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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.

Let function f be defined by
f(x) = |x - 1| + |x - 2| + |x - 3| + |x - 4| + |x - 5|
This function is continuous everywhere.
Also it is differentiable everywhere except at x = 1, 2, 3, 4, 5.
Hence the result.

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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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