Exponential functions
Exponential functions are useful for modeling animal populations, investments, drugs in the body, etc, etc!
Population example
City A has 60 thousand people, and is expected to increase by 10 000 people per year. What is its population after 1 year? 2 years? 3 years? $t$ years? 2.5 years?
| Year | 0 | 1 | 2 | 3 | 4 | 5 |
|---|---|---|---|---|---|---|
| City A | 60 000 | 70 000 | 80 000 | 90 000 | 100 000 | 110 000 |
City B has 60 thousand people, and is expected to increase by 10% each year. What is its population after 1 year? 2 years? 3 years? $t$ years? 2.5 years?
City C has 60 thousand people, and is expected to decrease by 10% each year. What is its population after 1 year? 2 years? 3 years? $t$ years? 2.5 years?
Describe the shape of each population vs. time graph:
- increasing or decreasing?
- concave up, concave down, or linear
Change language
My favorite cactus grew from 10 feet to 13 feet this year!
What was its
- absolute change in height?
- relative change?
- percentage change?
Absolute change = $h_2-h_1$.
Relative (or fractional change)=$\frac{h_2-h_1}{h_1}$.
Percentage change=$\frac{h_2-h_1}{h_1}*100.$
Over one day, the volume of water in my dog's water dish dropped from 500 ml to 350 ml.
What was the
- absolute change in volume?
- relative change?
- percentage change?
Calculating percentage changes
"Last week my dog weighed 120 lbs. This week he weighs 4% more!"
Two ways to calculate... $$120 + (0.04)120=120+4.8=124.8\text{ lbs}.$$ Or... $$\begineq 120+(0.04)120&=& (1)120+(0.04)120= (1.+0.04)120\\ &=&(1.04)120=124.8\text{ lbs}.\endeq$$
"124.8 is 104% of 120".
"Oh boy, my dog has been gaining weight, 4% every week for the last 8 weeks!"
| week | 0 | 1 | 2 | 3 | ... | 8 |
| weight (lbs) | $120$ | $124.8$ $=(1.04) 120$ |
$129.8$ $=(1.04)124.8$ $=(1.04)^2 120$ |
$135$ $=(1.04)129.8$ $=(1.04)^3 120$ | ... | ?? |
What is the dog's weight after 2.5 weeks?
Basic concepts
A function is linear if it changes at a constant absolute rate: $$y(x)=b+mx$$
- $b$ is the initial value / $y$-intercept,
- $m$ is the rate of change.
A function is exponential if it changes at a constant percentage rate: $$y(x)=a b^x=a(1+r)^x$$
- $a=y(0)$ is the initial value,
- $b$ is the "growth factor".
- If $1\lt b$ then $y$ is growing exponentially.
- If $0\lt b \lt 1$ then $y$ is decreasing exponentially, or "decaying" exponentially.
- $r$ is the growth rate, or relative growth rate.
- If $0\lt r$ then $y$ is growing exponentially.
- If $-1\lt r \lt 0$ then $y$ is decaying exponentially.
Investment example
Suppose an investment earns a fixed return of 8% per year, and that you initially invest $500.
- What will your investment, $I(t)$ be worth $t$ years after the initial investment?
- Graph this function.
- Based on your graph, how long will it take to double the investment?
- How long will it take to double the investment again?
Bacteria example
Suppose the population of bacteria (in millions) is $P(t)=4.3(0.76)^t$ after $t$ hours.
- What was the initial ($t=0$) population of bacteria?
- What is the percent decrease per hour?
Folding paper
A single sheet of paper is about 0.1 mm thick.
How thick would a piece of paper that has been folded $n$ times be?
Image credits: Ozzy Delaney, Mr TGT, Beau Rogers, Frank Wittig