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alittlebitofcalculus-pdf-january2011-111112001007-phpapp01

# 1 the ball may bounce finitely many times and then

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Unformatted text preview: ion y(t) = u . s i n θ . t -- 1 g t 2 -2 h2 t T vertical speed y ’(t) = u . s i n θ -- g t t2 INDEFINITE INTEGRAL y(t) = ∫ y ’(t)dt = ∫ { u . s i n θ -- g t }dt = u . s i n θ . t -- 1 g t 2 -2 = ANTI-DERIVATIVE of y ’(t) = area under y ’(t) t2 ∫ t1 = expression of CHANGE in y(t) . y ’(t)dt = y (t2) -- y(t1) = shaded area under vertical speed curve y ’(t) = CHANGE in y(t) from t1 to t2 = h2 − h1 = CHANGE in height from t1 to t2 . DEFINITE INTEGRAL y(t) over [t1, t2 ] = t2 ∫ t1 y ’(t)dt “area curv The concept or view of integration as “ar ea under the cur ve” is useful in “ar getting started. The most general and correct concept is : the indefinite integr al indefinite integ indef F(x) is always the expression of CHANGE. When we evaluate the indefinite integral indefinite F(x) over a given interval and in a given direction, the value or number we get is called the change . change 168 We have three points of view : Calculation Calculation : ANTIDERIVATIVE = ∫ DERIVATIVE . y(t) = ANTI-DERIVATIVE of y ’(t) = ∫ y’(t).dt F(x) = ∫ f (x)dx where f(x) = F’(x) FC(x) = F(x) + C Geometric : area curv area under the cur ve of the INSTANTANEOUS RATE OF CHANGE ar area curv y(t) = area under the cur ve y’(t) = ar y’(t). dt . area curv F(x) = area under the cur ve ar Analytical Analytical : ∫ ∫ f (x)dx f(x) = EXPRESSION OF CHANGE = ∫ INSTANTANEOUS RATE OF CHANGE t2 CHANGE in y(t) over [t1, t2] = y(t2) − y(t1) = ∫ t y’(t). dt . 1 F(x) = ∫ f (x)dx the indefinite integral indefinite b ∫ a f(x)dx = F(b) − F(a) = CHANGE in F(x) from a to b is the definite integral . definite A function F(x) is an e x p r ession of CHANGE. In w o r k i n g with INFINITESIMALS we have a relationship between INSTANT ANTANEOUS RATE we know C H A N G E a n d INSTANTANEOUS RATE OF CHANGE. If w e kno w expr xpression INSTANT ANTANEOUS RATE we f (x) the e xpr ession of the INSTANTANEOUS RATE OF CHANGE w e expr xpression INTEGRATE ind can INTEGRATE f (x) and ffind F (x) the e xpr es...
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