Antiderivatives from Slope and the Indefinite Integral
We have seen that the definite integral of a derivative gets us back to the original function. In other words, if f ' is the derivative of f, then f is an antiderivative of f '. For example, if f (x) = x² then f '(x) = 2x. So an antiderivative of 2x is x². But, since the derivative of a constant is zero, x² + 1 is also an antiderivative of 2x, and so is x² + C, where C is a constant. Hence the general antiderivative of a function is a family of functions, which all differ by a constant. This applet explores finding the antiderivative graphically, thinking of the integrand as representing the slope of the antiderivative. The notation for the general antiderivative of a function, g, is , called the indefinite integral because there are no limits. Using our example from above, we would write .
Try the following:
- The applet shows a graph on the left of the integrand f ' (x) = 2, a constant function. On the right is the graph of the antiderivative, . Think of the graph on the left (the integrand) as representing the slope of the graph on the right (the antiderivative). Note that since the left-hand graph is constant, so is the slope of the right hand graph, and we get a line with slope 2. Move the C slider; what happens to the graph? If you work backwards, thinking that the graph on the left is the derivative of the graph on the right, you see why changing C has no effect on the left-hand graph.
- Select the second example from the drop down menu. Now the integrand changes value from -1 to 1 at x = 0. The antiderivative on the right therefore changes slope from -1 to 1 at x = 0. Move the x slider, causing a crosshair to move on the left-hand graph (it's hard to see on this example) and a point and a black tangent line to move on the right-hand graph. You can see from the tangent line that the slope of the antiderivative changes.
- Select the third example from the drop down menu. In this example, the integrand function changes from a constant value of 1 to a downward sloping line at x = 1. Move the x slider and watch what happens to the tangent line on the antiderivative graph. It starts out with a slope of 1, but then the tangent line's slope decreases to zero. Where does this happen on the left-hand graph? What happens if you move the x slider past this point? Think about the graph of the integrand as telling you the slope of the antiderivative at any given x value. The result is a family of antiderivatives, and the specific member of this family that is graphed depends on the value of C (move the C slider to change the graph to a different member of the family).
- Select the fourth example, showing an integrand made up of two lines. Move the x slider to observe how the value of the integrand (on the left) tells you the slope of the antiderivative (on the right).
This work by Thomas S. Downey is licensed under a Creative Commons Attribution 3.0 License.