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

space space space space space space space space space space space space space space straight x equals straight a space secө comma space space straight y equals straight b space tanө space therefore space space straight b dx over dθ equals straight a space secθ space tanθ comma space dy over dθ equals straight b space sec squared 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 b space sec squared over denominator straight a space secθ space tanθ end fraction equals fraction numerator straight b space secθ over denominator straight a space tan space straight theta end fraction equals straight b over straight a fraction numerator begin display style 1 over cosθ end style over denominator begin display style fraction numerator sin space straight theta over denominator cos space straight theta end fraction end style end fraction equals straight b over straight a fraction numerator 1 over denominator cos space straight theta end fraction cross times fraction numerator cos space straight theta over denominator sin space straight theta end fraction equals straight b over straight a fraction numerator 1 over denominator sin space straight theta end fraction equals straight b over straight a cosec space straight theta

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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

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

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