alittlebitofcalculus-pdf-january2011-111112001007-phpapp01

1 area of a trapezoid is s um of the parallel sides

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Unformatted text preview: must be a polynomial o f the form 1 ax 2 + b x + C polynomial pol -2 For : dy(t) = -- g t + u.si n θ dt 1 y(t) = ----- g t 2 + u .si n θ . t + may be some constant C. 2 if at time t = 0 the ball was thrown from the ground level ( = 0) : C = 0 if at time t = 0 the ball was thrown from a height of 5 meters: C = + 5m This mechanical method of calculation wor ks when the functions are INTEGRABLE. It is known as finding the ANTIDERIVATIVE. There are tables which you can look up to find the matching ANTIDERIVATIVE of the given DERIVATIVE. Analyticall ytically Analytically, if f(x) is the DERIVATIVE of the function F(x), then the function F(x) is called the ANTIDERIVATIVE of the function f(x). We may write: d F(x) f(x) = dx If we write f(x)dx = dF(x) then we call f(x) the DIFFERENTIAL COEFFICIENT. 129 Integ Below is a partial Table of Integr als of the more frequently encountered functions deriv tiv integ antideriv tiv integ in standard form. The derivative = integr and and antiderivative = integr al . deri antideri DERIVA ANTIDERIVA F(x) DERIVATIVE f(x) ANTIDERIVATIVE F(x) n+1 x xn n ≠ −1 n+1 1 log |x| /x ex ex ax x a a > 0, a ≠ 1 log a sin (x) − cos x cos (x) sin (x) sec2 (x) tan (x) cosec2 (x) − cot (x) tan (x) . sec (x) sec (x) cot (x) . cosec (x) − cosec (x) tan (x) log |sec (x)| cot (x) log |sin (x)| sec (x) log|sec (x) + tan (x)|= log|tan (π 4 + x/ 2)| / cosec (x) log|cosec (x) − cot (x)|or log|tan x/ 2| 1 a + x2 1 2 a − x2 1 tan −1 ( x ) a ≠ 0 /a /a a+x 1 --- log | | 2a a−x 1 x − a2 1 x−a --- log | | 2a x+a 2 2 1 √a2 − x2 x sin −1(|a|) x2 < a2 130 26. 26. F(x) = area under f(x) = ∫ f(x)dx f(x)dx Let us look at the original concept of the integral - finding the area under a integral area curv cur ve . For the time being we denote the area function as F(x). area curv F(x) = ar ea under the cur v e of f(x) In the next chapter we shall prove that : {f(x) f(x)} F(x ANTIDERIVA f(x)dx F( x ) = ANTIDERIVATIVE {f(x)} = ∫ f(x)dx area We now present 2 methods to find...
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