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# mt2Sample - Math 16A Spring 2010 MT2 Sample 1 Describe the...

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Math 16A, Spring 2010 MT2 Sample 1. Describe the x -values at which the fcts are diﬀerentiable and explain why example f ( x ) = | x 2 - 9 | . W have to remember that when the abso- lute value changes the sign of the fct, there we may have a non diﬀerentiable point. We know now how to derive this fct; to this aim let’s rewrite it as f ( x ) = x 2 - 9 when x ≤ - 3 - ( x 2 - 9) when - 3 < x < 3 x 2 - 9 when x 3 from which we get f ( x ) = 2 x when x < - 3 - 2 x 2 when - 3 < x < 3 2 x when x > 3 From the previous eq. reveals to be clear that when x approaches - 3 from the left and from the right we get two diﬀerent result (6 and - 6 respectively). We may conclude that the fct in not diﬀerentiable in x = - 3. For the same reason the function is not diﬀerentiable in x = 3. example2 f ( x ) = ( x +5) 2 / 3 . In this case the tangent line to this func- tion becomes vertical when x approaches - 5. Using the power- chain rule ( d dx f ( x ) n

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mt2Sample - Math 16A Spring 2010 MT2 Sample 1 Describe the...

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