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Chapter 10: Exponential functions 10.1 Unlimited growth and doubling
Bunnies
A pair of bunnies makes 4 babies in 1 generation.
Suppose a bunny weighs 1 kg, and the earth weighs 6× 1024 kg.When will the bunnies weigh as much as the earth, if a generation is 3months?
Chapter 10: Exponential functions 10.1 Unlimited growth and doubling
Exponential Functions
Definition
An exponential function is a function of the form f (x) = ax , where a issome constant.Contrast: a power function is a function of the form f (x) = xa, where ais some constant.
exponential:
f (x) = 2x
f (x) = 5x
f (x) = 12x = 0.5x
not exponential:
f (x) = x2
f (x) = x5
f (x) =√x = x0.5
Exponential functions and power functions look similar, but they behave infundamentally different ways.
Chapter 10: Exponential functions 10.1 Unlimited growth and doubling
Graphing Exponential Functions
x
y
10
1
20
2
30
3
40
4
50
5
y = 2xy = 3x
y =(
12
)x
Chapter 10: Exponential functions 10.1 Unlimited growth and doubling
Graphing Exponential Functions
0.2
−1
0.4
−2
0.6
−3
0.8
−4
1
−5
y =(
12
)x
y = 2x
y = 3x
Chapter 10: Exponential functions 10.1 Unlimited growth and doubling
x
y
y = ax
y = bx y = cx
Order a, b, c , and 1 from smallest to largest.Order 1
a , b, c , and 1 from smallest to largest.
Chapter 10: Exponential functions 10.2 Derivatives of exponential functions and the function ex
Derivatives of Exponential Functions
x
y
1
f (x) = ax
small large
Consider ddx {a
x}.f (x) is always increasing, so f ′(x) is always positive.f ′(x) might look similar to f (x).
Chapter 10: Exponential functions 10.2 Derivatives of exponential functions and the function ex
Exponential Functions
ddx {a
x} = limh→0
ax+h − ax
h
= limh→0
axah − ax
h
= limh→0
ax(ah − 1)
h
= ax limh→0
(ah − 1)
h
= ax(times a constant)
Given what you know about ddx {a
x}, is it possible that limh→0
ah − 1
h=0∞?
A. Sure, there’s no reason we’ve seen that would make it impossible.
B. No, it couldn’t be 0, that wouldn’t make sense.
How could we find out what this limit is?
Chapter 10: Exponential functions 10.2 Derivatives of exponential functions and the function ex
Exponential Functions
In general, ddx {a
x} = ax limh→0
ah − 1
hfor any positive number a.
Euler’s Number
We define e to be the unique number satisfying limh→0
eh − 1
h= 1
e ≈ 2.71828182845904523536028747135266249775724709369995...(Wikipedia)
Derivatives of Exponential Functions
Using the definition of e,
d
dx{ex} = ex lim
h→0
eh − 1
h= ex
In general, limh→0
ah − 1
h= ln(a), so d
dx {ax} = ax ln(a) [proof]
Chapter 10: Exponential functions 10.2 Derivatives of exponential functions and the function ex
Quick Practice
Things to Have Memorized
d
dx{ex} = ex
When a is any constant,
d
dx{ax} = ax loge(a)
Let f (x) =ex
3x5. When is the tangent line to f (x) horizontal?
Horizontal tangent line ⇔ slope of tangent line is zero
Chapter 10: Exponential functions 10.2 Derivatives of exponential functions and the function ex
Evaluate ddx
{e3x}
.
Chapter 10: Exponential functions 10.2 Derivatives of exponential functions and the function ex
Example 1:
According to the collision theory of bimolecular gas reactions, a reactionbetween two molecules occurs when the molecules collide with energygreater than some activation energy, Ea, referred to as the Arrheniusactivation energy. Ea > 0 is constant for the given substance. The fractionof bimolecular reactions in which this collision energy is achieved is
F = e−EaRT ,
where T is temperature (in degrees Kelvin) and R > 0 is the gas constant.
Suppose that the temperature increases at some constant rate, C , per unittime.
Determine the rate of change of the fraction F of collisions that result in asuccessful reaction.
Chapter 10: Exponential functions 10.3 Inverse functions and logarithms
Invertibility: Grading Code
Key
message codeGood Job! GJNice Idea! NIAlgebra Mistake AMCreative Strategy! CSWrite your Name WYNUse a Logarithm LOGPlease don’t submit papers with coffee stains CS
The longer code is not uniquely translatable; viewed as a function, it is notinvertible.
Chapter 10: Exponential functions 10.3 Inverse functions and logarithms
Functions are Maps
domain range
f (x)
f −1(x)
Chapter 10: Exponential functions 10.3 Inverse functions and logarithms
x
y
A. invertible B. not invertible
Chapter 10: Exponential functions 10.3 Inverse functions and logarithms
Relationship between f (x) and f −1(x)
Let f be an invertible function.What is f −1(f (x))?
A. xB. 1C. 0D. not sure
domain range
f (x)
f −1(x)
5 25
What is f (f −1(x))?
A. xB. 1C. 0D. not sure
Chapter 10: Exponential functions 10.3 Inverse functions and logarithms
Invertibility
In order for a function to be invertible , different x values cannot map tothe same y value.We call such a function one-to-one, or injective.
Example 2:
Suppose f (x) = 3√
19 + x3. What is f −1(3)? (simplify your answer)
What is f −1(10)? (do not simplify)
What is f −1(x)?
Chapter 10: Exponential functions 10.3 Inverse functions and logarithms
Example 3:
Let f (x) = x2 − x .
