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Topic: [ap-calculus] FTC in plain language
Replies: 2   Last Post: Feb 17, 2012 9:57 AM

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BCrombie@AOL.COM

Posts: 108
Registered: 12/8/04
Re: [ap-calculus] FTC in plain language
Posted: Feb 17, 2012 9:57 AM
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The answer to why a thing doesn't matter is often answered by how it does
matter. The lower limit of integration in the FTC is a good example of
this. It also helps students to understand these relationships if they are
related to mathematical concepts which act as a solid foundation for the
Calculus. The solid foundation for the Calculus is not what students studied
during the last semester. It's the algebra and geometry they've been learning
for the past four years.

It helps to draw pictures for the graphs described below.

For simplicity consider the graph of a continuous function, y(x), in the
first quadrant. Denote the integral with a more algebraic notation: the area
from a to x of the function y, as A_a,x( y ) . We can write the equation
of the tangent line to the area function at a point x=b using the
point-slope form for the equation of a straight line.

A_t = m(x-b) + A_a,b( y )

The first term, m(x-b), has the form of the area of a rectangle. And since
we know that the constant component of the boundary function produces the
linear component of the area function, the slope, m, is just the height of
the boundary function. m=y(b).

So in considering the tangent line to the area graph at x=b,


A_t = y(b)(x-b) + A_a,b( y )


the lower limit of integration enters through the value of area function
at the point of tangency,

( b, A_a,b( y ) )

and the slope of the tangent line, m( A( y(x) ) )=y(x), is the central
content of the FTC.

Bill


In a message dated 2/15/2012 8:57:01 A.M. Eastern Standard Time,
med7172@lausd.net writes:

Hi everyone,

Can someone please explain to me, in layman's terms, why the lower limit
of integration doesn't matter when using the Fundamental Theorem of Calc to
evaluate the definite integral as a function? I get the "what" of the FTC,
but I can't get a handle on the "why" of the irrelevance of the value of a.

Thanks!! I'm SUCH a newbie! :)
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