70.For h51:[22, 3] by [25, 5]For h50.7:[22, 3] by [25, 5]For h50.3:[22, 3] by [25, 5]As h→0, the second curve (the difference quotient)approaches the first (y5 22xsin (x2)). This is because 22xsin (x2) is the derivative of cos (x2), and the secondcurve is the difference quotient used to define the derivativeof cos (x2). As h→0, the difference quotient expressionshould be approaching the derivative.71. (a)Let f(x)5)x).Then }ddx})u)5}ddx}f(u)5f9(u) }ddux}51}ddu})u)21}ddux}25})uu)}u9.The derivative of the absolute value function is 11 forpositive values,21 for negative values, and undefined at 0. So f9(u)5hBut this is exactly how the expression })uu)}evaluates.(b)f9(x)53}ddx}(x229)4?})xx222299)}5}(2x)x)(2x2229)9)}g9(x)5}ddx}()x)sin x)5)x)}ddx}(sin x)1(sin x) }ddx} )x)5)x)cos x1}xs)xin)x}Note: The expression for g9(x) above is undefined atx50, but actually g9(0)5limh→0}g(01hh)2g(0)}5limh→0})h)shinh}50.
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