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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

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

Here space 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 since space straight f space is space continous space at space straight x equals straight c therefore space straight f space is space continous space at space straight x equals straight c therefore space Lt with straight x rightwards arrow straight c to the power of minus below space straight f left parenthesis straight x right parenthesis equals Lt with straight x rightwards arrow straight c to the power of plus below straight f left parenthesis straight x right parenthesis equals straight f left parenthesis straight c right parenthesis rightwards double arrow space Lt with straight x rightwards arrow straight c to the power of minus below space straight x squared equals Lt with straight x rightwards arrow straight c to the power of plus below left parenthesis straight a space straight x plus straight b right parenthesis equals straight c squared rightwards double arrow space straight c squared equals straight a space straight c plus straight b minus straight c to the power of 2 space space space space space end exponent rightwards double arrow ac plus straight b equals straight c squared since space straight f space is space continous space at space straight x equals straight c space therefore Lt with straight x rightwards arrow straight c to the power of minus below fraction numerator straight f left parenthesis straight x right parenthesis minus straight f left parenthesis straight c right parenthesis over denominator straight x minus straight c end fraction equals Lt with straight x rightwards arrow straight c to the power of plus below fraction numerator straight f left parenthesis straight x right parenthesis minus straight f left parenthesis straight c right parenthesis over denominator straight x minus straight c end fraction rightwards double arrow space Lt with straight x rightwards arrow straight c to the power of minus below fraction numerator left parenthesis straight x right parenthesis squared minus left parenthesis straight c right parenthesis squared over denominator straight x minus straight c end fraction equals Lt with straight x rightwards arrow straight c to the power of plus below fraction numerator left parenthesis straight a space straight x plus straight b right parenthesis minus straight c squared over denominator straight x minus straight c end fraction rightwards double arrow space Lt with straight h rightwards arrow 0 below fraction numerator left parenthesis straight c minus straight h right parenthesis squared minus straight c squared over denominator straight c minus straight h minus straight c end fraction equals Lt with straight h rightwards arrow 0 below fraction numerator left curly bracket straight a left parenthesis straight c plus straight h right parenthesis plus straight h right curly bracket minus straight c squared over denominator straight c plus straight h minus straight c end fraction rightwards double arrow space Lt with straight h rightwards arrow 0 below fraction numerator straight h squared minus 2 space straight c space straight h over denominator negative straight h end fraction equals Lt with straight h rightwards arrow 0 below fraction numerator straight a space straight c plus straight b plus straight a space straight h minus straight c squared over denominator straight h 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 space space space space space space space space space space space space space space space space space space space space space space space space left square bracket because space of space left parenthesis 1 right parenthesis right square bracket rightwards double arrow space Lt with straight h rightwards arrow 0 below fraction numerator straight h squared minus 2 space straight c space straight h over denominator negative straight h end fraction equals Lt with straight h rightwards arrow 0 below fraction numerator straight a space straight h over denominator straight h end fraction rightwards double arrow space Lt with straight h rightwards arrow 0 below left parenthesis negative straight h plus 2 space straight c right parenthesis equals Lt with straight h rightwards arrow 0 below straight a rightwards double arrow 0 minus plus 2 space straight c equals straight a space space space space space space space space space space space space space space space space space space rightwards double arrow space straight a equals 2 straight c therefore space from space left parenthesis 1 right parenthesis comma space 2 straight c. straight c plus straight b equals straight c squared space space space space rightwards double arrow space straight b equals negative straight c squared Here space straight a equals 2 space straight c comma space straight b equals negative straight c squared

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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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