Lesson 15: Exponential Growth and Decay (Section 021 handout)

Section 3.4Exponential Growth and Decay

V63.0121.021, Calculus I

New York University

October 28, 2010

Announcements

I Quiz 3 next week in recitation on 2.6, 2.8, 3.1, 3.2

Announcements

I Quiz 3 next week inrecitation on 2.6, 2.8, 3.1,3.2

V63.0121.021, Calculus I (NYU) Section 3.4 Exponential Growth and Decay October 28, 2010 2 / 40

Objectives

I Solve the ordinarydifferential equationy ′(t) = ky(t), y(0) = y0

I Solve problems involvingexponential growth anddecay


Notes

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1

Section 3.4 : Exponential Growth and DecayV63.0121.021, Calculus I October 28, 2010

Outline

Recall

The differential equation y ′ = ky

Modeling simple population growth

Modeling radioactive decayCarbon-14 Dating

Newton’s Law of Cooling

Continuously Compounded Interest


Derivatives of exponential and logarithmic functions

y y ′

ex ex

ax (ln a) · ax

ln x1

x

loga x1

ln a· 1

x


Outline

Recall







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2


What is a differential equation?

Definition

A differential equation is an equation for an unknown function whichincludes the function and its derivatives.

Example

I Newton’s Second Law F = ma is a differential equation, wherea(t) = x ′′(t).

I In a spring, F (x) = −kx , where x is displacement from equilibriumand k is a constant. So

−kx(t) = mx ′′(t) =⇒ x ′′(t) +k

mx(t) = 0.

I The most general solution is x(t) = A sinωt + B cosωt, whereω =

√k/m.


Showing a function is a solution

Example (Continued)

Show that x(t) = A sinωt + B cosωt satisfies the differential equation

x ′′ +k

mx = 0, where ω =

√k/m.

Solution

We have

x(t) = A sinωt + B cosωt

x ′(t) = Aω cosωt − Bω sinωt

x ′′(t) = −Aω2 sinωt − Bω2 cosωt

Since ω2 = k/m, the last line plus k/m times the first line result in zero.


The Equation y ′ = 2

Example

I Find a solution to y ′(t) = 2.

I Find the most general solution to y ′(t) = 2.

Solution

I A solution is y(t) = 2t.

I The general solution is y = 2t + C .

Remark

If a function has a constant rate of growth, it’s linear.


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The Equation y ′ = 2t

Example

I Find a solution to y ′(t) = 2t.

I Find the most general solution to y ′(t) = 2t.

Solution

I A solution is y(t) = t2.

I The general solution is y = t2 + C .


The Equation y ′ = y

Example

I Find a solution to y ′(t) = y(t).

I Find the most general solution to y ′(t) = y(t).

Solution

I A solution is y(t) = et .

I The general solution is y = Cet , not y = et + C .

(check this)


Kick it up a notch: y ′ = 2y

Example

I Find a solution to y ′ = 2y .

I Find the general solution to y ′ = 2y .

Solution

I y = e2t

I y = Ce2t


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In general: y ′ = ky

Example

I Find a solution to y ′ = ky .

I Find the general solution to y ′ = ky .

Solution

I y = ekt

I y = Cekt

Remark

What is C ? Plug in t = 0:

y(0) = Cek·0 = C · 1 = C ,

so y(0) = y0, the initial value of y .


Constant Relative Growth =⇒ Exponential Growth

Theorem

A function with constant relative growth rate k is an exponential functionwith parameter k. Explicitly, the solution to the equation

y ′(t) = ky(t) y(0) = y0

isy(t) = y0ekt


Exponential Growth is everywhere

I Lots of situations have growth rates proportional to the current value

I This is the same as saying the relative growth rate is constant.

I Examples: Natural population growth, compounded interest, socialnetworks


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5


Outline

Recall







Bacteria

I Since you need bacteria tomake bacteria, the amountof new bacteria at anymoment is proportional tothe total amount of bacteria.

I This means bacteriapopulations growexponentially.


Bacteria Example

Example

A colony of bacteria is grown under ideal conditions in a laboratory. At theend of 3 hours there are 10,000 bacteria. At the end of 5 hours there are40,000. How many bacteria were present initially?

Solution

Since y ′ = ky for bacteria, we have y = y0ekt . We have

10, 000 = y0ek·3 40, 000 = y0ek·5

Dividing the first into the second gives4 = e2k =⇒ 2k = ln 4 =⇒ k = ln 2. Now we have

10, 000 = y0e ln 2·3 = y0 · 8

So y0 =10, 000

8= 1250.


Notes

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6


Outline

Recall







Modeling radioactive decay

Radioactive decay occurs because many large atoms spontaneously give offparticles.

This means that in a sample of abunch of atoms, we can assume acertain percentage of them will“go off” at any point. (Forinstance, if all atom of a certainradioactive element have a 20%chance of decaying at any point,then we can expect in a sampleof 100 that 20 of them will bedecaying.)


Radioactive decay as a differential equation

The relative rate of decay is constant:

y ′

y= k

where k is negative. So

y ′ = ky =⇒ y = y0ekt

again!It’s customary to express the relative rate of decay in the units of half-life:the amount of time it takes a pure sample to decay to one which is onlyhalf pure.


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Computing the amount remaining of a decayingsample

Example

The half-life of polonium-210 is about 138 days. How much of a 100 gsample remains after t years?

Solution

We have y = y0ekt , where y0 = y(0) = 100 grams. Then

50 = 100ek·138/365 =⇒ k = −365 · ln 2

138.

Thereforey(t) = 100e−

365·ln 2138

t = 100 · 2−365t/138.

Notice y(t) = y0 · 2−t/t1/2 , where t1/2 is the half-life.


