HWQ 1/12/15HWQ 1/12/15

2

1

1 lne xdx

x

Evaluate the definite integral:No calculator please.

7

3

2

Section 5.3 Section 5.3 2014320143

Derivatives of Inverse FunctionsDerivatives of Inverse Functions

Objective: To find the derivative Objective: To find the derivative of the inverse of a function.of the inverse of a function.

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:

Inverse Functions - ReviewInverse Functions - Review

gxxxgf ofdomain in the allfor )( and

fxxxfg ofdomain in the allfor )(

The inverse function exists (without any constraints) if it satisfies one condition:

1. 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.

Inverse FunctionsInverse Functions The graph of the inverse of

f (x), f -1(x), is the reflection of f (x) across the line y = x.

Example 1 – Example 1 – Verifying Inverse Verifying Inverse FunctionsFunctions

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:

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

Example 1 – Example 1 – SolutionSolutioncont'd

The idea of a reflection of the graph of f in the line y = x is generalized in the following theorem.

Figure 5.12

Inverse FunctionsInverse Functions

Inverse FunctionsInverse Functions

Finding an Inverse FunctionFinding an Inverse Function

2 3f x x 1Find f x

2

1 3

2

xf x

Derivative of an Inverse FunctionDerivative of an Inverse Function

116 3 3

3f f

6Find f

16 3Find f and f

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.

16 3, 3 6f f

Derivative of an Inverse FunctionDerivative 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.

x

y

x

y

The Derivative of the InverseThe Derivative of the Inverse Let’s try to get a better understanding of

what this formula is actually stating:

0)(,)(

1)(

xgf

xgfxg

Let this function be f (x)

The inverse function will be reflected across the line y = x

f (x)

g (x)

Let this function be f -1 (x) = g (x)

x

y

The Derivative of the InverseThe Derivative of the Inverse Let’s try to get a better understanding of

what this formula is actually stating:

0)(,)(

1)(

xgf

xgfxg

f (x)

g (x)

Let’s say we wish to find the slope of the tangent line of g (x) at x = 5

5

The coordinate will be (5, g (5))

(5, g (5))

The slope of the tangent line will be some constant; we’ll call it g ‘(x)

x

y

The Derivative of the InverseThe Derivative of the Inverse Let’s try to get a better understanding of

what this formula is actually stating:

0)(,)(

1)(

xgf

xgfxg

f (x)

g (x)

Let’s say we wish to find the slope of the tangent line of g (x) at x = 5

5

We know that if there exists a point on g (x), then there is an inverse point on f (x)

(5, g (5))

This coordinate will be the reverse or the inverse (g (5), 5)

(g (5), 5)

x

y

The Derivative of the InverseThe Derivative of the Inverse Let’s try to get a better understanding of

what this formula is actually stating:

0)(,)(

1)(

xgf

xgfxg

f (x)

g (x)

5

According to this formula, the reciprocal of the slope of the tangent line of f(x) at x = g(5) is the SAMESAME as the slope of the tangent line of g(x) at x = 5

(5, g (5))

(g (5), 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)

The Derivative of the InverseThe Derivative of the Inverse Let’s try to get a better understanding of

what this formula is actually stating:

0)(,)(

1)(

xgf

xgfxg

x

yf (x)

g (x)

a

(a,b)

(b, a)

If we let the coordinate be more generic such as (a, b), then we could say:

0,1

)(

bfbf

ag

Graphs of inverse functions have reciprocal slopes at inverse points.

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.

ExamplesExamples

31 )( if )8( Find.1 xxff

First, let’s try the first option. Find the inverse by switching x and y and solving for y:

3/23

1

x

ExamplesExamples

yx 3/13yx 3/11 )( xxf

12

1 3/21

3

1

xf 43

1

3/21

83

18

f

Find the derivative of f -1:

31 )( if )8( Find.1 xxff

Now, let’s try the second option. Find where f(x) = 8, find the derivative there, and reciprocate.

f ( x ) h a s t h e p o i n t ( ? ,8 )

ExamplesExamples

31 )( if )8( Find.1 xxff

1f (x ) h a s th e p o in t (8 ,? )

( 2 ) 1 2f 1 1(8)

12f

This is usually the easiest method.

ExamplesExamples

23)( xxf

31 )( if )8( Find.1 xxff

3)( xxf 2

1

)(3

1)(

xgxf

12

1

2)8()8()8( 3/11 fg

43

1

223

1

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)

Now, let’s try to find the derivative of the inverse by using the formula:

0)(,)(

1)(

xgf

xgfxg

2

1

)8(3

1)8(

gf

ExamplesExamples

12)( if )2( Find.2 31 xxxff

1 1(2)

5f

1 has the point 2,?f x has the point ?, 2f x

has the point 1, 2f x

1 = 5,Since f

ExamplesExamples

0 ,6

)( if )5( Find.3 31 x

xxxff

1 2(5)

27f

ExamplesExamples

0 ,ln)( if )( Find.4 1 xxxxfef

1 1( )

2f e

ExamplesExamples

5. Find (f –1)'(3) if

1 1(3)

4f

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).

Derivative of an Inverse Derivative of an Inverse FunctionFunction

HWQHWQ

3 2 12 1, find 13f x x x f

1 113

20f

HWQHWQ

11 1If (2) 3, 2 , 3 , find 3

4 3h h h h

1 3 4h

ExamplesExamples

2 ,41

1)( if )0( Find.5

22

1

xdtt

xFFx

1 (0) 17F

HomeworkHomework Inverse Functions

Day 1: P. 347: 1, 3, 9, 11, 71-89 odd

Day 2: Derivatives of Inverses W/S

HomeworkHomework

Derivatives of Inverses W/S

AP FRQAP FRQ

2007 #3