Bunnies - University of British Columbia Department of ...elyse/102/2017/Chap10Condensed.pdfChapter...

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?


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.


Graphing Exponential Functions

x

y

10

1

20

2

30

3

40

4

50

5

y = 2xy = 3x

y =(

12

)x


Graphing Exponential Functions

0.2

−1

0.4

−2

0.6

−3

0.8

−4

1

−5

y =(

12

)x

y = 2x

y = 3x


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).



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?



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]

http://tutorial.math.lamar.edu/Classes/CalcI/DiffExpLogFcns.aspx


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


Evaluate ddx

{e3x}

.


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.


Functions are Maps

domain range

f (x)

f −1(x)


x

y

A. invertible B. not invertible


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


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)?


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)?


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) =


x

y y = ex

y = ln(x)

(0, 1)

(1, 0)

(1, e)

(e, 1)(−1, 1/e)

(1/e,−1)

y = x


Logs of Other Bases

log10 108 =

A. 0

B. 8

C. 10

D. other

log2 16 =

A. 1

B. 2

C. 3

D. other


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)


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)


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?


https://xkcd.com/1162/

Log scale in action: https://xkcd.com/482/




World Populations (wikipedia)

20 million

40 million

60 million

80 millionGER TUR

CAN

AUS

GAM

2m

illio

n

SEY

95th

ousa

nd


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


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

http://biology-forums.com/index.php?action=gallery;sa=view;id=6156

http://biology-forums.com/index.php?action=gallery;sa=view;id=6156


Logarithmic Differentiation - A Fancy Trick

Differentiate:

f (x) =

((x15 − 9x2)10(x + x2 + 1)

(x7 + 7)(x + 1)(x + 2)(x + 3)

)5

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