Let f be differentiable at a, then evaluate from Mathematics Co

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 Multiple Choice QuestionsShort Answer Type

121.

Find the values of a and b such that the function defined by
straight f left parenthesis straight x right parenthesis equals open curly brackets table row cell 5 comma end cell row cell ax plus straight b comma end cell row cell 21 comma end cell end table close table row cell if space straight x less or equal than 2 space space space space space space space space space space end cell row cell if space 2 less than straight x less than 10 end cell row cell if space straight x greater or equal than 10 space space space space space space space space end cell end table
is a continous function.

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122. Show that the function defined by f(x) = sin (x2 ) is a continuous function.
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123. Show that the function defined by f(x) =  cos(x2) is a continuous function.
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124. Examine that sin | x | is a continuous function.
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125. Show that the function defined by f(x) = | cos x | is a continuous function.
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126. Show that the function f defined by f(x) = |1–x+|x||, where x is any real number, is a continuous function.
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127.

Find all the points of discontinuity of f defined by
f(x) =|x|–|x+1|.

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128. If f is differentiable at r = a, then prove that
Lt with straight x rightwards arrow straight a below fraction numerator straight x space straight f left parenthesis straight a right parenthesis minus straight a space straight f left parenthesis straight x right parenthesis over denominator straight x minus straight a end fraction equals straight f left parenthesis straight a right parenthesis equals straight a space straight f left parenthesis straight a right parenthesis
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129. Let a function f r be continuous at a. Show that the function g (x) = (x - a)f(x) is differentiable at x = a.
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130. Let f be differentiable at a, then evaluate

Lt with straight x rightwards arrow straight a below fraction numerator straight x squared space straight f left parenthesis straight a right parenthesis minus straight a squared space straight f left parenthesis straight x right parenthesis over denominator straight x minus straight a end fraction


Put space straight x equals straight a plus straight h space so space that space straight h rightwards arrow 0 space as space straight x rightwards arrow straight a
therefore space Lt with straight x rightwards arrow straight a below fraction numerator straight x squared space straight f left parenthesis straight a right parenthesis minus straight a squared space straight f left parenthesis straight x right parenthesis over denominator straight x minus straight a end fraction equals Lt with straight h rightwards arrow 0 below fraction numerator left parenthesis straight a plus straight h right parenthesis squared space straight f left parenthesis straight a right parenthesis minus straight a squared space straight f left parenthesis straight a plus straight h right parenthesis over denominator straight a plus straight h minus straight a end fraction
space space space space space space space space space space space space space space space space space equals Lt with straight h rightwards arrow 0 below fraction numerator left parenthesis straight a squared plus 2 space straight a space straight h plus straight h right parenthesis squared space straight f left parenthesis straight a right parenthesis minus straight a squared space straight f left parenthesis straight a plus straight h right parenthesis over denominator straight h end fraction
space space space space space space space space space space space space space space space space space equals Lt with straight h rightwards arrow 0 below fraction numerator straight a squared space straight f left parenthesis straight a right parenthesis plus 2 space straight a space straight h space straight f left parenthesis straight a right parenthesis plus straight h squared space straight f left parenthesis straight a right parenthesis minus straight a squared space straight f left parenthesis straight a plus straight h right parenthesis over denominator straight h end fraction

space space space space space space space space space space space space space space space space space equals Lt with straight h rightwards arrow 0 below fraction numerator straight h left curly bracket 2 space straight a space straight f left parenthesis straight a right parenthesis plus straight h space straight f left parenthesis straight a right parenthesis right curly bracket minus straight a squared left curly bracket straight f left parenthesis straight a plus straight h right parenthesis minus straight f left parenthesis straight a right parenthesis right curly bracket over denominator straight h end fraction
space space space space space space space space space space space space space space space space space equals Lt with straight h rightwards arrow 0 below open square brackets 2 space straight a space straight f left parenthesis straight a right parenthesis plus straight h space straight f left parenthesis straight a right parenthesis minus straight a squared 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 close square brackets
space space space space space space space space space space space space space space space space space equals Lt with straight x rightwards arrow straight a below 2 space straight a space straight f left parenthesis straight a right parenthesis plus Lt with straight h rightwards arrow 0 below straight h space straight f left parenthesis straight a right parenthesis minus straight a squared space Lt with straight h rightwards arrow 0 below 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
space space space space space space space space space space space space space space space space space equals 2 space straight a space straight f left parenthesis straight a right parenthesis plus 0 straight f left parenthesis straight a right parenthesis minus straight a squared straight f left parenthesis straight a right parenthesis equals 2 space straight a space straight f left parenthesis straight a right parenthesis minus straight a squared straight f left parenthesis straight a right parenthesis.
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