Derivatives as Rates of Change
Derivatives as Rates of Change
Objectives
- Determine a new value of a quantity from the old value and the amount of change.
- Calculate the average rate of change and explain how it differs from the instantaneous rate of change.
- Apply rates of change to displacement, velocity, and acceleration of an object moving along a straight line.
- Predict the future population from the present value and the population growth rate.
- Use derivatives to calculate marginal cost and revenue in a business situation.
Summary
The derivative of a given function \(y=f(x)\) measures the instantaneous rate of change of the output variable with respect to the input variable.
The units on the derivative function \(y = f'(x)\) are units of \(y\) per unit of \(x\text{.}\) Again, this measures how fast the output of the function \(f\) changes when the input of the function changes.
The central difference approximation to the value of the first derivative is given by
\begin{equation*} f'(a) \approx \frac{f(a+h) - f(a-h)}{2h}\text{.} \end{equation*}This quantity measures the slope of the secant line to \(y = f(x)\) through the points \((a-h, f(a-h))\) and \((a+h, f(a+h))\text{.}\) The central difference generates a good approximation of the derivative's value.