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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.

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34.
If f is derivable at x=a, evaluate $space space space space space space space space space stack space lim with straight x rightwards arrow straight a below fraction numerator xf left parenthesis straight a right parenthesis minus af left parenthesis straight x right parenthesis over denominator straight x minus straight a end fraction$ .

Since f is derivable at x =a

∴ $space space space space space space space space space space space space space space straight f apostrophe left parenthesis straight a right parenthesis space space space space equals space limit as straight h rightwards arrow straight a of fraction numerator straight f left parenthesis straight a plus straight h right parenthesis minus straight f left parenthesis straight a right parenthesis over denominator straight h end fraction$

Now, $space space space space space space space space space space space space limit as straight x rightwards arrow 0 of fraction numerator xf left parenthesis straight a right parenthesis minus af left parenthesis straight x right parenthesis over denominator straight x minus straight a end fraction equals limit as straight h rightwards arrow 0 of fraction numerator left parenthesis straight a plus straight h right parenthesis space straight f left parenthesis straight a right parenthesis space minus af left parenthesis straight a plus straight h right parenthesis over denominator straight a plus straight h minus straight a end fraction$

[By putting x =a +h as x $rightwards arrow$ a, h $rightwards arrow$ 0]

space space space space space space space space space space space space space space space space space equals space space space limit as straight h rightwards arrow 0 of fraction numerator af left parenthesis straight a right parenthesis plus hf left parenthesis straight a right parenthesis minus af left parenthesis straight a plus straight h right parenthesis over denominator straight h end fraction equals limit as straight h rightwards arrow 0 of fraction numerator straight a left square bracket straight f left parenthesis straight a right parenthesis minus straight f left parenthesis straight a plus straight h right parenthesis plus right square bracket plus hf left parenthesis straight a right parenthesis over denominator straight h end fraction

space space space space space space space space space space space space space equals negative straight a space limit as straight h rightwards arrow 0 of open square brackets fraction numerator straight f left parenthesis straight a plus straight h right parenthesis minus straight f left parenthesis straight a right parenthesis over denominator straight h end fraction plus straight f left parenthesis straight a right parenthesis close square brackets

space space space space space space space space space space space space space space equals space minus straight a space limit as straight h rightwards arrow 0 of fraction numerator straight f left parenthesis straight a plus straight h right parenthesis minus straight f left parenthesis straight a right parenthesis over denominator straight h end fraction plus limit as straight h rightwards arrow 0 of straight f left parenthesis straight a right parenthesis

= -af'(a) + f(a) [By using (1)]

= f(a)-af'(a)

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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)