First Fundamental Theorem of Calculus
Accumulation Function
We suggest that you first explore the Second Fundamental Theorem of Calculus via Dr. Jackson's GeoGebra activity before proceeding. https://www.geogebra.org/m/cn2khzjf
Differentiating and then Integrating
In the App
Start by typing in any formula for a function f(x) in the input box. If you check the f(x) checkbox in the right window the graph of f(x) will appear in the right window in blue.
The formula for f '(x) is displayed, along with the graph of f '(x) in red in the left window.
For now, toggle off the graph of f(x) to clear out most of the right window.
Choose a value for a via the slider or input box in the left window. Similarly pick a value for x.
Start the value of x the same as the value for a and slowly slide the slider for x to the right. You will see area accumulating between the graph of f '(x) and the x-axis. Green areas accumulate positively and red areas accumulate negatively. The green area minus the red area is the value of the accumulation function for that value of x.
Click the checkbox for (x, A(x)) in the right window to see this value graphed there. Move x around via the slider to see it change. Now click on the checkbox for A(x) to see the graph. Again move x around to investigate.
Now deselect (x, A(x)) to hide that portion of the illustration. Does the graph of A(x) look familiar?
Select the checkbox for f(x). How does the graph of A(x) compare to the graph of f(x)?
Does they look like vertical shifts of each other?
Select the checkbox for Shift in the right window. This will show how far apart the two graphs are vertically for a particular x-value. Move this point on the graph of f(x) around. Does the vertical distance stay constant? How does this distance compare to f(x)?
You should see that, yes, the graphs of A(x) and f(x) are vertical shifts of each other and that the amount of the vertical shift is f(a).
What does this tell use about an alternate way to express ?
First Fundamental Theroem of Calculus
(Fundamental Theorem of Calculus Part 1)
If f is any function differentiable on the interval including a and b and any points between them, then
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First differentiating and then integrating produces the original function, possibly with a vertical shift.
One of the consequences of this is that if we are integrating a function g(x) and we can find a function f(x) so that g(x) = f '(x) (i.e. we find any antiderivative f for the original function g), then we can find an exact value for the definite integral by finding the total change in the antiderivative over the interval:
.