Fundamental Theorem of Calculus Part II: The Derivative of an Integral Form

Kathryn Amejka – Cousino High SchoolKrystal Krygowski – Cousino High School

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Beginnings of the Fundamental Theorem

In a letter to Gottfried Wilhelm Leibniz (1646--1716), Newton stated the two most basic problems of calculus were

   "1. Given the length of the space continuously [i.e., at every instant of time], to find the     speed of motion [i.e., the derivative] at any time proposed.    2. Given the speed of motion continuously, to  find the length of the space [i.e., the integral or the antiderivative] described at any time proposed."

This indicates his understanding (but not proof) of the Fundamental Theorem of Calculus. Instead of using derivatives, Newton referred to fluxions of variables, denoted by x, and instead of

antiderivatives, he used what he called fluents.   Newton considered lines as generated by points in motion, planes as generated by lines in motion and bodies as generated by planes in motion, and he called these fluents.  He used the term fluxions to refer to the velocity of these fluents.

Newton began thinking of the traditional geometric problems of calculus in algebraic terms. Newton’s three calculus monographs were circulated to his colleagues of the Royal Society, but they were not published until much later, after his death.

 Leibniz’s ideas about integrals, derivatives, and calculus in general were derived from close analogies with finite sums and differences.   Leibniz also formulated an early statement of the  Fundamental Theorem of Calculus, and then later  in a 1693 paper Leibniz stated, "the general  problem of quadratures can be reduced to the finding of a curve that has a given law of tangency.

http://www.saintjoe.edu/~karend/m441/Cauchy.html

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Background: Who Started It?

We can thank Isaac Newton and Gottfried Wilhelm Leibniz for the Fundamental

Theorem of Calculus. Their ideas helped form the basis of the theorem.

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Theorem in Formal Mathematical Language

If g(x) = , where a stands for a constant, and f is continuous in the neighborhood

of a, then g’(x) = f(x).

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Another Way to State the Theorem

If F(x) is the anti-derivative of f(x), that is F’(x) = f(x), then .

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What Does This Mean? The derivative of an integral of a

function is that original functionOR

Differentiation undoes the result of integration

OR The derivative and the integral are

inverse operations

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ConditionsThe function must be continuous in the

interval.

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Proof g(x) =

g(x)= F(x) – F(a) g’(x)= F’(x) – 0

g’(x)= f(x)

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Background The FTC shows the rate at which the

area under a function increases as we move the upper bound to the right.http://www.physicsinsights.org/calculus_fundamental-1.html

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The Graphs

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Khan Academy Explanationhttp://

www.khanacademy.org/math/calculus/integral-calculus/fundamental-theorem-of-calculus/v/applying-the-fundamental-theorem-of-calculus

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Example of FTC: #1

Plug in 7x wherever t appears in the equation and multiply the whole equation

by the derivative of 7x using the chain rule. Since the two is a constant, the derivative is zero so it disappears.

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Example of FTC: #2

Plug in wherever t appears in the equation and don’t forget to multiply by

the derivative of , which is equal to using the chain rule. Once again, since 2 is a

constant, it’s derivative is 0 so it disappears.

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Example of FTC: Application FTC can be used in finding speed. Use

example from: http://www.physicsinsights.org/calculus_fundamental-1.html

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Works Cited

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Any Questions?