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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

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148. Use the Chain rule to find the derivatives of the following:
$straight f left parenthesis straight t right parenthesis equals open parentheses fraction numerator 2 straight t cubed plus 1 over denominator 3 straight t squared plus 1 end fraction close parentheses squared$

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 Let space space space space space space space space space space space space space space space straight y equals straight f left parenthesis straight t right parenthesis minus straight u squared comma space where space straight u equals fraction numerator 2 straight t cubed plus 1 over denominator 3 straight t squared plus 1 end fraction. Then space space space space space space space space space dy over du equals 2 straight u and space space space space space space space space space space space du over dt equals fraction numerator left parenthesis 3 straight t squared plus 1 right parenthesis begin display style straight d over dt end style left parenthesis 2 straight t cubed plus 1 right parenthesis minus left parenthesis 2 straight t cubed plus 1 right parenthesis begin display style straight d over dt end style left parenthesis 3 straight t squared plus 1 right parenthesis over denominator left parenthesis 3 straight t squared plus 1 right parenthesis squared end fraction space space space space space space space space space space space space space space space space space space space space space space space space equals fraction numerator left parenthesis 3 straight t squared plus 1 right parenthesis left parenthesis 6 straight t squared right parenthesis minus left parenthesis 2 straight t cubed plus 1 right parenthesis left parenthesis 6 straight t right parenthesis over denominator left parenthesis 3 straight t squared plus 1 right parenthesis squared end fraction equals fraction numerator 18 straight t to the power of 4 plus 6 straight t squared minus 12 straight t to the power of 4 minus 6 straight t over denominator left parenthesis 3 straight t squared plus 1 right parenthesis squared end fraction space space space space space space space space space space space space space space space space space space space space space space space space equals fraction numerator 6 straight t to the power of 4 plus 6 straight t squared minus 6 straight t over denominator left parenthesis 3 straight t squared plus 1 right parenthesis squared end fraction equals fraction numerator 6 left parenthesis straight t to the power of 4 plus straight t squared minus straight t right parenthesis over denominator left parenthesis 3 straight t squared plus 1 right parenthesis squared end fraction By space Chain space Rule comma dy over dt equals dy over du du over dt therefore space space space space space space space space space space space space space space space space space space dy over dt equals 2 straight u. fraction numerator 6 left parenthesis straight t to the power of 4 plus straight t squared minus 1 right parenthesis over denominator left parenthesis 3 straight t squared plus 1 right parenthesis squared end fraction equals 2 open parentheses fraction numerator 2 straight t cubed plus 1 over denominator 3 straight t squared plus 1 end fraction close parentheses. fraction numerator 6 left parenthesis straight t to the power of 4 plus straight t squared minus straight t right parenthesis over denominator left parenthesis 3 straight t squared plus 1 right parenthesis squared end fraction space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals fraction numerator 12 left parenthesis 2 straight t squared plus 1 right parenthesis left parenthesis straight t to the power of 4 plus straight t squared minus straight t right parenthesis over denominator left parenthesis 3 straight t squared plus 1 right parenthesis cubed end fraction

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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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