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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 $space space space space space space space space space stack space lim with straight x rightwards arrow straight a below fraction numerator xf left parenthesis straight a right parenthesis minus af left parenthesis straight x right parenthesis over denominator straight x minus straight a end fraction$ .

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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)

Here, $space space space space space space space space space space space space space space space space space space space space space space space space space space straight f space left parenthesis straight x right parenthesis space equals space straight x to the power of 100 over 100 plus straight x to the power of 99 over 99 plus..... plus straight x squared over 2 plus straight x plus 1$

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 $space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals straight d over dx open parentheses straight x to the power of 100 over 100 close parentheses plus straight d over dx open parentheses straight x to the power of 99 over 99 close parentheses plus..... plus straight d over dx open parentheses straight x squared over 2 close parentheses plus straight d over dx plus left parenthesis straight x right parenthesis plus straight d over dx plus left parenthesis 1 right parenthesis$

$space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals 1 over 100 straight x 100 straight x to the power of 99 plus 1 over 99 space straight d over dx straight x to the power of 99 plus..... plus 1 half space straight d over dx straight x squared plus straight d over dx straight x to the power of 1 plus 0$

$space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals 1 over 100 straight x 100 straight x to the power of 99 plus 1 over 99 straight x 99 straight x to the power of 98 plus...... plus 1 half straight x 2 straight x to the power of 1 plus 1 straight x to the power of 0$

$rightwards double arrow$ $space space space space space space space space space space space space space space space space space space space space space space space space space space space straight f left parenthesis x right parenthesis space equals space straight x to the power of 99 space end exponent plus straight x to the power of 98 plus...... plus space straight x to the power of 1 plus 1$

L.H.S. = f(1) = (1)⁹⁹+ (1)⁹⁸+ ....... +(1)¹+ 1 = 1+1+ ....+1+1 = 100

R.H.S. = [100f'(0) = 100(0)⁹⁹+(0)⁹⁸+ ..... + (0)¹+1] = 100