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63 we shall use the words differentiate or find the

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Unformatted text preview: te of change of f(x) at particular instant x1 ? chang hange instantaneous ra We shall define this in three steps. VERAGE step: 1. AVERAGE step: If we take a par ticular small inter val, say x1 t o x2 , w e could be more precise. AVERAGE RATE OF CHANGE from instant x1 to x2 is : CHANGE in function CHANGE in variable ∆f f(x ) − f(x ) = ∆x = 2 −x 1 x2 1 step: 2. TENDS TO step: the smaller the interval x2 -- x1 the more accurate the calculation of the AVERAGE RATE OF CHANGE at par ticular instant x1. So we fix particular par x1 and let x2 get closer and closer to x1. The difference ∆ x between x1 and x2 becomes smaller and smaller. It becomes infinitely small. This kind of difference we denote using the Greek symbol δ. As x2→ x1 we may say : x2 = x1 + δ x . So : δ x = x2 − x1 = (x1 + δ x) − x1 and δ f = f(x2) − f(x1) = f(x1 + δ x) − f(x1) Hence, δf δx = f(x1 + δ x) − f(x1) (x1 + δ x) − x1 61 step: 3. LIMIT step: Finally, when we let x2 coincide with x1, we get the INSTANTANEOUS particular RATE OF CHANGE of f(x) at par ticular instant x1. This we denote by : par df LIMIT δf LIMIT δf = x → x δx = δx→ 0 δx dx 2 1 “ra chang hange” This “r a te of c hange” got by taking the LIMIT is called the INSTANTANEOUS “r SPEED or INSTANTANEOUS RATE OF CHANGE at the particular instant x1 . Limit of AVERAGE RATE OF CHANGE of f(x) over the interval [ x1, x2 ] as x2 TENDS TO x1 = INSTANTANEOUS RATE OF CHANGE of f(x) at the instant x1 f (x1 + δ x) -- f (x1) L imit = δx → 0 ( x1 + δ x) -- x1 L imit δx → 0 f (x1 + δ x) -- f(x1) δx This is denoted by f ’ (x1). f (x1 + δ x) -- f(x1) δx This is called the FIRST DERIVATIVE of f(x1). There are several other notations for the FIRST DERIVATIVE : f ’ ( x1) = L imit δx → 0 --(df )x , dx 1 . f’ (x1), Df(x1), f (x1) Now, r ather than c hoosing the par ticular instant x1, we w ant the p ar INSTANTANEOUS RATE OF CHANGE of the function f(x) at any general eneral instant x . 62 L Using this same method of...
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