3.5: Continuous Growth - Revision history

Nwakefield2: Created page with " Prior Lesson | Next Lesson ==Objectives:== * Introduce the notion of compounding continuously and the number $e$
2020-06-01T14:42:07Z

Created page with " Prior Lesson | Next Lesson ==Objectives:== * Introduce the notion of compounding continuously and the number <math>e</..."

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[[3.4: Compound Growth | Prior Lesson]] | [[4.1: Inverse Functions | Next Lesson]]
==Objectives:==

* Introduce the notion of compounding continuously and the number <math>e</math>
* Understand the difference between annual and continuous growth rates

==Important Items==
===Definitions:===

continuous growth

==Lesson Guide==

===Warm-Up===

Have students do Problem 1

===Introduce the notion of compounding continuously and the number <math>e</math>===

'''Introduction''': In the last lesson, we learned about compound interest and looked at how investments can grow with interest compounded annually, monthly, weekly, or even daily. We noted (hopefully!) that as interest of a given rate is compounded more and more frequently, we earn more money overall, since the interest earned in one period earns interest itself in the next. This raises an interesting question: How much money could be earned if we compounded interest non-stop? Let's build a chart and see what happens as we make the compounding period smaller and smaller.

{{Ex-begin}}
Choose a principal, <math>P=1</math>, and an interest rate, <math>r=1</math>, for an account. Have the students help to compute the value of the account after one year if the interest is compounded <math>n</math> times per year. Recall from the previous lesson that the amount one year later (i.e., when <math>t=1</math>), will be <math>A(1)=P\left(1+\frac{r}{n}\right)^{n}.</math>

'''Note:''' if you choose <math>P=1</math> and <math>r=1,</math> increasing <math>n</math> will make the value of the account after one year converge to <math>e.</math>

{| class="wikitable"
|-
|Compound frequency
|Value of account after one year
|-
|n=1
|2
|-
|n=2
|2.25
|-
|n=4
|2.441406
|-
|n=12
|2.613035
|-
|n=365
|2.714567
|-
|n=8760 (hourly)
|2.718127
|-
|n=525,600 (each minute)
|2.718279
|-
|n=31,536,000 (each second)
|2.718282
|}

Discuss how compounding at a higher and higher frequency starts to appear as though you are "always" compounding, i.e., compounding continuously.

What does this table show us? You may find it interesting to note that while the number of times we compounded got bigger and bigger, the amount we earn gets bigger too - but not too big! We never ended up with more than 2.75 dollars in our account. Explain to your students that this value, 2.718282, is very close to the number <math>e</math> and that we use <math>e</math> to represent continuous growth, or in this case, interest compounded continuously.

{{Ex-end}}

If a bank has an annual interest rate of <math>r</math> that is [[compounded continuously]],
the value <math>A(t)</math> of the account <math>t</math> years after the initial deposit of the
principal <math>P</math> is <math>A(t)=Pe^{rt}</math>

Have students do Problems 2-4.

===Understand the difference between annual and continuous growth rates===

For many instructors these are terms you may have not heard before. Make sure to look up these terms in the book so that you know what they are!

Unless the problem says [[continuous growth rate]], the student should assume it is not continuous.

Do at least one example that explores continuous vs. non-continuous growth rates.

{{Ex-begin}}
As of 2011, Lincoln, NE, has a population of 262,341 people. Suppose the population grows at a continuous growth rate of 2.1% per year.

Find a formula for <math>P(t)</math>, the population of Lincoln, NE, <math>t</math> years after 2011.
'''Answer:''' <math>P(t)=262341e^{0.021t}</math>.
By what percent does the population increase each year?
'''Answer:'''
<math>
262341e^{0.021t}=262341(1+r)^t
</math>

<math>
262341e^{0.021}=262341(1+r), t=1
</math>

<math>
e^{0.021}=1+r
</math>

<math>
e^{0.021}-1=r
</math>

<math>
0.02122\approx r
</math>

So the city would be growing at approximately 2.122% each year.
Under this model, predict the population in 2020.
'''Answer:''' <math>P(9)=262341e^{(.021)(9)}=316918.6715\approx 316,918</math>

{{Ex-end}}

Be sure to make a clear distinction between the [[continuous growth rate]] of 2.1\% and the [[growth rate]] of 2.122\%. (Remember nominal vs. effective interest rates? It's sort of the same idea.)

Have students do Problems 5-6.

Tie the sections of Chapter 3 together by comparing the formulas in Sections 3.4 and 3.5 with an exponential function of the form <math>f(t)=a(b)^t.</math> Have students help you fill in the following table:

{| class="wikitable"
|-
|Exponential Formula
|Initial Value
|Growth Factor
|-
|<math>f(t)=a(b)^t</math>
|
|
|-
|<math>A(t)=P{\underbrace{(1+r)}_{b}} ^t</math>
|
|
|-
|<math>A(t)=P{\underbrace{\left[\left(1+\frac{r}{n}\right)^n\right]}_{b}} ^{t}</math>
|
|
|-
|<math>A(t)=P{\underbrace{(e^r)}_{b}} ^t</math>
|
|
|}

Remind students how the growth factor relates to the effective annual rate (or annual growth rate). Do an example where you find a formula <math>A(t)=P(b)^t</math> given continuous growth rate <math>A(t)=Pe^{rt}</math> (i.e., solve for <math>b</math> given <math>e^r</math>) and hint that we may want to do the reverse as well, which we'll talk about in Chapter 4 (i.e., solve for <math>r</math> given <math>b</math>).

Work Problem 7 in groups. Let them move on to the Synthesis Problem only if time allows. (But it is a good review problem and a nice teaser for anyone going on to calculus!)

Work on the Focus Problem. ''This will likely be challenging for students as they have not seen functions like this before. After they have had some time to get comfortable with the function and its graph, lead them to applying terms such as "horizontal asymptote" to what they are seeing. In particular, note that this model might be more realistic than others we have considered because populations tend to reach the "carrying capacity" of their environment and stabilize at that population.

==Comments==
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