from Mathematics Continuity and Differentiability

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Continuity and Differentiability

Short Answer Type

361.

If space straight x equals straight a left parenthesis straight theta plus sinθ right parenthesis comma space straight y equals straight a left parenthesis 1 minus cosθ right parenthesis comma space fimd space dy over dx at space straight theta equals straight pi over 2.

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362. 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.

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363.

FInd space dy over dx x equals straight a open parentheses cos space straight t plus log space tan straight t over 2 close parentheses comma space straight y equals straight a space sin space straight t.

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364.

Find space dy over dx space when space straight x space equals straight a space cosө comma space straight y space equals straight b space cosө space

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365.

Find space dy over dx space when space straight x equals sin space straight t comma space straight y space equals cos space 2 straight t

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366.

Find space dy over dx when space straight x equals straight a space secө comma space straight y equals straight b space tanө space

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367.

Find space dy over dx when space straight x equals cosө minus cos 2 straight ө comma space straight y equals sinө minus sin 2 straight ө

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368.

Find space dy over dx when space straight x equals straight a left parenthesis straight ө minus sinө right parenthesis space comma straight y equals straight a left parenthesis 1 plus cosө right parenthesis

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369.

Find space dy over dx when space straight x space equals straight a space left parenthesis cosө plus straight ө space sinө right parenthesis comma space straight y space equals space straight a space left parenthesis sinө space minus space straight ө space cosө right parenthesis

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Long Answer Type

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370. $Find space dy over dx when space straight x equals fraction numerator sin cubed straight t over denominator square root of cos space 2 straight t end root end fraction comma space straight y equals fraction numerator cos cubed straight t over denominator square root of cos space 2 straight t end root end fraction$

space space space space space space straight x equals fraction numerator sin cubed straight t over denominator square root of cos space 2 straight t end root end fraction comma space space straight y equals fraction numerator cos cubed straight t over denominator square root of cos space 2 straight t end root end fraction dx over dt equals fraction numerator square root of cos space 2 straight t end root.3 space sin squared straight t. begin display style fraction numerator negative 2 sin space 2 straight t over denominator 2 square root of cos space 2 straight t end root end fraction end style over denominator cos space 2 straight t end fraction space space space space space space space space equals space fraction numerator begin display style fraction numerator 3 sin squared straight t space cos space straight t space square root of cos space 2 straight t space end root over denominator 1 end fraction end style plus begin display style fraction numerator sin cubed straight t space sin space 2 straight t over denominator square root of cos space 2 straight t end root end fraction end style over denominator cos space 2 straight t end fraction space space space space space space space space equals fraction numerator begin display style 3 sin squared straight t space cos space straight t space. cos space 2 straight t plus sin cubed straight t space sin space 2 straight t end style over denominator open parentheses cos space 2 straight t close parentheses to the power of begin display style 3 over 2 end style end exponent end fraction dy over dt equals fraction numerator square root of cos space 2 straight t end root. left parenthesis negative 3 cos squared straight t space sin space straight t right parenthesis minus cos cubed straight t. begin display style fraction numerator negative 2 sin space 2 straight t over denominator 2 square root of cos space 2 straight t end root end fraction end style over denominator cos space 2 straight t end fraction space space space space space space space space equals fraction numerator negative begin display style fraction numerator 3 cos squared straight t space sin space straight t square root of cos space 2 straight t end root over denominator 1 end fraction end style plus begin display style fraction numerator cos cubed straight t space sin space 2 straight t over denominator square root of cos space 2 straight t end root end fraction end style over denominator cos space 2 straight t end fraction dy over dt equals fraction numerator negative 3 cos squared straight t space sin space straight t space cos space 2 straight t plus cos cubed straight t space sin space 2 straight t over denominator open parentheses cos space 2 straight t close parentheses to the power of 3 over 2 end exponent end fraction 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 cos squared straight t space sin space straight t space cos space 2 straight t plus cos cubed straight t space sin space 2 straight t over denominator open parentheses cos space 2 straight t close parentheses to the power of 3 over 2 end exponent end fraction cross times fraction numerator open parentheses cos space 2 straight t close parentheses to the power of 3 over 2 end exponent over denominator 3 sin squared straight t space cos space straight t space. cos space 2 straight t plus sin cubed straight t space sin space 2 straight t end fraction

equals fraction numerator negative 3 cos squared straight t space sin space straight t space cos space 2 straight t plus cos cubed straight t space sin space 2 straight t over denominator 3 sin squared straight t space cos space straight t space. cos space 2 straight t plus sin cubed straight t space sin space 2 straight t end fraction equals fraction numerator negative 3 cos squared straight t space sin space straight t left parenthesis 2 cos squared straight t minus 1 right parenthesis plus cos cubed straight t space.2 sin space straight t space cos space straight t over denominator 3 sin squared straight t space cos space straight t space left parenthesis 1 minus 2 sin squared straight t right parenthesis. negative plus 2 sin space straight t space cos space 2 straight t plus sin cubed straight t space end fraction equals fraction numerator negative 6 cos to the power of 4 straight t space sin space straight t plus 3 space cos squared straight t space sin space straight t plus 2 sint space cos to the power of 4 straight t over denominator 3 sin squared straight t space cos space straight t minus 6 space sin to the power of 4 straight t space cos space straight t plus space 2 sin to the power of 4 straight t space cos space straight t end fraction equals fraction numerator negative 4 cos to the power of 4 straight t space sin space straight t plus 3 cos squared straight t space sin space straight t over denominator 3 sin squared straight t space cos space straight t minus 4 sin to the power of 4 straight t space cos space straight t end fraction equals fraction numerator sin space straight t space cos space straight t left parenthesis 3 cos space straight t minus 4 cos cubed straight t right parenthesis over denominator negative sin space straight t space cos space straight t left parenthesis 4 sin cubed straight t minus 3 sint right parenthesis end fraction equals negative fraction numerator cos space 3 straight t over denominator sin space 3 straight t end fraction equals negative cot space 3 straight t.

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