alittlebitofcalculus-pdf-january2011-111112001007-phpapp01

Discr an approximation of the actual area function fx

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Unformatted text preview: ine it. 127 In Algebra we are familiar with the concept of discrete summation of a finite Algebra discrete number of terms. For example : n 1 Σ /2 i where n is finite. i=0 We are also familiar with the discrete summation of an infinite number of terms. discrete summation discr For example, the geometric series : ∞ 1 Σ /2 i i=0 We could have defined a more general case like : ∞ Σ f(x i ) i=0 Notice that even though the summation involves infinitely many terms, they are COUNTABLE. Hence we may identify each term using a discrete subscript. Recall discrete discr the difference between the set of natural numbers N that is COUNTABLE, and the continuous contiguous set of real numbers R that is UNCOUNTABLE. In the continuous contin summation summation over an interval [a, b] we must include each and every real number or point or instant in [a, b]. Since there are UNCOUNTABLY many of them it does NOT make sense to use a subscript. We know only the end points a and b of the interval [a, b]. So we evolve the notation for the continuous summation in 3 steps : continuous n Σ i=0 discrete summation of FINITELY many terms ⇒ n →∞ Σ i=0 discrete summation of COUNTABLY many terms ⇒ b ∫ a continuous summation of UNCOUNTABLY many terms For the continuous summation to be of practical value it must converge to some continuous definite finite quantity. This fundamental property, in an intuitive way, is known as integ integr able. We give an informal definition of this. 128 25. F(x ANTIDERIVA {f(x) f(x)} 25. F( x ) = ANTIDERIVATIVE {f(x)} Suppose we know the function that expresses the INSTANTANEOUS RATE OF CHANGE, are we able to determine the function that expresses the CHANGE ? Suppose we know that the function y’(t) that describes the ver tical speed i.e. the INSTANTANEOUS RATE OF CHANGE in height is : dy(t) . = u. si n θ -- g t dt Are we able to determine the function y(t) = u. si n θ . t -- 1 g t 2 -. 2 y’(t) = that describes the CHANGE in height ? polynomial We could mechanically say: dy(t) is a polynomial o f the form ax + b pol dt So y(t)...
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