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Algebra and Calculus 8-1 Copyright © Genetic Computer School 2007 Lesson 8 Fundamentals of Calculus
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Algebra and Calculus 8-2 Copyright © Genetic Computer School 2007 Lesson 8 Overview Limits Power series Continuity Intermediate – value theorem Derivatives Antiderivatives and definite integrals Fundamental theorem of calculus Mean - value theorem Table of derivatives Table of antiderivatives
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Algebra and Calculus 8-3 Copyright © Genetic Computer School 2007 Limits Lim a n = L n
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Algebra and Calculus 8-4 Copyright © Genetic Computer School 2007 Power series E.g. e x = 1+x+x 2 /2!+x 3 /3!+x 4 /4!+….
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Algebra and Calculus 8-5 Copyright © Genetic Computer School 2007 Continuity A function f(x) is said to be continuous at x = c if lim f(x) = f(c) x c
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Algebra and Calculus 8-6 Copyright © Genetic Computer School 2007 Intermediate – value theorem If f(x) is a continuous function on the interval [a,b] and f(a) f(b), then for every number y between f(a) and f(b) there exists a number x where x [a,b] and f(x) = y.
![Algebra and Calculus 8-6 Copyright © Genetic Computer School 2007 Intermediate – value theorem If f(x) is a continuous function on the interval [a,b] and f(a) f(b), then for every number y between f(a) and f(b) there exists a number x where x [a,b] and f(x) = y.](http://images.slideplayer.com/39/10977631/slides/slide_6.jpg "Algebra and Calculus 8-6 Copyright © Genetic Computer School 2007 Intermediate – value theorem If f(x) is a continuous function on the interval [a,b] and f(a) f(b), then for every number y between f(a) and f(b) there exists a number x where x [a,b] and f(x) = y.")
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Algebra and Calculus 8-7 Copyright © Genetic Computer School 2007 Derivatives The derivative of the function f is that function, denoted by f, such that its value at any number x in the domain of f is given by f (x) = lim { f(x+ x) – f(x) }/ x x 0 if this limit exists.
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Algebra and Calculus 8-8 Copyright © Genetic Computer School 2007 Antiderivatives and definite integrals y = f(x) ab
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Algebra and Calculus 8-9 Copyright © Genetic Computer School 2007 Fundamental theorem of calculus Suppose that f(x) is continuous function on the closed interval [a,b] and that F(x) is some antiderivative of f(x). Then:
![Algebra and Calculus 8-9 Copyright © Genetic Computer School 2007 Fundamental theorem of calculus Suppose that f(x) is continuous function on the closed interval [a,b] and that F(x) is some antiderivative of f(x).](http://images.slideplayer.com/39/10977631/slides/slide_9.jpg "Then:.")
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Algebra and Calculus 8-10 Copyright © Genetic Computer School 2007 Mean - value theorem If f(x) is continuous on the closed interval [a,b] and differentiable on the open interval ]a,b[ then there exists some ]a,b[ such that d/dx f( ) = { f(b) – f(a) } / (b-a)
![Algebra and Calculus 8-10 Copyright © Genetic Computer School 2007 Mean - value theorem If f(x) is continuous on the closed interval [a,b] and differentiable on the open interval ]a,b[ then there exists some ]a,b[ such that d/dx f( ) = { f(b) – f(a) } / (b-a)](http://images.slideplayer.com/39/10977631/slides/slide_10.jpg "Algebra and Calculus 8-10 Copyright © Genetic Computer School 2007 Mean - value theorem If f(x) is continuous on the closed interval [a,b] and differentiable on the open interval ]a,b[ then there exists some ]a,b[ such that d/dx f( ) = { f(b) – f(a) } / (b-a)")
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Algebra and Calculus 8-11 Copyright © Genetic Computer School 2007 Table of derivatives d/dx(a) = 0 d/dx(au) = adu/dx d/dx(x n ) = nx n-1 d/dx(u.v) = v.d/dx(u)+u.d/dx(v) Etc.
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Algebra and Calculus 8-12 Copyright © Genetic Computer School 2007 Table of antiderivatives a dx = ax au dx = a u dx (u + v)dx = u dx + v dx x a dx = x a+1 /(a+1) (a -1) Etc.
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