from Mathematics Continuity and Differentiability

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Continuity and Differentiability

Short Answer Type

571.

If space straight y equals left parenthesis cos to the power of negative 1 end exponent space straight x right parenthesis squared comma space prove space that space left parenthesis 1 minus straight x squared right parenthesis space straight y subscript 2 minus straight x space straight y subscript 1 minus 2 equals 0.

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572.

If space space space straight y equals left parenthesis sin to the power of negative 1 end exponent straight x right parenthesis squared comma space prove space that space left parenthesis 1 minus straight x 2 right parenthesis straight y subscript 2 minus straight x space straight y subscript 1 minus 2 equals 0. space

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573.

If space space straight y equals open square brackets log open parentheses straight x plus square root of straight x squared plus 1 end root close parentheses close square brackets squared comma space prove space that space left parenthesis 1 space plus space straight x squared right parenthesis straight y subscript 2 space plus space straight x space straight y subscript 1 space equals space 2.

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574.

If space space straight y equals sin left parenthesis straight m space sin to the power of negative 1 end exponent straight x right parenthesis comma space prove space that space left parenthesis 1 minus straight x squared right parenthesis straight y subscript 2 minus straight x space straight y subscript 1 plus straight m squared space straight y equals 0.

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575. If y = e^{m sin₁ x} , prove that (1 - x²) y₂ - x y₁ = m² y.

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576. If y = e^{a cos_{-1 x}} , show that

left parenthesis 1 minus straight x squared right parenthesis fraction numerator straight d squared straight y over denominator dx squared end fraction minus straight x dy over dx minus straight a squared straight y equals 0.

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577.

If space space straight f left parenthesis straight x right parenthesis equals vertical line straight x vertical line cubed comma space show space that space straight f " left parenthesis straight x right parenthesis space exists space for space all space real space straight x space and space find space it.

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578. If x = f(t), y = g (t) possess second order derivative for all t in (a, b) and f"(t) is inevitable then prove that

fraction numerator straight d squared straight y over denominator dx squared end fraction equals fraction numerator straight f apostrophe left parenthesis straight t right parenthesis space straight g apostrophe apostrophe left parenthesis straight t right parenthesis minus straight g apostrophe left parenthesis straight t right parenthesis space straight f apostrophe apostrophe left parenthesis straight t right parenthesis over denominator left parenthesis straight f apostrophe left parenthesis straight t right parenthesis right parenthesis squared end fraction

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Long Answer Type

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579. $If space left parenthesis straight x minus straight a right parenthesis squared space plus left parenthesis straight y minus straight b right parenthesis squared space equals straight c squared space comma space prove space that space space fraction numerator open square brackets 1 plus open parentheses begin display style dy over dx end style close parentheses squared close square brackets to the power of begin display style 3 over 2 end style end exponent over denominator begin display style fraction numerator straight d squared straight y over denominator dx squared end fraction end style end fraction space is space straight a space constant space independent space of space straight a space and space straight b.$

The space given space equation space is space left parenthesis straight x minus straight a right parenthesis squared space plus left parenthesis straight y minus straight b right parenthesis squared space equals straight c squared 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 parenthesis 1 right parenthesis Differentiating space it space straight w. straight r. straight t. straight x space successively space two space times comma 2 left parenthesis straight x minus straight a right parenthesis plus 2 left parenthesis straight y minus straight b right parenthesis dy over dx equals 0 or space left parenthesis straight x minus straight a right parenthesis plus left parenthesis straight y minus straight b right parenthesis dy over dx equals 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 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 parenthesis 2 right parenthesis rightwards double arrow space 1 plus left parenthesis straight y minus straight b right parenthesis fraction numerator straight d squared straight y over denominator dx squared end fraction plus open parentheses dy over dx close parentheses squared equals 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 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 parenthesis 3 right parenthesis From space left parenthesis 3 right parenthesis comma space left parenthesis straight y minus straight b right parenthesis fraction numerator straight d squared straight y over denominator dx squared end fraction equals negative open square brackets 1 plus open parentheses dy over dx close parentheses squared close square brackets comma space space therefore space straight y minus straight b equals negative fraction numerator open square brackets 1 plus open parentheses begin display style dy over dx end style close parentheses squared close square brackets to the power of begin display style 3 over 2 end style end exponent over denominator begin display style fraction numerator straight d squared straight y over denominator dx squared end fraction end style end fraction space space space space space space... left parenthesis 4 right parenthesis

