Exponential functions, growth, and decay
In an exponential function, the variable is in the exponent. Exponential models describe anything that grows or shrinks by a constant percentage, from savings to bacteria to radioactive atoms.
Exponential functions
An exponential function has the form \(f(x) = ab^x\), with \(a \neq 0\), base \(b > 0\), and \(b \neq 1\). Linear functions add a constant amount each step; exponential functions multiply by a constant factor b. If b > 1 the function grows; if 0 < b < 1 it decays.
Graph features
The graph of \(y = b^x\) passes through (0, 1), has domain all real numbers and range y > 0, and approaches the x-axis (its horizontal asymptote, y = 0) on one side without touching it. Growth graphs rise steeply to the right; decay graphs fall towards zero.
Percentage growth and decay
Growth at rate r per period gives \(A = A_0(1 + r)^t\); decay gives \(A = A_0(1 - r)^t\). A town of 20,000 growing 3% a year has \(20{,}000(1.03)^{10} \approx 26{,}900\) people after 10 years. A car worth $30,000 losing 15% a year is worth \(30{,}000(0.85)^4 \approx 15{,}660\) dollars after 4 years.
Doubling time and half-life
Exponential processes have a fixed doubling time or half-life. Carbon-14 has a half-life of about 5,730 years, so after \(t\) years a fraction \(\left(\tfrac{1}{2}\right)^{t/5730}\) remains; after 11,460 years, one quarter is left. The rule of 70 estimates doubling time: at 7% growth, about 70/7 = 10 years.
Compound interest
Interest compounded n times a year gives \(A = P\left(1 + \frac{r}{n}\right)^{nt}\). $1,000 at 6% compounded monthly for 10 years grows to \(1000(1.005)^{120} \approx 1{,}819\) dollars. More frequent compounding earns slightly more.
The number e
As compounding becomes continuous, the formula becomes \(A = Pe^{rt}\), where \(e \approx 2.71828\) is a constant studied by Leonhard Euler in the 1700s. $1,000 at 6% compounded continuously for 10 years becomes \(1000e^{0.6} \approx 1{,}822\) dollars. The natural exponential \(e^x\) is central in calculus because its rate of change equals itself.
Linear versus exponential
Exponential growth eventually overtakes any linear growth. Starting with $1 and doubling every day, you have over $500 by day 10 and over $500,000 by day 20, whereas adding $50 a day gives only $1,001 by day 20. This is why early stages of epidemics and compound interest are easy to underestimate.
Worked example
A culture of 500 bacteria doubles every 3 hours, so \(N(t) = 500 \cdot 2^{t/3}\). After 12 hours, \(N(12) = 500 \cdot 2^4 = 8{,}000\) bacteria. After 1 day, \(N(24) = 500 \cdot 2^8 = 128{,}000\).
Key takeaways
Exponential functions f(x) = a b^x multiply by a constant factor per step, growing when b > 1 and decaying when 0 < b < 1. Their graphs pass through (0, a), stay above the asymptote y = 0, and have range y > 0 when a > 0. Percentage change uses (1 + r)^t or (1 - r)^t. Half-life and doubling time are constant; carbon-14's half-life is about 5,730 years, and the rule of 70 estimates doubling time. Compound interest uses P(1 + r/n)^(nt), and continuous compounding uses Pe^(rt) with e about 2.718. Exponential growth always overtakes linear growth.
Flashcard
f(x) = a b^x
Check your understanding
A culture of 500 bacteria doubles every 3 hours. How many after 12 hours?