3A-2. The Difference Quotient
Instructions:
- Use the input boxes for f(x) and a and b to define the function and its domain.
- Use the input box and slider tool for c to move c across the domain of f.
- Use the slider tool for h to adjust how far apart the two points are that are being used to calculate average rates of change.
- Use the Plot Slope checkbox to graph the slope of the secant line as a function.
- Use the Trace and Trace Slope buttons to leave traces of these objects on the graph.
The Difference Quotient
The difference quotient is the expression used to calculate an average rate of change over progressively smaller intervals in order to estimate the instantaneous rate of change.
This expression calculates an average rate of change. However, if we repeat the calculation for smaller and smaller values of h (i.e., let h \to 0), then we get the instantaneous rate of change, called the derivative: