If f is differentiable at r = a, then prove that from Mathemati
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    Continuity and Differentiability

    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.
    81 Views
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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

    straight L. straight H. straight S equals 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 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 a right parenthesis plus straight a 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 space space space space space space space space space equals Lt with straight x rightwards arrow straight a below fraction numerator left parenthesis straight x minus straight a right parenthesis space straight f left parenthesis straight a right parenthesis minus straight a left curly bracket straight f left parenthesis straight x right parenthesis minus straight f left parenthesis straight a right parenthesis right curly bracket over denominator straight x minus straight a end fraction equals Lt with straight x rightwards arrow straight a below open square brackets fraction numerator left parenthesis straight x minus straight a right parenthesis space straight f left parenthesis straight a right parenthesis over denominator straight x minus straight a end fraction minus straight a fraction numerator straight f left parenthesis straight x right parenthesis minus straight f left parenthesis straight a right parenthesis over denominator straight x minus straight a end fraction close square brackets space space space space space space space space space equals Lt with straight x rightwards arrow straight a below straight f left parenthesis straight a right parenthesis minus straight a space Lt with straight x rightwards arrow straight a below fraction numerator straight f left parenthesis straight x right parenthesis minus straight f left parenthesis straight a right parenthesis over denominator straight x minus straight a end fraction equals straight f left parenthesis straight a right parenthesis minus straight a space straight f left parenthesis straight a 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 space space space space space space space space space space space space space space space space space space space space space space space left square bracket because straight f left parenthesis straight x right parenthesis space is space differentiable space aty space straight x equals straight a right square bracket
    73 Views
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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
    82 Views
    Answer
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