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Continuity And Differentiability

Question

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If x=a sin 2t(1+os2t) and y=b cos2t(1-cos2t), show that
$open parentheses dy over dx close parentheses subscript straight t equals straight pi over 2 end subscript equals straight b over straight a.$

Solution

space space space space space space space space space space space straight x equals straight a space sin space 2 straight t left parenthesis 1 plus cos space 2 straight t right parenthesis therefore space dx over dt equals straight a open square brackets sin space 2 straight t. straight d over dt left parenthesis 1 plus cos space 2 straight t right parenthesis plus left parenthesis 1 plus cos space 2 straight t right parenthesis. straight d over dt left parenthesis sin space 2 straight t right parenthesis close square brackets space space space space space space space space space space space space space equals straight a left square bracket sin space 2 straight t. left parenthesis negative 2 sin space space 2 straight t right parenthesis plus left parenthesis 1 plus cos space 2 straight t right parenthesis. left parenthesis 2 space cos space 2 straight t right parenthesis right square bracket space space space space space space space space space space space space space equals straight a left square bracket negative 2 sin squared space 2 straight t plus 2 cos squared space 2 straight t plus 2 cos space 2 straight t right square bracket space space space space space space space space space space space space space equals 2 straight a left square bracket left parenthesis cos squared space 2 straight t minus sin squared space 2 straight t right parenthesis plus cos space 2 straight t right square bracket space space space space space space space space space space space space space equals 2 straight a left square bracket cos space 4 straight t plus cos space 2 straight t right square bracket equals 2 straight a left square bracket 2 space cos space 3 straight t space cos space straight t right square bracket equals 4 straight a space cos space 3 straight t space cos space straight t space space space space space space space space space space space straight y equals straight b space cos space 2 straight t left parenthesis 1 minus cos space 2 straight t right parenthesis therefore space dy over dt equals straight b open square brackets cos space 2 straight t. straight d over dt left parenthesis 1 minus cos space 2 straight t right parenthesis plus left parenthesis 1 minus cos space 2 straight t right parenthesis. straight d over dt left parenthesis cos space 2 straight t right parenthesis close square brackets space space space space space space space space space space space space space equals straight b left square bracket cos space 2 straight t. left parenthesis 2 space sin space 2 straight t right parenthesis plus left parenthesis 1 minus cos space 2 straight t right parenthesis left parenthesis negative 2 space sin space 2 straight t right parenthesis right square bracket space space space space space space space space space space space space space equals straight b left square bracket 2 space sin space 2 straight t space cos space 2 straight t plus 2 space sin space 2 straight t space cos space 2 straight t minus 2 space sin space 2 straight t right square bracket space space space space space space space space space space space space space equals 2 straight b left square bracket 2 space sin space 2 straight t minus sin space 2 straight t right square bracket equals 2 straight b left parenthesis 2 space cos space 3 straight t space sin space straight t right parenthesis therefore space dy over dt equals 4 straight b space cos space 3 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 4 straight b space cos space 3 straight t space sin space straight t over denominator 4 straight a space cos space 3 straight t space cos space straight t end fraction equals straight b over straight a tan space straight t At space straight t equals straight pi over 4 comma space dy over dx equals straight b over straight a tan straight pi over 4 equals straight b over straight a cross times 1 equals straight b over straight a

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