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Continuity And Differentiability

Question
CBSEENMA12035433
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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.

Solution
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

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