Transcript pwrpt 6.2

6.2
Exponential Functions and
Their Derivatives
Copyright © Cengage Learning. All rights reserved.
Warm Up:
Watch Video First!!!
https://www.youtube.com/watch?v=lDfbJt3-K1U
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Exponential Functions and Their Derivatives
The function f(x) = 2x is called an exponential function
because the variable, x, is the exponent.
In general, an exponential function is a function of the
form
f(x) = ax
where a is a positive constant.
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Exponential Functions and Their Derivatives
A representation of this graph is shown in Figure 1.
Representation of y = 2x, x rational
Figure 1
The graph is not solid because the irrational numbers cause holes
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Exponential Functions and Their Derivatives
We want to enlarge the domain of y = 2x to include both
rational and irrational numbers.
There are holes in the graph in Figure 1 corresponding to
irrational values of x. We want to fill in the holes by defining
f(x) = 2x, where x 
so that f is an increasing continuous
function.
For example, the irrational number
1.7 <
satisfies
< 1.8
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Exponential Functions and Their Derivatives
Since:
21.7 <
< 21.8
and we know what 21.7 and 21.8 mean because 1.7 and 1.8
are rational numbers.
Similarly, if we use better approximations for
better approximations for
:
, we obtain
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Exponential Functions and Their Derivatives
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Exponential Functions and Their Derivatives
We define
to be this number. Using the preceding
approximation process we can compute it correct to six
decimal places:
 3.321997
Similarly, we can define 2x (or ax, if a > 0 ) where x is any
irrational number.
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Exponential Functions and Their Derivatives
Figure 2 shows how all the holes in Figure 1 have been
filled to complete the graph of the function f(x) = 2x, x 
.
y = 2x, x real
Figure 2
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Exponential Functions and Their Derivatives
In general, if a is any positive number, we define
Works where x is irrational
This definition makes sense because any irrational number
can be approximated as closely as we like by a rational
number.
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Exponential Functions and Their Derivatives
The graphs of members of the family of functions y = ax are
shown in Figure 3 for various values of the base a.
Member of the family of exponential functions
Figure 3
Notice that all of these graphs pass through the same point (0, 1) because a0 = 1
for a ≠ 0
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Exponential Functions and Their Derivatives
There are basically three kinds of exponential functions
y = ax.
y = ax, 0 < a < 1
y = 1x
y = a x, a > 1
Because (1/a)x = 1/ax = a–x, the graph of y = (1/a)x is just
the reflection of the graph of y = ax about the y-axis.
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Exponential Functions and Their Derivatives
Laws of Exponents:
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Exponential Functions and Their Derivatives
Limit Laws for Exponential Functions:
Note: If a ≠ 1, then the x-axis is a horizontal asymptote of
the graph of the exponential function y = ax.
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Example 1
(a) Find
.
(b) Sketch the graph of the function y = 2–x – 1.
Solution:
(a)
=0–1
= –1
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Example 1 – Solution
cont’d
(b) Write
as in part (a). The graph of
is shown in Figure 3.
.
Shift it down one unit to obtain the graph of
shown in Figure 7.
Part (a) shows that the line y = –1 is a horizontal asymptote.
Member of the family of exponential functions
Figure 3
Figure 7
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Derivatives of
Exponential Functions
&
The natural number
“e”
(The Euler Number)
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Natural Base e
Like Л and ‘i’, ‘e’ denotes a number.
Called The Euler Number after
Leonhard Euler (1707-1783)
It can be defined by:
e= 1 + 1 + 1 + 1 + 1 + 1 +…
0! 1! 2! 3! 4! 5!
= 1 + 1 + ½ + 1/6 + 1/24 + 1/120+...
≈ 2.71828182845904523536…. 18
Natural Base e
The number e is irrational – its’ decimal
representation does not terminate or
follow a repeating pattern.
The previous sequence of e can also
be represented:
n
(1+1/n)
As n gets larger (n→∞),
gets
closer and closer to 2.71828…..
Which is the value of e.
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Examples: Natural Base e
3
e
7
e
·
4
e
=
3
10e =
-4x
2
(3e )
5e2
(-4x)2
9e
3-2
=
2e
2e
9e-8x
9
e8x
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Using a calculator
7.389
2
Evaluate e using a
graphing calculator
Locate the ex button
you need to use the second
button
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Graphing
f(x) =
rx
ae
is a natural base exponential
function
If a>0 & r>0 it is a growth function
If a>0 & r<0 it is a decay function
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Graphing examples
Graph y=ex
Remember the rules
for graphing
exponential
functions!
The graph goes thru
(0,a) and (1,e)
(1,2.7)
(0,1)
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Graphing Example
Graph y=2e0.75x
State the Domain &
Range
Because a=2 is
positive and r=0.75,
the function is
exponential growth.
Plot (0,2)&(1,4.23) and
draw the curve.
(1,4.23)
(0,2)
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Derivatives of Exponential Functions
Of all the possible exponential functions y = ax, the function
f(x) = ex is the one whose tangent line at (0, 1) has a slope
f(0) that is exactly 1.
Figure 10
Figure 11
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Derivatives of Exponential Functions
We call the function f(x) = ex the natural exponential
function.
If we put a = e and, therefore, f(0) = 1 in Equation 4, it
becomes the following important differentiation formula.
Key Formula!!!
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Derivatives of Exponential Functions
Thus the exponential function f(x) = ex has the property that
it is its own derivative.
The geometrical significance of this fact is that the slope of
a tangent line to the curve y = ex at any point is equal to the
y-coordinate of the point.
Figure 11
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Example 2
Differentiate the function y = etan x.
Solution:
To use the Chain Rule, we let u = tan x. Then we have
y = eu, so
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Derivatives of Exponential Functions
In general if we combine Formula 8 with the Chain Rule, as
in Example 2, we get
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Exponential Graphs
This function is just a special case of the exponential
functions considered in Theorem 2 but with base a = e > 1.
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Example 6
Find
Solution:
We divide numerator and denominator by e2x
(to get one on top)
=1
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Example 6 – Solution
We have used the fact that
and so
cont’d
as
=0
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Integration
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Integration
Because the exponential function y = ex has a simple
derivative, its integral is also simple:
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Example 8
Evaluate
Solution:
We substitute u = x3. Then du = 3x2 dx , so x2 dx = du
and
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Classwork
Page 401 # 24-42 even, 80-88 even
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Homework: Warm Up for Lesson 4.4
Page 408 # 10-15 all, 28,30
I’ve included the powerpoint for 4.3 on my webpage if you need a refresher from
Pre-Cal on Logarithms.
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