CP1 Calculus

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Warm-up

Review of Inverse FunctionsA one-to-one function (passes the horizontal and vertical line test) has an inverse function, which “undoes” or “reverses” the original function. The graph of a function’s inverse function can be found by reflecting over the line y = x.

Example: f ( x )=√ x x ³0 f−1( x )=x2

Use the following table of select values of function f to answer the questions below.x -2 0 1 3 7f(x) 0 3 12 5 -2

Find: f−1(−2)= f−1(0 )= f−1(3 )=

Intro to Derivatives of InversesFor each problem on this page, find the equation of the tangent line at the given point. Then find the inverse and the equation of the tangent line to the inverse at the given point.1. f ( x )=x2

at x = 2 tangent line:

f−1( x )= at x = 4 tangent line:

Introduction to Derivatives of Inverse Functions page 2

2. g( x )= 2

x+3 at x = –4 tangent line:

g−1 (x )= at x = –2 tangent line:

3. h( x )=√ x3+1 at x = 1 tangent line:

h−1 ( x )= at x = 2 tangent line:

Summarize your findings about slopes of functions and their inverses. Using what you know about inverses justify why this makes sense.

Introduction to Derivatives of Inverse Functions page 3

4. At the right is the graph of f ( x )=x5+2 x−1 with the tangent line drawn at (1, 2).

a. Find f'( x ). Notice that it is always positive. What

does this tell you about the function?

b. Find the equation of the tangent line. Call it T(x).

c. You cannot find the inverse of f(x) but you do know that if f(1) = 2 then f−1( ____ )=_____ .

d. Find the inverse of the tangent line T(x) and the equation of the tangent line using your summary.

5. If we know f(a) = b and the slope of f(x) at a is

dfdx

|(a,b )= f ' (a )=m, and we know f

−1(b )=a

with a slope at b of

df−1

dx|(b, a )=( f −1 )¢ (b)

, can you say how

dfdx

|(a,b ) and

df−1

dx|(b,a )

are related?

dfdx

|(a, b )=m df−1

dx|(b,a )=

Introduction to Derivatives of Inverse Functions page 4

Practice

1. For each of the following, find f(x) andf '( x ) at x = a and then find the slope of the inverse at

the corresponding point. Notationally, you are finding f(a) = b,

dfdx

|(a,b ) and

df−1

dx|(b,a )

.

a. f ( x )=3 x2+2 x+5 at x = 2

b. f ( x )=(2 x−3 )( x5+2 )3 at x = –1

c. f ( x )= x+1

x2 at x = 1

2. For each of the following, find the equation of the tangent line of the inverse at the corresponding point. Basically, this is the same as question 1 but now you are writing an equation too.

a. f ( x )=(2√ x+1 )3 at x = 9

b. f ( x )= x3+1

2 x2+2 at x = 2

c. f ( x )=x2√ x+1 at x = 8

Introduction to Derivatives of Inverse Functions page 5

3. Use the table of values to find the following derivatives.

x f(x) f '( x ) g(x) g' ( x )

–1 4 2 3 5

1 6 3 –1 –3

3 10 4 1 –2

a.

ddx ( f−1( x ))|(6,1)

b.

ddx (g−1 ( x ))|(3 ,−1)

c. h( x )=( f ( x )+2g( x )) , find

ddx (h−1( x ))|(10 ,−1)

d. j( x )=(3 f (x )⋅g ( x )) , find

ddx ( j−1( x ))|(30 ,3 )

e. k (x )=( f (x )

g( x ) ), find

ddx (k−1( x ))|(−6,1)

f. t ( x )=( f ( g( x )) ) , find

ddx (t−1( x ))|(4,1)

Answer Key

1. a.

df−1

dx|(21 , 2)=

114 b.

df−1

dx|(−5 ,−1)=- 1

73 c.

df−1

dx|(2,1 )=- 1

3

2. a. y−9= 149 (x−343) b. y−2= 25

12 ( x−0 . 9) c. y−8= 3176 (x−192)

Introduction to Derivatives of Inverse Functions page 6

3. a.13 b.

15 c.

112 d.

− 148 e.

115 f.

−16