The Accumulation Function and the FTC
- Ken Schwartz
What is the connection between the area under a function curve and the antiderivative of that function?
- When the app is first started (or reset), the function is shown (in red). is defined to be . This means that the value of is the same as the area under between a fixed point and another point .
- By the FTC, . Check the "Show f(t)" box to display , the derivative of (in blue).
- You can drag the point on the black cursor on the function (red) to analyze its derivative (blue), such as finding sign changes of at extrema of .
- We can also use a function graph to analyze its antiderivative .
- Check the "Show FTC" box. Do not move point "a" at this time.
- Drag the point x so that it coincides with a. What is the area under between a and x? What is the value of at this point?
- Drag x to various points left and right of a. The signed area under between a and x is the -value of at x. This is because the area under the graph of a function's rate of change on an interval (for example, velocity) gives the amount of change of the function (position) on that interval. For our example, we could say that : the area under from a to x is equal to the change in the value of from a to x.
- Now leave x fixed, and move the point a along the x-axis. What do you observe? This happens because changing the value of a changes the constant of integration of the function .