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Continuity and Differentiability

Short Answer Type

141. Examine the derivability of the following function:

straight f left parenthesis straight x right parenthesis equals open curly brackets table attributes columnalign left end attributes row cell straight x minus 1 comma space space space straight x less than 2 end cell row cell 2 straight x minus 3 comma space straight x greater or equal than 2 end cell end table close at space straight x equals 2

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Long Answer Type

142. For what choices of a and b is the function

straight f left parenthesis straight x right parenthesis equals open curly brackets table attributes columnalign left columnspacing 1.4ex end attributes row cell straight x squared comma end cell cell straight x less or equal than straight c end cell row cell straight a space straight x plus straight b comma end cell cell straight x greater than straight c end cell end table close differentiable space at space straight x equals straight c ?

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143. Write an example of a function which is everywhere continuous but not differentiable at exactly 3 points.

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Short Answer Type

144. Does there exist a function which is continuous everywhere but not differentiable at exactly two points ? Justify your answer.

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145. Write an example of a function which is continuous everywhere but fails to be differentiable at exactly five points.

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146. Use Chain rule to find the derivative of (3x² + 2)².

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147. Use Chain rule to find the derivative of $open parentheses fraction numerator 3 space straight x minus 1 over denominator 2 space straight x plus 1 end fraction close parentheses squared$

Let space straight y equals open parentheses fraction numerator 3 space straight x minus 1 over denominator 2 space straight x plus 1 end fraction close parentheses squared equals straight u squared space where space straight u equals fraction numerator 3 space straight x minus 1 over denominator 2 space straight x plus 1 end fraction therefore space space space dy over du equals 2 space straight u space space space space space space space du over dx equals fraction numerator left parenthesis 2 straight x plus 1 right parenthesis. begin display style straight d over dx end style left parenthesis 3 straight x minus 1 right parenthesis minus left parenthesis 3 straight x minus 1 right parenthesis. begin display style straight d over dx end style left parenthesis 2 straight x plus 1 right parenthesis over denominator left parenthesis 2 straight x plus 1 right parenthesis squared end fraction space space space space space space space space space space space space space space space equals fraction numerator left parenthesis 2 straight x plus 1 right parenthesis.3 minus left parenthesis 3 straight x minus 1 right parenthesis.2 over denominator open parentheses 2 straight x plus 1 close parentheses squared end fraction equals fraction numerator 6 straight x plus 3 minus 6 straight x plus 2 over denominator left parenthesis 2 straight x plus 1 right parenthesis squared end fraction therefore space space du over dx equals fraction numerator 5 over denominator left parenthesis 2 straight x plus 1 right parenthesis squared end fraction By space Chain space Rule comma space space space space space dy over du equals dy over du dy over dx equals left parenthesis 2 space straight u right parenthesis open square brackets fraction numerator 5 over denominator left parenthesis 2 straight x plus 1 right parenthesis squared end fraction close square brackets equals fraction numerator 10 over denominator left parenthesis 2 straight x plus 1 right parenthesis squared end fraction cross times fraction numerator 3 straight x minus 1 over denominator 2 straight x plus 1 end fraction space space space space space space space space space space space space equals fraction numerator 10 left parenthesis 3 straight x minus 1 right parenthesis over denominator left parenthesis 2 straight x plus 1 right parenthesis cubed end fraction

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148. Use the Chain rule to find the derivatives of the following:

straight f left parenthesis straight t right parenthesis equals open parentheses fraction numerator 2 straight t cubed plus 1 over denominator 3 straight t squared plus 1 end fraction close parentheses squared

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149. Differentiate the following w.r.t.x:

straight e to the power of negative straight x end exponent

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150. Differentiate the following w.r.t.x:

straight e to the power of straight x squared end exponent

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