The Fundamental Theorem of Calculus
We usually think
about derivatives, nDeriv(Y1,X,X),
as functions, and about integrals, fnInt(Y1,X,A,B) as numbers. But in fnInt, when we replace the upper
bound B by a variable, then fnInt(Y1,X,A,X) becomes a function of X that can be graphed and used
as any other function.
Example
Define,
\Y1=.2X+1
\Y2=fnInt(Y1,X,0,X)
Set the window to ZOOM standard, and graph.
Compute a value of
Y2 for X = 2.33.
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Y2 |
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ENTER |
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2.87289 |
Now enter a
function of your choice for Y1,
and set the window to view it. (The function should be defined for
all values seen in the window.) In order to speed up graphing set
Xres=3.
| \Y1 = | Function \Y1 (We selected Y1 = X^5-2X^2+1, with Xmin = -3, Xmax = 3, Ymin = -10, Ymax=10.) |
| -oY2 = nDeriv(fnInt(Y1,X,A,X),X,X) | derivative of integral of 1 |
 Y3= Y1(A) +
fnInt(nDeriv(Y1,X,X),X,A,X) |
integral of derivative of Y1, plus constant Y1(A) |
Remarks
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Y2 is going to be traced by a moving circle, and Y3 is going
to be drawn with a wide line. |
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Y2 is the derivative of the integral of Y1. |
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Y3 is the integral of the derivative of Y1. And Y1 (A) = Y3(A) because
fnInt(nDeriv(Y1,X,X),X,A,A) = 0. (You may check this from the home
screen.) |
Graphing
After you graph Y1,
Y2, and Y3, you will see that they
represent the same function. This means that the operations of taking a
derivative and computing an integral cancel each other. In math jargon the
derivative is the inverse of the integral, and the integral is the inverse of
the derivative. This fact is called the fundamental theorem of calculus.
A word of warning
The fundamental theorem is less general than one may think, because it requires
that all integrals and derivatives involved are well-defined. So,
for example, Y3 is not defined for a function Y1 that doesn't have
a derivative.
Calculus Index