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

space space space space space space straight x equals straight a left parenthesis cosө plus straight ө space sinө right parenthesis comma space straight y equals straight a left parenthesis sinө minus straight ө space cosө right parenthesis dx over dθ equals straight a left parenthesis negative sinθ plus straight theta space cosθ plus sinθ right parenthesis equals straight a space straight theta space cosθ dy over dθ equals straight a left parenthesis cosθ plus straight theta space sinθ minus cosθ right parenthesis equals straight a space straight theta space sinθ 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 straight theta space sinθ over denominator straight a space straight theta space cosθ end fraction equals tanθ

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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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