Key Concepts

• The basic exponential function is [latex]f\left(x\right)=a{b}^{x}\\[/latex]. If b > 1, we have exponential growth; if 0 < b < 1, we have exponential decay.
• We can also write this formula in terms of continuous growth as [latex]A={A}_{0}{e}^{kx}\\[/latex], where [latex]{A}_{0}\\[/latex] is the starting value. If [latex]{A}_{0}\\[/latex] is positive, then we have exponential growth when k > 0 and exponential decay when k < 0.
• In general, we solve problems involving exponential growth or decay in two steps. First, we set up a model and use the model to find the parameters. Then we use the formula with these parameters to predict growth and decay.
• We can find the age, t, of an organic artifact by measuring the amount, k, of carbon-14 remaining in the artifact and using the formula [latex]t=\frac{\mathrm{ln}\left(k\right)}{-0.000121}\\[/latex] to solve for t.
• Given a substance’s doubling time or half-time, we can find a function that represents its exponential growth or decay.
• We can use Newton’s Law of Cooling to find how long it will take for a cooling object to reach a desired temperature, or to find what temperature an object will be after a given time.
• We can use logistic growth functions to model real-world situations where the rate of growth changes over time, such as population growth, spread of disease, and spread of rumors.
• We can use real-world data gathered over time to observe trends. Knowledge of linear, exponential, logarithmic, and logistic graphs help us to develop models that best fit our data.
• Any exponential function with the form [latex]y=a{b}^{x}\\[/latex] can be rewritten as an equivalent exponential function with the form [latex]y={A}_{0}{e}^{kx}\\[/latex] where [latex]k=\mathrm{ln}b\\[/latex].

Glossary

carrying capacity

in a logistic model, the limiting value of the output

doubling time

the time it takes for a quantity to double

half-life

the length of time it takes for a substance to exponentially decay to half of its original quantity

logistic growth model

a function of the form [latex]f\left(x\right)=\frac{c}{1+a{e}^{-bx}}\\[/latex] where [latex]\frac{c}{1+a}\\[/latex] is the initial value, c is the carrying capacity, or limiting value, and b is a constant determined by the rate of growth

Newton’s Law of Cooling

the scientific formula for temperature as a function of time as an object’s temperature is equalized with the ambient temperature

order of magnitude

the power of ten, when a number is expressed in scientific notation, with one non-zero digit to the left of the decimal