SENIOR CLASS XII MATHEMATICS DIFFERENTIATION

INDIAN SCHOOL MUSCAT DEPARTMENT OF MATHEMATICS CLASS 12

CONTINUITY AND DIFFERENTIABILITY 1+𝑥 2 − 1−𝑥 2

1.Find 𝑑𝑦 if y = tan−1 ( 𝑑𝑥 2.Find 3.Find

𝑑𝑦

, if y = tan−1 (

𝑑𝑥 𝑑𝑦

5. Find

𝑑𝑦 𝑑𝑥 𝑑𝑦 𝑑𝑥

𝑐𝑜𝑠𝑥 1+𝑠𝑖𝑛𝑥

2 𝑥 +1

, if y = sin−1 (

𝑑𝑥

4. Find

1+𝑥 2 + 1−𝑥 2

1+4 𝑥

, if y = tan−1 (

8. Find 9. Find

)

)

2𝑥

1+15𝑥 2

)

, y = 𝑠𝑖𝑛𝑥 𝑥 +sin 𝑥 𝑥

6. 𝑥 𝑦 = 𝑒 𝑦−𝑥 .show that dy/dx= 7. Find

)

𝑑𝑦 𝑑𝑥 𝑑𝑦 𝑑𝑥 𝑑𝑦 𝑑𝑥

,𝑦 =

2−𝑙𝑜𝑔𝑥 1−𝑙𝑜𝑔𝑥 2

𝑥−3 𝑥 2 +3 3𝑥 2 +4𝑥+5

, (𝑠𝑖𝑛𝑥)𝑦 = 𝑠𝑖𝑛𝑦 𝑥 . , x=a(cos𝜃 +𝜃𝑠𝑖𝑛𝜃) , y=a(sin𝜃 − 𝜃cos𝜃)

10. x=asin2t (1+cos2t),y=bcos2t(1-cos2t) , Showthat dy/dx at t = 11. x = a(cost+tsint) , y = a(sint – tcost) . Find 1 − 𝑥 2 + 1 − 𝑦 2 = a(x – y) .P.T

12. If

13. Differentiate tan−1

2𝑥 1−𝑥 2

𝑑𝑦 𝑑𝑥

𝑑2𝑦 𝑑𝑥 2

=

𝜋 4

is b/a.

.

1−𝑦 2 1−𝑥 2

with respect to sin−1 (

2𝑥

1+𝑥 2

)

14. If y = [𝑙𝑜𝑔(𝑥 + 𝑥 2 + 1)]2 . Show that (1+x2)d2y/dx2 + x .dy/dx - 2 = 0 15. y =

𝑠𝑖𝑛𝑥 + 𝑠𝑖𝑛𝑥 + 𝑠𝑖𝑛𝑥 + … … … … . . P.T

𝑑𝑦 𝑑𝑥

=

𝐶𝑂𝑆 𝑥 2𝑦−1

16. Verify Rolles Theorem f(x) = x3-6x2+11x-6 on [1 , -3] 17. Verify Rolles Theorem 𝜋 5𝜋

f(x) = ex(sinx-cosx) on [ , 4

4

]

18. Verify MVT f(x) = x (x-1)(x-2) on [0 , 1/2] 19. Discuss the applicability of Rolle's Theorem:

𝜋

f(x) = tanx on [0 , ] 2

W.S (1)

20.If xmyn = (x + y)m+n. Prove that

𝑑𝑦

21. If x16y9 = (x2 + y)17. Prove that

𝑑𝑦

𝑑𝑥

𝑑𝑥

𝑦

= . 𝑥

=

2𝑦 𝑥

.

22.If y = (𝑥 + 𝑥 2 + 1)]𝑚 , prove that (x2+1) y2 + xy1 – m2y = 0. 1

23.If x = sin ( 𝑙𝑜𝑔𝑦) , show that (1 – x2) y2 – xy1 – a2y = 0 . 𝑎

24. f(x) = 𝑥 3 . Show that f is differentiable and find f '(x). 25. y=(𝑙𝑜𝑔𝑥)𝑥 + 𝑥 𝑙𝑜𝑔𝑥 , find

𝑑𝑦 𝑑𝑥

.

26. Find the values of k ,if the following function 2 𝑥 +2 −16

f(x) =

4 𝑥 −16

,𝑥 ≠ 2

𝑘 , 𝑖𝑓 𝑥 = 2

, 𝑖𝑠 𝑐𝑜𝑛𝑡𝑖𝑛𝑢𝑜𝑢𝑠 𝑎𝑡 𝑥 = 2

𝑎𝑥 2 + 𝑏, 𝑥 > 2 27. Find the value of a and b if ,f(x) = is continuous at x = 2. 2,𝑥 = 2 2𝑎𝑥 − 𝑏 , 𝑥 < 2 1−𝑠𝑖𝑛 3 𝑥

,𝑥 <

3𝑐𝑜𝑠 2 𝑥

𝜋

𝑎, 𝑥=

28. Find the value of a and b if , f(x) =

2 𝑏(1−𝑠𝑖𝑛𝑥 ) (𝜋−2𝑥)2

𝜋 2

at x = ,𝑥 >

2

𝑥

𝑏 𝑥3

𝑥 −4 𝑥 −4

30. Find the value of a and b if , f(x) =

,𝑥 < 0

𝑐 ,𝑥 = 0 𝑥 +𝑏𝑥 2 – 𝑥

2

𝜋

𝑠𝑖𝑛 𝑎+1 𝑥+𝑠𝑖𝑛𝑥

29. Find the value of a , b and c if , f(x) =

𝜋

is continuous

+ 𝑎 ,𝑥 < 4

𝑎 + 𝑏 , 𝑥 = 4 is continuous + 𝑏 ,𝑥 > 4

at x = 4

𝑥−4 𝑥−4

31. Find all the points of discontinuity. 2𝑥 , 𝑥 < 0 f(x) = 0 ,0 ≤ 𝑥 ≤ 1 4𝑥 , 𝑥 > 1 32. Show that f(x) = 𝑥 + 3 is continuous but not differentiable at x= -3 33. Find the points where the function f(x) = 𝑥 , -2≤ 𝑥 < 3 is not differentiable. 1+𝑝𝑥 − 1−𝑝𝑥

34. Find the value of p if , f(x) =

𝑥 2𝑥+1 𝑥−2

at x = 0

, 𝑥 >0

, −1 ≤ 𝑥 < 0

,0 ≤ 𝑥 ≤ 1

is continuous.