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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)

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36. For some constants a and b, find the derivatives of the following functions :

(x-a)(x-b)

Let $space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space straight y space equals space left parenthesis straight x minus straight a right parenthesis left parenthesis straight x minus straight b right parenthesis equals straight x squared space minus ax minus 0 bx plus ab$

$rightwards double arrow$ y = x² - (a+b)x + ab

∴ $space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space dx over dx equals straight d over dx straight x squared minus straight d over dx left parenthesis straight a plus straight b right parenthesis straight x plus straight d over dx left parenthesis ab right parenthesis equals 2 straight x minus left parenthesis straight a plus straight b right parenthesis straight d over dx straight x plus 0$

= 2x-(a+b).1=2x-(a+b)