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Continuity and Differentiability

Short Answer Type

561.

If space straight y equals log open parentheses straight x plus square root of 1 plus straight x squared end root close parentheses comma space show space that space left parenthesis 1 plus straight x squared right parenthesis fraction numerator straight d squared straight y over denominator dx squared end fraction plus straight x dy over dx equals 0.

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562.

If space straight y equals fraction numerator sin to the power of negative 1 end exponent straight x over denominator square root of 1 minus straight x squared end root end fraction comma space prove space that space left parenthesis 1 minus straight x squared right parenthesis straight y subscript 2 minus 3 space straight x space straight y subscript 1 minus straight y equals 0.

82 Views

563.

If space space space straight x equals straight a left parenthesis straight ө plus sin space straight ө right parenthesis comma space straight y equals straight a left parenthesis 1 minus cos space straight ө right parenthesis comma space find space dy over dx space at space straight theta equals straight pi over 2.

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564.

If space space straight x equals straight a left parenthesis cos space straight t plus straight t space sin space straight t right parenthesis comma space straight y equals straight a left parenthesis sin space straight t space minus straight t space cos space straight t right parenthesis comma space evaluate space fraction numerator straight d squared straight y over denominator dx squared end fraction space at space straight t equals straight pi over 4

79 Views

565. $If space straight x equals 3 space sin space straight t minus sin space 3 straight t comma space straight y equals 3 space cos space straight t minus cos space 3 space straight t comma space find space dy over dx space at space straight t equals straight pi over 3.$

space space space space space space space space space space space straight x equals 3 space sin space straight t minus sin space 3 straight t comma space space space space space straight y equals 3 space cos space straight t minus cos space 3 space straight t therefore space dx over dt equals 3 space cos space straight t minus 3 space cos space 3 straight t comma space dy over dt equals negative 3 space sin space straight t plus 3 space sin space 3 straight t Now space dy over dx equals fraction numerator begin display style dy over dt end style over denominator begin display style dx over dt end style end fraction equals fraction numerator negative 3 space sin space straight t plus 3 space sin space 3 straight t over denominator 3 space cos space straight t minus 3 space cios space 3 straight t end fraction equals fraction numerator sin space 3 straight t minus sin space straight t over denominator cos space straight t minus cos space 3 straight t end fraction equals fraction numerator 2 space cos begin display style fraction numerator 3 straight t plus straight t over denominator 2 end fraction end style sin begin display style fraction numerator 3 straight t minus straight t over denominator 2 end fraction end style over denominator 2 space sin space begin display style fraction numerator 3 straight t plus straight t over denominator 2 end fraction end style sin begin display style fraction numerator 3 straight t minus straight t over denominator 2 end fraction end style end fraction 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 fraction numerator cos space 2 straight t over denominator sin space 2 straight t end fraction equals cos space 2 straight t At space straight t equals straight pi over 3 comma space fraction numerator straight d squared straight y over denominator dx squared end fraction equals negative 2 over 3 fraction numerator cosec squared space begin display style fraction numerator 2 straight pi over denominator 3 end fraction end style over denominator cos begin display style straight pi over 3 end style minus cos space straight pi end fraction equals negative 2 over 3 fraction numerator cosec squared space begin display style straight pi over 3 end style over denominator cos begin display style straight pi over 3 end style minus cos space straight pi end fraction 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 negative 2 over 3. fraction numerator open parentheses begin display style fraction numerator 2 over denominator square root of 3 end fraction end style close parentheses squared over denominator begin display style 1 half end style plus 1 end fraction equals negative 2 over 3. fraction numerator begin display style 4 over 3 end style over denominator begin display style 3 over 2 end style end fraction equals negative 2 over 3 cross times 2 over 3 cross times 4 over 3 equals negative 16 over 27

87 Views

566.

If space space space straight x equals 2 space cos space straight ө minus cos space 2 straight ө space and space straight y equals 2 space sin space straight ө minus sin space 2 straight ө comma space find space open parentheses fraction numerator straight d squared straight y over denominator dx squared end fraction close parentheses subscript straight theta equals straight pi over 2 end subscript

81 Views

567.

If space straight x to the power of 1 half end exponent plus straight y to the power of 1 half end exponent equals straight a to the power of 1 half end exponent comma space find space the space value space of space fraction numerator straight d squared straight y over denominator dx squared end fraction space at space straight x equals straight a.

79 Views

568.

If space straight x squared over straight a squared plus straight y squared over straight b squared equals 1 comma space show space that space fraction numerator straight d squared straight y over denominator dx squared end fraction equals negative fraction numerator straight b 4 over denominator straight a squared straight y cubed end fraction.

82 Views

569.

If space straight y equals tan to the power of negative 1 end exponent space straight x comma space show space that space open parentheses 1 plus straight x squared close parentheses fraction numerator straight d squared straight y over denominator dx squared end fraction plus 2 space straight x dy over dx equals 0

93 Views

570.

If space straight y equals left parenthesis tan to the power of negative 1 end exponent space straight x right parenthesis squared comma space show space that space left parenthesis straight x squared space plus space 1 right parenthesis 2 space straight y subscript 2 plus 2 space straight x left parenthesis straight x squared plus 1 right parenthesis space straight y to the power of 1 equals 2.

97 Views