Carbon-14 Dating

The ratio of carbon-14 tocarbon-12 in an organism decaysexponentially:

p(t) = p0e−kt .

The half-life of carbon-14 isabout 5700 years. So theequation for p(t) is

p(t) = p0e−ln25700

t

Another way to write this wouldbe

p(t) = p02−t/5700


Computing age with Carbon-14 content

Example

Suppose a fossil is found where the ratio of carbon-14 to carbon-12 is 10%of that in a living organism. How old is the fossil?

Solution

We are looking for the value of t for which

p(t)

p(0)= 0.1

From the equation we have

2−t/5700 = 0.1 =⇒ − t

5700ln 2 = ln 0.1 =⇒ t =

ln 0.1

ln 2· 5700 ≈ 18, 940

So the fossil is almost 19,000 years old.


Notes

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8


Outline

Recall








I Newton’s Law of Coolingstates that the rate ofcooling of an object isproportional to thetemperature differencebetween the object and itssurroundings.

I This gives us a differentialequation of the form

dT

dt= k(T − Ts)

(where k < 0 again).


General Solution to NLC problems

To solve this, change the variable y(t) = T (t)− Ts . Then y ′ = T ′ andk(T − Ts) = ky . The equation now looks like

dT

dt= k(T − Ts) ⇐⇒ dy

dt= ky

Now we can solve!

y ′ = ky =⇒ y = Cekt =⇒ T − Ts = Cekt =⇒ T = Cekt + Ts

Plugging in t = 0, we see C = y0 = T0 − Ts . So

Theorem

The solution to the equation T ′(t) = k(T (t)− Ts), T (0) = T0 is

T (t) = (T0 − Ts)ekt + Ts


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Computing cooling time with NLC

Example

A hard-boiled egg at 98 ◦C is put in a sink of 18 ◦C water. After 5minutes, the egg’s temperature is 38 ◦C. Assuming the water has notwarmed appreciably, how much longer will it take the egg to reach 20 ◦C?

Solution

We know that the temperature function takes the form

T (t) = (T0 − Ts)ekt + Ts = 80ekt + 18

To find k, plug in t = 5:

38 = T (5) = 80e5k + 18

and solve for k.


Finding k

Solution (Continued)

38 = T (5) = 80e5k + 18

20 = 80e5k

1

4= e5k

ln

(1

4

)= 5k

=⇒ k = −1

5ln 4.

Now we need to solve for t:

20 = T (t) = 80e−t5ln 4 + 18


Finding t


20 = 80e−t5ln 4 + 18

2 = 80e−t5ln 4

1

40= e−

t5ln 4

− ln 40 = − t

5ln 4

=⇒ t =ln 4015 ln 4

=5 ln 40

ln 4≈ 13min


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10


Computing time of death with NLC

Example

A murder victim is discovered atmidnight and the temperature ofthe body is recorded as 31 ◦C.One hour later, the temperatureof the body is 29 ◦C. Assumethat the surrounding airtemperature remains constant at21 ◦C. Calculate the victim’stime of death. (The “normal”temperature of a living humanbeing is approximately 37 ◦C.)


Solution

I Let time 0 be midnight. We know T0 = 31, Ts = 21, and T (1) = 29.We want to know the t for which T (t) = 37.

I To find k:

29 = 10ek·1 + 21 =⇒ k = ln 0.8

I To find t:

37 = 10et·ln(0.8) + 21

1.6 = et·ln(0.8)

t =ln(1.6)

ln(0.8)≈ −2.10hr

So the time of death was just before 10:00pm.


Outline

Recall







Notes

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Interest

I If an account has an compound interest rate of r per yearcompounded n times, then an initial deposit of A0 dollars becomes

A0

(1 +

r

n

)ntafter t years.

I For different amounts of compounding, this will change. As n→∞,we get continously compounded interest

A(t) = limn→∞

A0

(1 +

r

n

)nt= A0ert .

I Thus dollars are like bacteria.


Continuous vs. Discrete Compounding of interest

Example

Consider two bank accounts: one with 10% annual interested compoundedquarterly and one with annual interest rate r compunded continuously. Ifthey produce the same balance after every year, what is r?

Solution

The balance for the 10% compounded quarterly account after t years is

B1(t) = P(1.025)4t = P((1.025)4)t

The balance for the interest rate r compounded continuously account aftert years is

B2(t) = Pert


Solving


B1(t) = P((1.025)4)t

B2(t) = P(er )t

For those to be the same, er = (1.025)4, so

r = ln((1.025)4) = 4 ln 1.025 ≈ 0.0988

So 10% annual interest compounded quarterly is basically equivalent to9.88% compounded continuously.


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Computing doubling time with exponential growth

Example

How long does it take an initial deposit of $100, compoundedcontinuously, to double?

Solution

We need t such that A(t) = 200. In other words

200 = 100ert =⇒ 2 = ert =⇒ ln 2 = rt =⇒ t =ln 2

r.

For instance, if r = 6% = 0.06, we have

t =ln 2

0.06≈ 0.69

0.06=

69

6= 11.5 years.


I-banking interview tip of the day

I The fractionln 2

rcan also be

approximated as either 70 or72 divided by the percentagerate (as a number between 0and 100, not a fractionbetween 0 and 1.)

I This is sometimes called therule of 70 or rule of 72.

I 72 has lots of factors so it’sused more often.


Summary

I When something grows or decays at a constant relative rate, thegrowth or decay is exponential.

I Equations with unknowns in an exponent can be solved withlogarithms.

I Your friend list is like culture of bacteria (no offense).


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Lesson 15: Exponential Growth and Decay (Section 021 handout)

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