); but since g (x) is the inverse of f (x), we know that the x and y values switch or (x, g (x)) → (g (x), x)

23. Examples

24. Examples
Find g (x). For this particular problem , we just need to solve f (x) = -2 to find g (x). This can be difficult at times, so try to plug in numbers or use the calculator if allowed.
This works when x = 1

25. Examples

26. Examples
Solve f (x) = 5 to find g (x)
This works when x = 2

27. Examples

28. Examples
Solve f (x) = e
x = e

29. Examples
5. Find (f –1)'(3) if

30. Derivative of an Inverse Function
In Example 5, note that at the point (2, 3) the slope of the graph of f is 4 and at the point (3, 2) the slope of the graph of f –1 is (see Figure 5.17).

31. Derivative of an Inverse Function
This reciprocal relationship can be written as shown below.
If y = g(x) = f –1(x), then f (y) = x and f'(y) = Theorem 5.9 says that

32. Example 6 – Graphs of Inverse Functions Have Reciprocal Slopes
Let f(x) = x2 (for x ≥ 0) and let Show that the slopes of the graphs of f and f –1 are reciprocals at each of the following points.
a. (2, 4) and (4, 2) b. (3, 9) and (9, 3)
Solution:
The derivative of f and f –1 are given by
f'(x) = 2x and
a. At (2, 4), the slope of the graph of f is f'(2) = 2(2) = 4.
At (4, 2), the slope of the graph of f –1 is

33. Example 6 – Solution
cont'd
b. At (3, 9), the slope of the graph of f is f'(3) = 2(3) = 6.
At (9, 3), the slope of the graph of f –1 is
So, in both cases, the slopes are
reciprocals, as shown in Figure 5.18.
Figure 5.18

34. HWQ

35. HWQ

36. Examples Calculators allowed – round to 3 decimal places
Solve f (x) = 3 using the calculator
x = 2.575

37. Examples

38. Examples
Solve F (x) = 0
This will only happen when x = 2

39. Homework Inverse Functions Day 1: P. 347: 1, 3, 9, 11, 71-89 odd
Day 2: Derivatives of Inverses W/S

40. Homework
Derivatives of Inverses W/S

41. AP FRQ
2007 #3