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Continuity and Differentiability

Short Answer Type

141. Examine the derivability of the following function:
$straight f left parenthesis straight x right parenthesis equals open curly brackets table attributes columnalign left end attributes row cell straight x minus 1 comma space space space straight x less than 2 end cell row cell 2 straight x minus 3 comma space straight x greater or equal than 2 end cell end table close at space straight x equals 2$

Here space 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 straight L. straight H. straight D equals Lt with straight x rightwards arrow 2 to the power of minus below fraction numerator straight f left parenthesis straight x right parenthesis minus straight f left parenthesis 2 right parenthesis over denominator straight x minus 2 end fraction equals Lt with straight x rightwards arrow 2 to the power of minus below space fraction numerator left parenthesis straight x minus 1 right parenthesis minus left parenthesis 4 minus 3 right parenthesis over denominator straight x minus 2 end fraction equals Lt with straight x rightwards arrow 2 to the power of minus below fraction numerator straight x minus 2 over denominator straight x minus 2 end fraction equals Lt with straight x rightwards arrow 2 to the power of minus below left parenthesis 1 right parenthesis equals 1 straight R. straight H. straight D equals Lt with straight x rightwards arrow 2 plus below fraction numerator straight f left parenthesis straight x right parenthesis minus straight f left parenthesis 2 right parenthesis over denominator straight x minus 2 end fraction equals Lt with straight x rightwards arrow 2 to the power of plus below space fraction numerator left parenthesis 2 straight x minus 3 right parenthesis minus left parenthesis 4 minus 3 right parenthesis over denominator straight x minus 2 end fraction equals Lt with straight x rightwards arrow 2 to the power of plus below fraction numerator 2 straight x minus 4 over denominator straight x minus 2 end fraction space space space space space space space space space space space stack equals Lt with straight x rightwards arrow 2 to the power of plus below fraction numerator 2 left parenthesis straight x minus 2 right parenthesis over denominator straight x minus 2 end fraction equals 2 Lt with straight x rightwards arrow 2 to the power of plus below left parenthesis 1 right parenthesis equals 2 cross times 1 equals 2 therefore space straight L. straight H. straight D not equal to straight R. straight H. straight D. therefore space straight f left parenthesis straight x right parenthesis space is space not space dervable space 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

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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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