Continuity and differentiability
- 1. CONTINUITY AND DIFFERENTIABILITY 2 MARKS QUESTIONS. 1. Write the value of k for which 𝑓 𝑥 = 3𝑠𝑖𝑛𝑥 2𝑥 + 𝑐𝑜𝑠𝑥, 𝑥 ≠ 0 𝑘, 𝑥 = 0 is continuous at x=0 2. Write two points at which 𝑓 𝑥 = 1 𝑥− 𝑥 is not continuous. 3. Write one point where f(x) = 𝑥 − 𝑥 + 1 is not differentiable 4. If y = 𝑒2𝑥3 , write 𝑑𝑦 𝑑𝑥 . 5. If x= cos𝜃 − 𝑐𝑜𝑠2𝜃, y = sin 𝜃- sin2 𝜃 , find 𝑑𝑦 𝑑𝑥 . 6. If 𝑠𝑖𝑛2 𝑦 + cos 𝑥𝑦 = 𝜋,find 𝑑𝑦 𝑑𝑥 . 7. If y = tan−1 5𝑥 1−6𝑥2 ,- 1 6 < x < 1 6 ,then show that 𝑑𝑦 𝑑𝑥 = 2 1+4𝑥2 + 3 1+9𝑥2 . 8. If x = 𝑎sin −1 𝑡 , 𝑦 = 𝑎cos −1 𝑡 , 𝑠ℎ𝑜𝑤 𝑡ℎ𝑎𝑡 𝑑𝑦 𝑑𝑥 = − 𝑦 𝑥 . 4 MARKS QUESTIONS 9. If f(x) = 1+𝑘𝑥 − 1−𝑘 𝑥 𝑥 − 1 ≤ 𝑥 < 0 2𝑥+1 𝑥−2 0 ≤ 𝑥 ≤ 1 is continuous at x = 0. Find the value of k 10. Find the value of ‘ a’ for which the function f(x) = 𝑎 𝑠𝑖𝑛 𝜋 2 𝑥 + 1 , 𝑥 ≤ 0 𝑡𝑎𝑛 𝑥−𝑠𝑖𝑛 𝑥 𝑥3 , 𝑥 > 0 , is continuous at x = 0. 11. Differentiate log ( xsin x + cot2 x) with respect to x. 12. . If 1 − 𝑥2 + 1 − 𝑦2 = a (x – y), Prove 𝑑𝑦 𝑑𝑥 = 1−𝑦2 1−𝑥2 13. If x = a sin t and y = a ( cos t + log tan 𝑡 2 ), find 𝑑2 𝑦 𝑑𝑥2 . 14. If x = a(cos t + t sin t), y = b(sin t – t cos t), Prove that 𝑑2𝑦 𝑑𝑥 2 = 𝑏 𝑠𝑒𝑐 3 𝑡 𝑎2 𝑡 . 15. Differentiate tan-1 1+ 𝑥2− 1− 𝑥2 1+ 𝑥2+ 1− 𝑥2 with respect to cos-1 x2 . 16. If x 1 + 𝑦 + y 1 + 𝑥 = 0, find 𝑑𝑦 𝑑𝑥 17. If y = sin −1 𝑥 1−𝑥2 , show that (1-𝑥2 ) 𝑑2 𝑦 𝑑𝑥2 − 3𝑥 𝑑𝑦 𝑑𝑥 − 𝑦 = 0 . 18. Differentiate cos-1 1− 𝑥2 1+ 𝑥2 with respect of tan-1 3𝑥 − 𝑥3 1−3 𝑥2 . 19. If x = a sin 2t(1 + cos 2t), y = b cos 2t( 1 – cos 2t) Show that 𝑑𝑦 𝑑𝑥 𝑡= 𝜋 4 = 𝑏 𝑎 20. If y = sin −1 𝑥 1−𝑥2 , show that (1-𝑥2 ) 𝑑2 𝑦 𝑑𝑥2 − 3𝑥 𝑑𝑦 𝑑𝑥 − 𝑦=0 21. IF x = a sec 3 , y = a tan3 𝜃,Prove that 𝑑2 𝑦 𝑑𝑥2 at 𝜃 = 𝜋 4 is 1/12a 22. If xm . yn = (x + y)m+n , Prove that 𝑑𝑦 𝑑𝑥 = 𝑦 𝑥 23. If y = 3 cos(logx) + 4 sin (logx), then show that x2 𝑑2 𝑦 𝑑𝑥2 + 𝑥 𝑑𝑦 𝑑𝑥 + 𝑦 = 0. 24. Find 𝑑𝑦 𝑑𝑥 , if yx + xy + xx = ab 25. if x = a cos t + b sin t , y = a sin t – b cos t , show that y2 𝑑2 𝑦 𝑑𝑥2 – x 𝑑𝑦 𝑑𝑥 + y = 0