FTC - The integral as a function
Suppose you have a function and you fix a number in the domain of . Notice that is a function of . We define
.
In this Interactive Figure, we explore the properties of and see how it is related to .
- Move the slider to see several examples of functions that come pre-loaded. You can modify the function using the textbox.
- Move one of the blue points to fix a value for , say .
- Start changing the value of . As you do so, notice two changes: (1) the definite integral at the bottom left updates its value, and (2) the point in the right pane has coordinates .
- As you drag the -value, the graph of will be traced out. Take some time to verify that the graph of makes sense, given the graph of and the value of .
- Does it appear that is an antiderivative of ? Or, put another way, does it appear that ? How does changing the fixed point affect the graph of ?
NOTE. Tips for using your own function :
- You must use the variable , not .
- You can pan the window by clicking and dragging it.
- Zoom in and out using the scroll wheel on your mouse.
- Use SHIFT + [click and drag an axis] to rescale an axis.
- Click on a graphics pane then use CTRL + M to return to "standard view."