alittlebitofcalculus-pdf-january2011-111112001007-phpapp01

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Unformatted text preview: he relationship between the “ar ea under the cur v e ” f(x) = F ’ (x) and the CHANGE area I N VA L U E o f t h e I n t e g r a l F(x). O n a c o m p u t e r t h i s c a n b e demonstr ated ver y elegantly. Dr aw the cur ve of the Fir st Deri vati ve f(x). Shade the area you ar e inte gr ating and simultaneously plot its value. You will get the cur ve of the Integral F C ( x) . Do this LEFT to RIGHT(in the postive direction ) p ostive from x 0 a nd then RIGHT to LEFT (in the negative direction ) from x 0 . n egative Choose a suitable ∆x . Exercise 1 : Given f(x) = 1 use the rectangular method to plot F0 ( x) rectangular over [ − 2, 0]. 2 0 f(x) = 1 98765432987654321 1 98765432987654321 1 98765432987654321 1 98765432987654321 1 98765432987654321 1 98765432987654321 1 98765432187654321 9 1 98765432187654321 9 98765432987654321 1 1 2 F 0 (x) = x 1 0 x 161 1 2 x Exercise 2: Given f(x) = x use the trapezoidal method to plot F0 (x) trapezoidal over [ − 2, 0]. F 0 (x) = 1/ 2 x 2 4 2 f(x f(x) = x 3 2 098765432987654321 1 098765432987654321 1 098765432987654321 1 098765432987654321 1 098765432987654321 1 098765432987654321 1 098765432987654321 1 098765432987654321 1 098765432987654321 1 0987654321 0987654321 0987654321 0987654321 0987654321 0987654321 0987654321 0987654321 1 0 1 2 1 x 0 1 2 3 4 x Exer Ex er cise 3 : On a graph paper draw the cur ve of the function f(x) = 2x over the inter val [ − 4, 4]. Let x 0 = 1 a nd F C ( x 0 ) = 2 . Plot F C ( x) over inter val [ − 4, 4]. Exer Ex er cise 4 : Repeat exercise 3 over the inter v al [ − 2, 2] using a smaller ∆x. Exercise 5 : Repeat exercises 3 and 4 this time using f(x) = x 2 . Compare 3 the graph you get with the graph of F(x)= x / 3 + 1 . In any kind of NUMERICAL INTEGRATION to find the CHANGE in F(x) or to find F C ( x) we must take into consideration : 1. The instants or points where f(x) CHANGES SIGN. 2. The direction of integration . d irection 162 31. Integ Constant of Integr ation Any continuous function f(x) has an infinity of ANTIDERIV...
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