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Key Concepts & Glossary

Key Equations

 Half-life formula If $\text{ }A={A}_{0}{e}^{kt}$, k < 0, the half-life is $t=-\frac{\mathrm{ln}\left(2\right)}{k}$. Carbon-14 dating $t=\frac{\mathrm{ln}\left(\frac{A}{{A}_{0}}\right)}{-0.000121}$.${A}_{0}$ A is the amount of carbon-14 when the plant or animal died t is the amount of carbon-14 remaining today is the age of the fossil in years Doubling time formula If $A={A}_{0}{e}^{kt}$, k > 0, the doubling time is $t=\frac{\mathrm{ln}2}{k}$ Newton’s Law of Cooling $T\left(t\right)=A{e}^{kt}+{T}_{s}$, where ${T}_{s}$ is the ambient temperature, $A=T\left(0\right)-{T}_{s}$, and k is the continuous rate of cooling.

Key Concepts

• The basic exponential function is $f\left(x\right)=a{b}^{x}$. If > 1, we have exponential growth; if 0 < < 1, we have exponential decay.
• We can also write this formula in terms of continuous growth as $A={A}_{0}{e}^{kx}$, where ${A}_{0}$ is the starting value. If ${A}_{0}$ is positive, then we have exponential growth when > 0 and exponential decay when < 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 $t=\frac{\mathrm{ln}\left(k\right)}{-0.000121}$ 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 $y=a{b}^{x}$ can be rewritten as an equivalent exponential function with the form $y={A}_{0}{e}^{kx}$ where $k=\mathrm{ln}b$.

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 $f\left(x\right)=\frac{c}{1+a{e}^{-bx}}$ where $\frac{c}{1+a}$ 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