The Difference Quotient (Illustrated)
The following applet shows the relationship between the difference quotient of a function f (evaluated from x = a to x = b) and the slope of the secant line (drawn through the graph of the function) connecting the points (a,f(a)) & (b,f(b)).
Feel free to type in any function in the green input box below.
You can use the blue slider to adjust the value of a. (You can also type on in its input box if you wish.)
You can use the blue slider to adjust the value of b. (You can also type on in its input box if you wish.)
Questions:
For any function f that's continuous over the interval [a,b]:
1) How does the difference quotient of this function relate to the average rate of change of this function?
2) How does the difference quotient of this function relate the slope of the secant line connecting (a,f(a)) & (b,f(b))?