3.5: Continuous Growth

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Objectives:

• Introduce the notion of compounding continuously and the number Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle e}
• 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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle e}

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.

Instructor Example

Suggested Example

This is a sentence that is in place to generate space in the table

Choose a principal, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P=1} , and an interest rate, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=1} , for an account. Have the students help to compute the value of the account after one year if the interest is compounded Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n} times per year. Recall from the previous lesson that the amount one year later (i.e., when Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t=1} ), will be Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A(1)=P\left(1+\frac{r}{n}\right)^{n}.} Note: if you choose Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P=1} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=1,} increasing Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n} will make the value of the account after one year converge to Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle e.} 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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle e} and that we use Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle e} to represent continuous growth, or in this case, interest compounded continuously.

If a bank has an annual interest rate of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r}
 that is compounded continuously, 
the value Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A(t)}
 of the account Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t}
 years after the initial deposit of the 
principal Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P}
 is Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A(t)=Pe^{rt}}

 

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.

Instructor Example

Suggested Example

This is a sentence that is in place to generate space in the table

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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P(t)} , the population of Lincoln, NE, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t} years after 2011. Answer: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P(t)=262341e^{0.021t}} . By what percent does the population increase each year? Answer: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 262341e^{0.021t}=262341(1+r)^t } Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 262341e^{0.021}=262341(1+r), t=1 } Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle e^{0.021}=1+r } Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle e^{0.021}-1=r } Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 0.02122\approx r } So the city would be growing at approximately 2.122% each year. Under this model, predict the population in 2020. Answer: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P(9)=262341e^{(.021)(9)}=316918.6715\approx 316,918}

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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(t)=a(b)^t.} Have students help you fill in the following table:

Exponential Formula	Initial Value	Growth Factor
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(t)=a(b)^t}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A(t)=P{\underbrace{(1+r)}_{b}} ^t}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A(t)=P{\underbrace{\left[\left(1+\frac{r}{n}\right)^n\right]}_{b}} ^{t}}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A(t)=P{\underbrace{(e^r)}_{b}} ^t}

Remind students how the growth factor relates to the effective annual rate (or annual growth rate). Do an example where you find a formula Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A(t)=P(b)^t} given continuous growth rate Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A(t)=Pe^{rt}} (i.e., solve for Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b} given Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle e^r} ) and hint that we may want to do the reverse as well, which we'll talk about in Chapter 4 (i.e., solve for Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r} given Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b} ).

• 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.

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