Lesson 15-16: Applications of Exponential and Log Functions

Hi everyone! Read through the material below, watch the videos, and send me your questions. Don’t forget to complete the Daily Quiz (below this post) before midnight to be marked present for the day.

Lesson Date: Thursday, April 2nd.

Topic: This lesson covers Chapter 15: Applications of Exponential and Logarithmic Functions and Chapter 16: Half-Life and Compound Interest.

WeBWorK: There is one WeBWorK assignment on today’s material, due next Thursday 4/9: Exponential Functions - Growth and Decay.

We study exponential functions because they are amazingly good at describing real-world phenomena. Today we will look at several different applications of exponential functions, including population growth and virus spread.

Exponential growth and decay

RECALL: If $f(x)$ is an exponential function, then $f(x)=c\cdot b^x$.

NOTE: You can determine the values of $c$ and $b$ from any two points (input-output) on the graph.

Example 15.1.
a) $f(0)=4, \quad f(1)=20$
c) $f(2)=160, \quad f(7)=5$

Example 15.3 The population size of a country was $12.7$ million in the year 2000, and $14.3$ million in the year 2010.
a) Assuming an exponential growth for the population size, find the formula for the population depending on the year $t$ (where $t=0$ in the year 2000.
b) What will the population size be in the year 2015, assuming the formula holds until then?
c) When will the population reach 18 million?

You will often see exponential growth and decay functions written in a slightly different (but equivalent) form, using the number $e = 2.718…$ as a base. It’s also traditional to use $t$ (for time) instead of $x$ as our independent variable.

FACT. If $f(t)$ is an exponential function, then $f$ can be written as $f(t)=Pe^{rt}$. In this form:

• $t$ represents time
• $P$ is the initial amount
• $r$ is the growth rate (if $r$ is positive, we have exponential growth, if $r$ is negative we have exponential decay)
• $f(t)$ is the amount remaining at time t
• $e=2.718…$

QUESTION: What’s the connection between $f(x)=c\cdot b^x$ and $f(t)=Pe^{rt}$?

Definition (exponential decay). The half-life of a substance is the time it takes for the amount to be cut in half.

EXAMPLE: A study published about two weeks ago, on March 17th, in the New England Journal of Medicine found experimentally that the half-life of the Covid-19 virus in the air is approximately 1.15 hours. A single cough by an infected person can release up to 6 billion coronavirus molecules into the air. Let’s consider what happens after a single cough by an infected person.

• a. Model the number of remaining virus molecules $V(t)$ in the air at time $t$ by an exponential function $V(t)=Pe^{rt}$ (find $P,r$).
• b. How many of virus molecules will remain viable 5 hours after the person coughed?
• c. How long will it take for the number of remaining molecules to reach $6$ million ($0.1\%$ of the original amount)?

That’s it for now. Take a look at the WeBWorK, and send me your questions!

-Prof. Reitz

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