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

space space space space space space space space space space space straight x equals straight a left parenthesis straight ө plus sin space straight ө right parenthesis therefore space dx over dθ equals straight a left parenthesis 1 plus cos space straight theta right parenthesis Also space straight y equals straight a left parenthesis 1 minus cos space straight ө right parenthesis therefore space dy over dθ equals straight a space sin space straight theta Now space dy over dx equals fraction numerator begin display style dy over dθ end style over denominator begin display style dx over dθ end style end fraction equals fraction numerator straight a space sin space straight theta over denominator straight a left parenthesis 1 plus cos space straight theta right parenthesis end fraction equals fraction numerator sin space straight theta over denominator 1 plus cos space straight theta end fraction equals fraction numerator 2 space sin begin display style straight theta over 2 end style cos begin display style straight theta over 2 end style over denominator 2 space cos squared begin display style straight theta over 2 end style end fraction therefore space dy over dx equals tan straight theta over 2 Again space differentiating space straight w. straight r. straight t. straight x comma fraction numerator straight d squared straight y over denominator dx squared end fraction equals sec squared straight theta over 2.1 half dθ over dx equals 1 half sec squared straight theta over 2. fraction numerator 1 over denominator straight a left parenthesis 1 plus cos space straight theta right parenthesis end fraction space space space space space space space space space space equals fraction numerator 1 over denominator 2 space straight a end fraction sec squared straight theta over 2 fraction numerator 1 over denominator 2 space cos squared end fraction equals fraction numerator 1 over denominator 4 straight a end fraction sec to the power of 4 straight theta over 2 When space straight theta equals straight pi over 4 comma space fraction numerator straight d squared straight y over denominator dx squared end fraction equals fraction numerator 1 over denominator 4 straight a end fraction sec to the power of 4 straight theta over 2 equals fraction numerator 1 over denominator 4 straight a end fraction open parentheses square root of 2 close parentheses to the power of 4 equals fraction numerator 1 over denominator 4 straight a end fraction cross times 4 equals 1 over straight a

89 Views

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.

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