1. Sketch a graph of f (x), and choose a domain over which it is invertible.
2. For the domain you chose, evaluate f −1(20).
3. For the domain you chose, evaluate f −1(x).
4. What are the domain and range of f −1(x)? What are the (restricted)domain and range of f (x)?
Chapter 10: Exponential functions 10.3 Inverse functions and logarithms
Domain and Range
f (x) = x2 − x , domain:[
12 ,∞
)f −1(x) =
−1 +√
1 + 4x
2
domain of f (x) range of f (x)
f (x)
f −1(x)
range of f −1(x) domain of f −1(x)
[12 ,∞
) [−1
4 ,∞)
Chapter 10: Exponential functions A.13 Logarithms
Exponents and Logarithms
f (x) = ex f −1(x) = ln(x) = log(x)
So, ln(ex) = x and e ln x = x .
x ex ln fact ↔ e fact
0 1 ln(1) = 0 ↔ e0 = 11 e ln(e) = 1 ↔ e1 = e−1 1
e ln( 1e ) = −1 ↔ e−1 = 1
en en ln(en) = n ↔ en = en
ln(1) =ln(e) =ln( 1
e ) =ln(en) =
Chapter 10: Exponential functions A.13 Logarithms
x
y y = ex
y = ln(x)
(0, 1)
(1, 0)
(1, e)
(e, 1)(−1, 1/e)
(1/e,−1)
y = x
Chapter 10: Exponential functions A.13 Logarithms
Logs of Other Bases
log10 108 =
A. 0
B. 8
C. 10
D. other
log2 16 =
A. 1
B. 2
C. 3
D. other
Chapter 10: Exponential functions A.13 Logarithms
Logarithm Rules
Let A and B be positive, and let n be any real number.ln(A · B) = ln(A) + ln(B)Proof: ln(A · B) = ln(e lnAe lnB) = ln(e lnA+lnB) = ln(A) + ln(B)ln(A/B) = ln(A)− ln(B)
Proof: ln(A/B) = ln(e ln A
e ln B
)= ln(e lnA−lnB) = lnA− lnB
ln(An) = n ln(A)Proof: ln(An) = ln
((e lnA
)n)= ln
(en lnA
)= n lnA
Simplify into a single logarithm:
f (x) = ln
(10
x2
)+ 2 ln x + ln(10 + x)
Chapter 10: Exponential functions A.13 Logarithms
Base Change
blogb(a) = a
⇒ ln(blogb(a)) = ln(a)
⇒ logb(a) ln(b) = ln(a)
⇒ logb(a) =ln(a)
ln(b)
In general, for positive a, b, and c:
logb(a) =logc(a)
logc(b)
Chapter 10: Exponential functions A.13 Logarithms
Base Change
In general, for positive a, b, and c:
logb(a) =logc(a)
logc(b)
Suppose your calculator can only compute logarithms base 10. Whatwould you enter to calculate ln(17)?
Suppose your calculator can only compute natural logarithms. What wouldyou enter to calculate log2(57)?
Suppose your calculator can only compute logarithms base 2. What wouldyou enter to calculate ln(2)?
Chapter 2: Differentiation 2.10 The Natural Logarithm
Differentiating the Natural Logarithm
Calculate ddx {ln x}.
One Weird Trick:
x = e ln x
d
dx{x} =
d
dx
{e ln x
}1 = e ln x · d
dx{ln x} = x · d
dx{ln x}
1
x=
d
dx{ln x}
Derivative of Natural Logarithm
d
dx{ln x} =
1
x(x > 0)
d
dx{ln |x |} =
1
x(x 6= 0)
Chapter 2: Differentiation 2.10 The Natural Logarithm
Derivative of Natural Logarithm
d
dx{ln |x |} =
1
x(x 6= 0)
Differentiate: f (x) = ln |x2 + 1|
Chapter 2: Differentiation 10.4 Applications of the logarithm
Manipulating Exponential Equations and Logarithms
Example 4:
(a) Write these expressions as exponential functions with base e: 2x , 15x .
(b) Find the derivative of f (x) = ax , where a is a positive constant, usingthe fact that d
dx [ex ] = ex .
(c) Find the zero(es) of the function eax − ebx2, where a and b are
constants.
(d) Suppose the quantity (in µg) of a radioactive isotope at time t
(measured in years) is given by Q(t) = 50e−t
700 . When is Q(t) = 40µg? How long does it take for half the substance to decay?
Chapter 2: Differentiation 10.4 Applications of the logarithm
https://xkcd.com/1162/
Log scale in action: https://xkcd.com/482/
Chapter 2: Differentiation 10.4 Applications of the logarithm
World Populations (wikipedia)
20 million
40 million
60 million
80 millionGER TUR
CAN
AUS
GAM
2m
illio
n
SEY
95th
ousa
nd
Chapter 2: Differentiation 10.4 Applications of the logarithm
World Populations–log scale
101
102
103
104
105
106
107
108
109CHI
log
10(1.4×
109)≈
9.1
IND
log
10(1.3×
109)≈
9.1
USA
log
10(3.2×
108)≈
8.5
GER
log
10(8.2×
107)≈
7.9
TUR
log
10(7.9×
107)≈
7.9
CAN
log
10(3.6×
107)≈
7.6
AUS
log
10(2.4×
107)≈
7.4
GAM
log
10(2×
106)≈
6.3
SEY
log
10(9.5×
104)≈
5.0
Chapter 2: Differentiation 10.4 Applications of the logarithm
Decibels: For a particular measureof the power P of a sound wave,the decibels of that sound is:
10 log10(P)
So, every ten decibels correspondsto a sound being ten times louder.
A lawnmower emits a 100dB sound.How much sound will twolawnmowers make?
A. 100 dB
B. 110 dB
C. 200 dB
D. other
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