Transcript 5 4 5 5 Exponential Functions

5-4 and 5-5: Exponential Functions
Objectives:
1. To define and use
exponential and
logarithmic functions
Assignment:
• P. 356: 1, 3, 7, 9, 13, 2124, 31, 32
• P. 366: 21, 27
• P. 356: 33-39 odd, 44, 45,
57, 61, 65, 85, 87, 95, 97,
2. To differentiate and
99, 101, 103
integrate exponential
• P. 366: 37-49 odd, 53, 55,
and logarithmic functions
61-67 odd, 79, 80, 91, 92,
94
Warm Up 1
Solve for 𝑥.
1. 7 = 𝑒 𝑥+1
2. ln 2𝑥 − 3 = 5
Warm Up 2
Let 𝑓 𝑥 = ln 𝑥 and 𝑔 𝑥 = 𝑒 𝑥 . Find 𝑓 𝑔 𝑥 .
Objective 1
You will be able to define
and use exponential
functions
Natural Exponential Function
The inverse function of the natural
logarithmic function 𝑓 𝑥 = ln 𝑥 is called the
natural exponential function and is
denoted by
𝑓 −1 𝑥 = 𝑒 𝑥
𝑦 = 𝑒𝑥
Exponential Form
if and only if
ln 𝑦 = 𝑥
Logarithmic Form
Exponential and Logarithmic
𝑦 = 𝑒𝑥
𝑦 = ln 𝑥
Domain: ℝ
Domain: 𝑥 > 0
Range: 𝑦 > 0
Range: ℝ
Continuous
Continuous
Increasing
Increasing
One-to-one
One-to-one
Concave Up
Concave Down
Exponential and Logarithmic
𝑦 = 𝑒𝑥
𝑦 = ln 𝑥
Horizontal Asymptote:
Vertical Asymptote:
𝑦=0
𝑥=0
lim 𝑒 𝑥 = ∞
𝑥→∞
lim 𝑒 𝑥 = 0
𝑥→−∞
lim ln 𝑥 = ∞
𝑥→∞
lim ln 𝑥 = −∞
𝑥→0+
Inverse Relationships
ln 𝑒 𝑥 = 𝑥
𝑒 ln 𝑥 = 𝑥
The inverse relationship between
𝑒 𝑥 and ln 𝑥 can be expressed
thusly:
Objective 2
You will be able to differentiate
and integrate exponential
functions
Exercise 1
Find the derivative of 𝑓 −1 𝑥 = 𝑒 𝑥 .
𝑓 −1
′ 𝑥 =
1
𝑓′ 𝑓 −1 𝑥
𝑓′ 𝑓 −1 𝑥
≠0
Derivative of
𝑥
𝑒
Let 𝑢 be a differentiable function of 𝑥.
Solution to the differential
equation 𝑦 ′ = 𝑦
𝑑 𝑥
𝑒 = 𝑒𝑥
𝑑𝑥
𝑒 𝑥 is its own
derivative
𝑑 𝑢
𝑑𝑢
𝑢
𝑒 =𝑒
𝑑𝑥
𝑑𝑥
Derivative of
𝑥
𝑒
Let 𝑢 be a differentiable function of 𝑥.
𝑑 𝑥
𝑒 = 𝑒𝑥
𝑑𝑥
𝑒 𝑥 is its own
derivative
The slope of
𝑦 = 𝑒 𝑥 at any
point is the
𝑦-coordinate of
that point.
Exercise 2
Find the derivative of each of the following.
1. 𝑦 = 𝑒 2𝑥−1
2. 𝑦 = 𝑒 −3/𝑥
Exercise 3
Show that the graph of
𝑦 = 𝑒 𝑥 is increasing and
concave up on its entire
domain.
Exercise 4
Find the relative
extrema of 𝑓 𝑥 = 𝑥𝑒 𝑥 .
Integration Rules
Let 𝑢 be a differentiable function of 𝑥.
𝑒 𝑥 𝑑𝑥 = 𝑒 𝑥 + 𝐶
𝑒 𝑢 𝑑𝑢 = 𝑒 𝑢 + 𝐶
Exercise 5
Integrate each of the following.
1.
𝑒 3𝑥+1 𝑑𝑥
2.
2
−𝑥
5𝑥𝑒
𝑑𝑥
Exercise 5
Integrate each of the following.
3.
𝑒 1/𝑥
𝑑𝑥
2
𝑥
4.
sin 𝑥 𝑒 cos 𝑥 𝑑𝑥
Exercise 6
Integrate
𝑒
𝑥2
𝑑𝑥.
Exercise 7
Evaluate each definite integral.
1.
1 −𝑥
𝑒
0
𝑑𝑥
2.
1 𝑒𝑥
𝑑𝑥
0 1+𝑒 𝑥
3.
0 𝑥
𝑒 cos
−1
𝑒 𝑥 𝑑𝑥
Exercise 8
Find the average value of 𝑦 = 𝑒
closed interval 0,1 .
𝑥2
on the
Exercise 9: AP
Let 𝑓 be the function given by 𝑓 𝑥 = 2𝑥𝑒 2𝑥 .
a) Find the absolute minimum of 𝑓. Justify that
your answer is an absolute minimum.
b) What is the range of 𝑓?
c) Consider the family of functions defined by
𝑦 = 𝑏𝑥𝑒 𝑏𝑥 , where b is a nonzero constant.
Show that the absolute minimum value of
𝑏𝑥𝑒 𝑏𝑥 is the same for all nonzero values of 𝑏.
Objective 1
You will be able to define
and use exponential
functions
Exercise 10
Simplify 𝑦 = 𝑒
ln 2 𝑥
.
Definition of
𝑥
𝑎
If 𝑎 is a positive
real number (𝑎 ≠ 1)
and 𝑥 is any real
number, then the
exponential
function to the base
𝑎 is denoted 𝑎 𝑥
and is defined by
𝑎𝑥 = 𝑒
ln 𝑎 𝑥
Exercise 11
Evaluate log 2 45.
Definition of log 𝑎 𝑥
If 𝑎 is a positive real
number (𝑎 ≠ 1) and 𝑥
is any positive real
number, then the
logarithmic function to
the base 𝑎 is denoted
by log 𝑎 𝑥 and is
defined by
ln 𝑥
log 𝑎 𝑥 =
ln 𝑎
1
log 𝑎 𝑥 =
ln 𝑥
ln 𝑎
Exercise 12
Solve for 𝑥.
1. 3𝑥 =
1
81
2. log 2 𝑥 = −4
Objective 2
You will be able to differentiate
and integrate exponential
functions
Exercise 12
Use the definition of 𝑎 𝑥 to find the derivative
of 𝑦 = 𝑎 𝑥 .
Exercise 13
Use the definition of log 𝑎 𝑥 to find the
derivative of 𝑦 = log 𝑎 𝑥.
Derivatives of
𝑥
𝑎
and log 𝑎 𝑥
Let 𝑎 be a positive real number 𝑎 ≠ 1 and let 𝑢 be
a differentiable function of 𝑥.
ln 𝑎 𝑎𝑢
𝑑 𝑥
𝑎 =
𝑑𝑥
𝑑𝑢
𝑑𝑥
𝑑
log 𝑎 𝑥 =
𝑑𝑥
𝑑 𝑢
𝑎 =
𝑑𝑥
ln 𝑎 𝑎 𝑥
1 𝑑𝑢
ln 𝑎 𝑢 𝑑𝑥
1
ln 𝑎 𝑥
𝑑
log 𝑎 𝑢 =
𝑑𝑥
Exercise 14
Find the derivative of each of the following.
1. 𝑦 = 2𝑥
2. 𝑦 = 23𝑥
3.
𝑦 = log10 cos 𝑥
Exercise 15
Integrate
2𝑥 𝑑𝑥.
Exponential Integration Rules
Let 𝑢 be a differentiable function of 𝑥.
1 𝑥
𝑎 𝑑𝑥 =
𝑎 +𝐶
ln 𝑎
𝑥
1 𝑢
𝑎 𝑑𝑢 =
𝑎 +𝐶
ln 𝑎
𝑢
Exponential Log Rules
Continuously Compounded Interest
Recall that the formula below is used to
calculate the amount of money in an
account after 𝑡 years with interest
compounded 𝑛 times per year.
𝑟
𝐴=𝑃 1+
𝑛
𝑛𝑡
Continuously Compounded Interest
When interest is compounded continuously,
the amount 𝐴 in an account after 𝑡 years is
given by the formula:
𝐴 = 𝑃𝑒
𝑟𝑡
Where 𝑃 is the principal
and 𝑟 is the interest
rate expressed as a
decimal.
Exercise 16
 r
Explain why A  Pe approximates A  P  1  
n
n


