Key Concepts & Glossary
Key Equations
Half-life formula | If , k < 0, the half-life is . |
Carbon-14 dating | . 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 , k > 0, the doubling time is |
Newton’s Law of Cooling | , where is the ambient temperature, , and k is the continuous rate of cooling. |
Key Concepts
- The basic exponential function is . 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 , where is the starting value. If 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 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 can be rewritten as an equivalent exponential function with the form where .
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 where 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