Putting space this space value space of space straight y minus straight b space in space left parenthesis 2 right parenthesis comma space we space get comma left parenthesis straight x minus straight a right parenthesis minus fraction numerator open square brackets 1 plus open parentheses begin display style dy over dx end style close parentheses squared close square brackets to the power of begin display style 2 end style end exponent over denominator begin display style fraction numerator straight d squared straight y over denominator dx squared end fraction end style end fraction. dy over dx equals 0 comma space space therefore space straight x minus straight a equals fraction numerator 1 plus open parentheses begin display style dy over dx end style close parentheses squared over denominator begin display style fraction numerator straight d squared straight y over denominator dx squared end fraction end style end fraction. dy over dx space space space space space space space space space space space space space space space space space space space space space... left parenthesis 5 right parenthesis

Putting space values space of space straight x minus straight a comma space straight y minus straight b space from space left parenthesis 5 right parenthesis space and space left parenthesis 4 right parenthesis space in space left parenthesis 1 right parenthesis comma space we space get. fraction numerator open square brackets 1 plus open parentheses begin display style dy over dx end style close parentheses squared close square brackets to the power of begin display style 2 end style end exponent over denominator begin display style fraction numerator straight d squared straight y over denominator dx squared end fraction end style end fraction. open parentheses dy over dx close parentheses squared plus fraction numerator open square brackets 1 plus open parentheses begin display style dy over dx end style close parentheses squared close square brackets to the power of begin display style 2 end style end exponent over denominator begin display style open parentheses fraction numerator begin display style straight d squared straight y end style over denominator begin display style dx squared end style end fraction close parentheses squared end style end fraction equals straight c squared or space space space open square brackets 1 plus open parentheses dy over dx close parentheses squared close square brackets squared open parentheses dy over dx close parentheses squared plus open square brackets 1 plus open parentheses dy over dx close parentheses squared close square brackets squared equals straight c squared open parentheses fraction numerator straight d squared straight y over denominator dx squared end fraction close parentheses squared or space space space open square brackets 1 plus open parentheses dy over dx close parentheses squared close square brackets squared open square brackets open parentheses dy over dx close parentheses squared plus 1 close square brackets equals straight c squared open parentheses fraction numerator straight d squared straight y over denominator dx squared end fraction close parentheses or space space space open square brackets 1 plus open parentheses dy over dx close parentheses squared close square brackets cubed equals straight c squared open parentheses fraction numerator straight d squared straight y over denominator dx squared end fraction close parentheses squared therefore space space open square brackets 1 plus open parentheses dy over dx close parentheses squared close square brackets to the power of 3 over 2 end exponent equals straight c fraction numerator straight d squared straight y over denominator dx squared end fraction space comma space space therefore space fraction numerator open square brackets 1 plus open parentheses begin display style dy over dx end style close parentheses squared close square brackets to the power of begin display style 3 over 2 end style end exponent over denominator begin display style fraction numerator straight d squared straight y over denominator dx squared end fraction end style end fraction equals straight c which space is space straight a space constant space independent space of space straight a space and space straight b.

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Short Answer Type

580.

Verify space Rolle apostrophe straight s space theorem space for space the space function straight f left parenthesis straight x right parenthesis equals fraction numerator 8 straight x squared over denominator 3 end fraction minus 2 straight x comma space straight x element of open square brackets 0 comma 3 over 4 close square brackets

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