m

n


as
. To do this, let
.
rt
r
n  mr

r 
1
 r

A  P 1    P 1 

P
1




m
 n
 mr 


nt
mrt
m



rt
As m → ∞
 Pe rt
nt
Limits Involving 𝑒
1
lim 1 +
𝑥→∞
𝑥
𝑥
𝑥+1
= lim
𝑥→∞
𝑥
𝑥
=𝑒
Limits Involving 𝑒
lim 𝑒 𝑥 = ∞
𝑥→∞
lim 𝑒 𝑥 = 0
𝑥→−∞
Limits Involving 𝑒
lim 𝑒 −𝑥 = 0
𝑥→∞
lim 𝑒 −𝑥 = ∞
𝑥→−∞
Exercise 17
A bacterial culture is growing according to
1.25
the logistic growth function 𝑦 =
1+0.25𝑒 −0.4𝑡
where 𝑡 is the time in hours (𝑡 ≥ 0) and 𝑦 is
the weight of the culture in grams.
a) What is the limiting weight of the culture?
b) What is the rate at which the weight of the
culture is changing at 𝑡 = 1 and 𝑡 = 10
hours?
5-4 and 5-5: Exponential Functions
Objectives:
1. To define and use
exponential and
logarithmic functions
Assignment:
• P. 356: 1, 3, 7, 9, 13, 2124, 31, 32
• P. 366: 21, 27
• P. 356: 33-39 odd, 44, 45,
57, 61, 65, 85, 87, 95, 97,
2. To differentiate and
99, 101, 103
integrate exponential
• P. 366: 37-49 odd, 53, 55,
and logarithmic functions
61-67 odd, 79, 80, 91, 92,
94