Download Lesson 3-8: Derivatives of Inverse Functions, Part 1

AP Calculus AB
Lesson 3-8: Derivatives of Inverse Functions, Part 1
Name_________________________________
Date _________________________________
Learning Goal:

I can calculate the numeric derivative of an inverse function
I. Review:
1. Let f  x   7 x  1 and g  x  
x 1
7
a. Find f  g  3 
  1 
b. Find f  g    
  2 
c. Find f  g  x  
d. How are the functions f  x  and g  x  related?
2. Let k  t   2t  5 . Find k 1  t  .
3. Let h and m be inverses of each other. Find h(m(
)
OVER 
Page 2
Let f  x  be some function that is one to one over its domain. Then some function g  x  is
Theorem:
the inverse of f  x  , written f 1  x  iff f  g  x    g  f  x    x .
II. Let f  x   x3
a. Find f 1  x 
b. Both f and f 1 are graphed to the right. Sketch the line y  x . Why would we add this line to the
graph?
c. Find
df
.
dx
g. Find
df 1

dx
df

dx
d.
e.
f.
df
dx
df
dx
df
dx

df 1
.
dx
h.
x 1
df 1
dx

x 1
x2
df 1
dx
x 8
x 3
df 1
dx
x  27


Conclusion:
i.
j.


Page 3
* See geometric “proof” on SMART Board. See explanation in text book on page 165.
df
Theorem:
If f is differentiable at every point of an interval I and
is never zero on I, then f has an
dx
inverse and f 1 is differentiable at every point of the interval f  I 
Example 1
Let f  x   x5  3x  2 , and let f 1 denote the inverse of f . Given that 1, 2  is on the graph of f,
find
df 1
dx
x2
Example 2
If f and g are inverses, then we know that f  g  x    x .


d
f  g  x    x . Solve for g '  x  . Explain how your answer makes sense with
dx
the conclusion from the previous page.
Use the chain rule to find
OVER 
Page 4
Practice #1
Let f  x   x5  2x3  x 1
(a) Find f 1 and f ' 1 .
(b) Find f 1  3 and
d 1
f  3 .
dx
Practice #2
Differentiable functions f and g have values as shown in the table.
a.
b.
c.
AP Calculus AB
Lesson 3-8 , Part 1 Homework
Name__________________________________
Date _________________________________
For the below problems:
a) Verify that the functions have inverses
b) Evaluate the derivative of the inverse at the given value
1.
2.
3.
4.
f  x   x3  7 x  2 contains 1,10  ; Find
df 1
dx
f  x   x5  3x3  7 x  2 contains 1,13 ; Find
f  x   x3  8x  cos  3x  contains  0,1 ; Find
f  x  e
2 x
x 10
df 1
dx
x 13
df 1
dx
x 1
df 1
 9 x  4 contains  0,5  ; Find
dx
3
x 5
OVER
5. Let f be the function defined by f  x   x3  x . If g  x   f 1  x  and g  2   1 , what is the value of
g '  2 ?
(A)
1
13
(B)
1
4
(C)
6. Let f be a function such that lim
7
4
(D) 4
f  2  h   f  2
h 0
I. f is continuous at x  2
h
(E) 13
 5 . Which of the following must be true?
II. f is differentiable at x  2
III. The derivative of f is differentiable at x  2
(A) I only
(B) II only
(C) I and II only
(D) I and III only
(E) II and III only