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The Fundamental Theorem of Calculus
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The FUNdamental Theorem of Calculus
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Each branch of mathematics has its own fundamental theorem(s).
If you check out the word fundamental in the dictionary, you will see that it relates to the foundation
or the base or is elementary.
Fundamental theorems are important foundations for the rest of the material to follow.
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The Fundamental Theorem of Arithmetic
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Every integer greater than one is either prime
or can be expressed as
an unique product of prime numbers.
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Fundamental Theorem of Algebra
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Every polynomial equation of degree n with complex coefficients
has n roots in the complex numbers.
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Fundamental Theorem of Calculus
Here we go! ....................................in the footsteps of Newton and Leibniz
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Isaac Newton Gottfried Wilhelm von Leibniz
http://calculustugowar.ytmnd.com/
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a b
f(x)
Find the exact area under the function f(x) and above the xaxisin the interval [a, b].
FUNDAMENTAL THEOREM OF CALCULUS
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a b
f(x)Sketch an antiderivative of f(x)
g(x) is an antiderivative of f(x)
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................n subintervals
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g(x1)g(a) = g'(c1) = f(c1)
Slopeof
secant line= =
Slopeof
tangent line
Heightof
Riemannrectangle
By Mean Value theorem
By definition of indefinite integral
c1
f(c1)
f(c2)
f(cn)
c2 c3
f(c3)
cn
.............................n subdivisions
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If: 1. f is an integrable function and2. g is an antiderivative of f,
Thena
b ∫f (x) dx = g(b) g(a)
Shazaam!Math Magic
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