Second Fundamental Theorem of Calculus

In the construction above, drag points A and B left/right to determine limits of integration a and b for the definite integral of f(x). Subsequent instructions assume you are using the default function . When dragging point A or B, observe that the "signed area" is sometimes positive and sometimes negative. Explain in your own words when the "signed area" equals 0 for the given default function f(x).

Now imagine that A is "fixed" (i.e. constant) at x=a, but B has a variable position at x=t. The definite integral becomes . As you vary the position of B, the upper limit of integration t also varies while the lower limit a stays fixed.. The "signed area" becomes a function of t in its own right. Let's call that function . Toggle on the "plot" checkbox to plot the point (t, F(t)). Toggle on the "trace" checkbox to give a sense of what the y=F(t) function would look like as you drag point B left/right. Note that at any time, you may toggle the "trace" checkbox off/on to clear any traced points. Consider three different values: a=-1, a=0, and a=1. Drag A such that we have a "fixed" value a=-1. Then drag B left/right with "trace" checked on and observe the F(t) graph. Drag A such that we have a "fixed" value a=0. Again drag B left/right and observe the F(t) graph. Drag A such that we have a "fixed" value a=1. Again drag B left/right and observe the F(t) graph. How does the F(t) graph compare for the different "fixed" values of a=-1, a=0, and a=1?

With "trace" still toggled on, also toggle on the "tangent" checkbox. Affirm your understanding that F'(t) would give the slope of the displayed tangent line segment at each value of t. With a graph of y=F(t) displayed via the traced points, consider what the graph of the derivative curve y=F'(t) would look like. How would you expect the F'(t) graph to look? How would the graphs of F'(t) compare for the different "fixed" values of a=-1, a=0, and a=1?

Because F(t) was defined as a definite integral with a variable limit of integration t, evaluating F'(t) involves taking the derivative of that integral with respect to t. Based on the answer to the previous question, write an equation stating what the derivative of this definite integral must be. This statement is referred to by many textbooks and resources as the "Second Fundamental Theorem of Calculus."

Consider the limit of integration t being replaced by a function g(t). Write a more general version of the Second Fundamental Theorem of Calculus that takes the chain rule into account. (The GeoGebra construction above does not facilitate visualizing how this works, so you'll have to rely on your skill in applying the chain rule.)

In addition to the limit of integration t being replaced by a function g(t), consider the previously-fixed value a being replaced with a variable function h(t). Write an even more general version of the Second Fundamental Theorem of Calculus that takes the chain rule into account for both variable limits of integration. (The GeoGebra construction above does not facilitate visualizing how this works, so you'll have to rely on your skill in applying the chain rule.)