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Continuity And Differentiability

Question
CBSEENMA12035435
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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

Solution
space space space space space space space space space space space space space 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 space space space space straight y equals straight a left parenthesis sin space straight t space minus straight t space cos space straight t right parenthesis therefore space space space space space dy over dt equals straight a left parenthesis negative sin space straight t plus straight t space cos space straight t plus sin space straight t right parenthesis equals straight a space straight t space cos space straight t and space space dy over dt equals straight a left parenthesis cos space straight t plus straight t space sin space straight t minus cos space straight t right parenthesis equals straight a space straight t space sin space 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 straight a space straight t space sin space straight t over denominator straight a space straight t space cos space straight t end fraction equals tan space straight t therefore space space space space fraction numerator straight d squared straight y over denominator dx squared end fraction equals sec squared space straight t. dt over dx equals sec squared space straight t. fraction numerator 1 over denominator straight a space straight t space cos space straight t end fraction equals fraction numerator 1 over denominator straight a space straight t end fraction sec cubed space straight t At space space space space straight t 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 straight a. begin display style straight pi over 4 end style end fraction sec cubed space straight pi over 4 equals fraction numerator 4 over denominator straight a space straight pi end fraction cross times left parenthesis square root of 2 right parenthesis cubed equals fraction numerator 8 square root of 2 over denominator straight a space straight pi end fraction

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