## Key Equations

Half-life formula | If [latex]\text{ }A={A}_{0}{e}^{kt}[/latex], k < 0, the half-life is [latex]t=-\frac{\mathrm{ln}\left(2\right)}{k}[/latex]. |

Carbon-14 dating |
[latex]t=\frac{\mathrm{ln}\left(\frac{A}{{A}_{0}}\right)}{-0.000121}[/latex].[latex]{A}_{0}[/latex]
is the age of the fossil in years |

Doubling time formula | If [latex]A={A}_{0}{e}^{kt}[/latex], k > 0, the doubling time is [latex]t=\frac{\mathrm{ln}2}{k}[/latex] |

Newton’s Law of Cooling | [latex]T\left(t\right)=A{e}^{kt}+{T}_{s}[/latex], where [latex]{T}_{s}[/latex] is the ambient temperature, [latex]A=T\left(0\right)-{T}_{s}[/latex], and k is the continuous rate of cooling. |

## 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