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Limits and Derivatives

Short Answer Type

31.

Let f(x) $space space space space space equals space open curly brackets table attributes columnalign left end attributes row cell 1 minus straight x squared comma space straight x less or equal than 0 end cell row cell 1 plus straight x squared comma space straight x space greater than 0 end cell end table close$ find Lf'(0) and Rf'(0)

Is f derivable at x = 0?

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32. Find the derivative of the function f(x) = 99x at x= 100.

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33.
Let $space space space straight f left parenthesis straight x right parenthesis space equals space open curly brackets table attributes columnalign left end attributes row cell straight x plus straight a comma space straight x less or equal than 1 end cell row cell ax squared plus 1 comma space straight x greater than 1 end cell end table close$

find a so that f may be derivable at x =1.

Here, f(1) = 1+a

$space space space space space space space space space space space Lf apostrophe left parenthesis 1 right parenthesis space equals space limit as straight h rightwards arrow 0 to the power of minus of fraction numerator straight f left parenthesis 1 plus straight h right parenthesis minus straight f left parenthesis 1 right parenthesis over denominator straight h end fraction equals limit as straight h rightwards arrow 0 to the power of minus of fraction numerator left parenthesis 1 plus straight h right parenthesis plus straight a minus left parenthesis 1 plus straight a right parenthesis over denominator straight h end fraction$

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$space space space space space space Rf apostrophe left parenthesis 1 right parenthesis space space equals space limit as straight h rightwards arrow 0 to the power of plus of fraction numerator straight f left parenthesis 1 plus straight h right parenthesis minus straight f left parenthesis 1 right parenthesis over denominator straight h end fraction$ (here h >0 $rightwards double arrow$ 1 + h > 1)

$space space space space space space equals space limit as straight h rightwards arrow 0 to the power of plus of fraction numerator straight a left parenthesis 1 plus straight h right parenthesis squared plus 1 minus left parenthesis 1 plus straight a right parenthesis over denominator straight h end fraction$

 $space space space space space space space space space equals space limit as straight h rightwards arrow 0 to the power of plus of fraction numerator ah squared plus 2 ah over denominator straight h end fraction equals limit as straight h rightwards arrow 0 to the power of plus of left parenthesis ah plus 2 straight a right parenthesis equals 0 plus 2 straight a equals 2 straight a$

Since f is derivable at x =1

∴ Lf'(1) = Rf'(1)

$rightwards double arrow$ 1 = 2a $rightwards double arrow$    $space space space space straight a space equals space 1 half$

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34.

If f is derivable at x=a, evaluate $space space space space space space space space space stack space lim with straight x rightwards arrow straight a below fraction numerator xf left parenthesis straight a right parenthesis minus af left parenthesis straight x right parenthesis over denominator straight x minus straight a end fraction$ .

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35.

For the function :

$space space space space space space space space straight f left parenthesis straight x right parenthesis space equals straight x to the power of 100 over 100 plus straight x to the power of 99 over 99 plus.... plus fraction numerator straight x squared 2 over denominator 2 end fraction plus straight x plus 1$ Prove that f(1) = 100 f'(0)