Suppose you have a function [math]f\left(t\right)[/math] and you fix a number [math]a[/math] in the domain of [math]f[/math]. Notice that [math]\int_a^xf\left(t\right)dt[/math] is a function of [math]x[/math]. We define[br] [math]F\left(x\right)=\int_a^xf\left(t\right)dt[/math]. [br]In this Interactive Figure, we explore the properties of [math]F\left(x\right)[/math] and see how it is related to [math]f\left(t\right)[/math].[br][list][*]Move the slider to see several examples of functions [math]f\left(t\right)[/math] that come pre-loaded. You can modify the function using the textbox.[/*][*]Move one of the blue points to fix a value for [math]a[/math], say [math]a=1[/math].[/*][*]Start changing the value of [math]x[/math]. 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 [math]\left(x,F\left(x\right)\right)[/math].[/*][*]As you drag the [math]x[/math]-value, the graph of [math]F\left(x\right)[/math] will be traced out. Take some time to verify that the graph of [math]F\left(x\right)[/math] makes sense, given the graph of [math]f\left(x\right)[/math] and the value of [math]a[/math].[/*][*]Does it appear that [math]F\left(x\right)[/math] is an antiderivative of [math]f\left(x\right)[/math]? Or, put another way, does it appear that [math]\frac{d}{dx}F\left(x\right)=f\left(x\right)[/math]? How does changing the fixed point [math]a[/math] affect the graph of [math]F\left(x\right)[/math]?[/*][/list]

[b][color=#1e84cc]NOTE[/color][/b]. [b]Tips for using your own function[/b] [math]f\left(t\right)[/math]:[br][list][*]You must use the variable [math]t[/math], not [math]x[/math].[/*][*]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."[/*][/list]