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alittlebitofcalculus-pdf-january2011-111112001007-phpapp01

# If we write fxdx dfx then we call fx the differential

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Unformatted text preview: d f ” ( x) = 6x At x = 0 : f(0) = +1 and f ’(0) = 0, implying x=0 may be a maximum or a minim um point. From the graph we can see it is neither. f(x) is minimum concav continuously increasing. However, the curve is changing shape from concave down concav up. do wn to conca v e up At x=0 : f ”(0) = 0 and ore positiv after. neg tiv befor f ”(x) is ne g a ti v e befor e and positi v e after. Consider the function f(x) = -- x 3 + 1 . Please draw the graph. f ’(x) = -- 3x 2 a nd f ”(x) = -- 6x At x = 0 : f(0) = +1 and f ’(0) = 0, implying x = 0 may be a maximum or a minim um point. From the graph we can see it is neither. f(x) is minimum continuously decreasing. However, the curve is changing shape from concave concav down. up to concave down At x = 0: f ” ( 0) = 0 and f ” ( x) is positi v e bef or e and ne g a ti v e after positiv befor ore neg tiv after. 121 y f ' (x) = 3x 2 f " (x) = 6x f(x) = x3 + 1 4 3 2 11 x -- 3 -- 2 -- 1 0 -- 1 -- 2 -- 3 -- 4 122 122 1 2 3 In both cases, f(x) = x 3 + 1 a nd f(x) = − x 3 + 1 , we have f ’ (0) = 0. But x = 0 is neither a maxim um nor a minim um How do we determine maximum minimum um. what kind of point x is ? f ’ ( x) = 0 and f ” ( x) negative ⇒ maximum maximum. m aximum f ’ ( x) = 0 and f ” ( x) positive ⇒ minimum minimum. m inimum f ’ ( x) = 0 and f ” ( x) = 0 and changing sign ⇒ inflexion inflexion. inflexion Let us now summarize in tabular form the results of these last few chapters. For 2 any well-behaved function y = f(x), with dy/ dx = f ’ (x) and d y/ dx2 = f” (x), we have: Type of point dy tan θ = slope of tangent s lope = dx before after d2y dx2 Maximum 0 + − − or 0 Minimum 0 − + + or 0 Inflexion (turning) 0 ± ± 0 and changing sign 123 ... Reviewing the example of the bouncing ball at the instants t1, t2, t3, .... w e . can analyse and say: 1. The function (height) is decreasing before and increasing after . 2. The first derivatives are ne g a ti v e before and positi v e after neg tiv positiv i.e. changing sign. 3. f ’ (x) = 0. W hat can you...
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