Calculus Notes 7.1 Inverse Functions & 7.2 Exponential Functions and Their Derivatives

Start up:1. (AP question) A continuous f(x) has one local maximum and two local minima.

Which of the following is true about its inverse function, ?(A) has one local maximum and two local minima.

(B) has two local maxima and one local minimum.

(C) has three local extrema; there is not enough information to tell if they are maxima or minima.

(D) has no local extrenum.

(E) does not exist.

2. (AP question) If your wealth was dollars after years, after how many years would you become a billionaire?

xf 1

xf 1

xf 1

xf 1

xf 1

xf 1

te t

Answers:(1) E(2) . te0000000001 ,,,

yearst

t

72320

109

.

ln

Calculus Notes 7.1 Inverse Functions

Definition: A function f is called a one-to-one function if it never takes on the same value twice; that is, 2121 xxwheneverxfxf

Binxeveryforxxff

Ainxeveryforxxff

1

1

Horizontal Line Test: a function is one-to-one if and only if no horizontal line intersects its graph more than once.

xyfyxf 1

yxfxyf 1 xf 1

Definition: Let f be a one-to-one function with domain A and range B. Then its inverse function has domain B and range A and is defined by

for any y in B.

xfy 1

How to Find the Inverse Function of a One-to-one Function f:

Step 1: Write y=f(x)Step 2: Solve for x in terms of y (if possible)Step 3: interchange x and y and write

The graph of is obtained by reflecting the graph of f about the line y=x.

1f

Calculus Notes 7.1 Inverse Functions

Theorem: If f is a one-to-one continuous function defined on an interval, then its inverse function is also continuous.

01 agfandfg 'Theorem: If f is a one-to-one differentiable function with inverse function

then the inverse function is differentiable at a and

agfag

''

1 xgf

xg'

'1

Calculus Notes 7.2 Exponential Functions and Their Derivatives

,0Theorem: If a>0 and a≠1, then is a continuous function with domain

and range . In particular, for all x. If 0<a<1, is a decreasing function; if a>1, f is an increasing function. If a,b>0 and then

yxyx aaa .1

limx r

r xa a

0xa

xaxf

yx , xaxf

y

xyx

a

aa .2 xyyx aa .3 xxx baab .4

Calculus Notes 7.2 Exponential Functions and Their Derivatives

dx

duee

dx

d uu xafxf 0''

1, lim lim 0

0 1, lim 0 lim

x x

x x

x x

x x

If a then a and a

If a then a and a

69314702 .x

dx

d

09861213 .x

dx

d

xx

dx

d26902 .

xx

dx

d31013 .

Definition: e is the number such that0

1lim 1

h

h

e

h

Derivative of the Natural Exponential Function:

x xde e

dx

x xe dx e C

0xe

,0 xexf Properties of the Natural Exponential Function: The exponential function

is an increasing continuous function with domain and range . Thus, for all x. Also . So the x-axis is

a horizontal asymptote of .

lim 0 limx x

x xe e

xexf

Calculus Notes 7.1 Inverse FunctionsExample 1: Given (a) Is f one-to-one? What is the Domain and Range of f and its inverse?

(b) Find its inverse.

(c) Graph the function, its inverse, and y=x.

xy

xxf

xxf

0yxfofR :

0xxfofD :

01 yxfofR :

01 xxfofD :

yx

xy

2

2

21 xxf

Example 2: Given find 2235 axxxxf agoraf ''1

235 24 xxxf' agfag

''

1 2?f

22111

12111 35

f

f

121 f

211

ff' 2

12

gfg

'' 1

1'f

21315

124

41

Calculus Notes 7.2 Exponential Functions and Their Derivatives

Example 3: Starting with the graph of find the equations of the graph that results from:

(a) Reflecting about the line y=4. (b) Reflecting about the line x=2.

xey

Example 4: Find the domain of each function. (a) .

(b) .

tey sin

ttg 21

xofD sin

021 ttgofD :

xeofD xeofDso sin

t21

t0

8xy e 4xy e

Calculus Notes 7.2 Exponential Functions and Their Derivatives

Example 6: Differentiate the function ueuf1

1 ueuf 21

1

ueuf u'

'' tetf t

21 1 uudu

d

2

1

u

e u

Example 5: Find the limit of 3 2

2lim x

xe

Let 3

2t

x

as 2x

t so 3 2

2lim limx t

txe e

0 By 11

Calculus Notes 7.2 Exponential Functions and Their DerivativesExample 7: The flash unit on a camera operates by storing charge on a capacitor and

releasing it suddenly when the flash is set off. The following data describe the charge Q remaining on the capacitor (measured in microcoulombs, ) at time t (measured in seconds).

C

xtQ 000045001100 ..

64670040 ..' Q

t 0.00 0.02 0.04 0.06 0.08 0.10

Q 100.00 81.87 67.03 54.88 44.93 36.76

A

(a) Use a graphing calculator to find an exponential model for the charge.

(b) The derivative Q’(t) represents the electric current (measured in microamperes, ) flowing from the capacitor to the flash bulb. Use part (a) to estimate the current when t=0.04s. Compare with the result of Example 2 in Section 2.1 (-670 )A

Close to the estimate from chapter 2.

7.1 pg.420 #2, 3, 6, 9, 12, 20, 21, 22, 23, 26, 27, 32, 33, 42, 44 (15)7.2 pg.431 #3, 4, 7, 8, 11, 13, 15, 17, 24, 29, 31, 32, 38, 45, 48, 60, 75, 78,81 (19)