Assignment Sheet on Continuity and Differentiability
1𝑀 × 1 + 2𝑀 × 3 + 3𝑀 × 2 + 5𝑀 × 1 + 4𝑀 × 1 = 22 𝑀𝑎𝑟𝑘𝑠
1 Mark Questions [Question 5]
𝑑𝑦
1. Find 𝑑𝑥 , if 𝑦 = sin(𝑎𝑥 + 𝑏).
𝑑𝑦
3
2. If 𝑦 = 𝑒 𝑥 , find
.
𝑑𝑥
2 ).
𝑑𝑦
3. Find 𝑑𝑥 , if 𝑦 = sin(𝑥
𝑑𝑦
4. If 𝑦 = cos √𝑥, find 𝑑𝑥 .
𝑑𝑦
5. Find 𝑑𝑥 , if 𝑦 = tan(2𝑥 + 3).
𝑑𝑦
1
6. Find 𝑑𝑥 , if 𝑦 = 𝑎2 log𝑎 cos 𝑥 .
𝑑𝑦
7. Find 𝑑𝑥 , if 𝑦 = sin(𝑥 2 + 5).
𝑑𝑦
8. Find 𝑑𝑥 , if 𝑦 = cos(1 − 𝑥).
𝑑𝑦
9. Find 𝑑𝑥 , if 𝑦 = log(sin 𝑥).
10. Write the points of discontinuity for the function 𝑓(𝑥) = [𝑥], −3 < 𝑥 < 3.
𝑑𝑦
11. If 𝑦 = 𝑒 3 log 𝑥 , then show that 𝑑𝑥 = 3𝑥 2
12. Differentiate log(cos 𝑒 𝑥 ) w.r.t. 𝑥.
13. The greatest integer function is not differentiable at integral points give reason.
14. Differentiate sin √𝑥 w.r.t. 𝑥.
15. Find the derivative of cos(𝑥 2 ) w.r.t. 𝑥.
1
16. The function 𝑓(𝑥) = 𝑥−5 is not continuous at 𝑥 = 5. Justify the statement.
17. Differentiate (3𝑥 2 − 9𝑥 + 5)9 w.r.t. 𝑥.
𝑑𝑦
18. If 𝑦 = sin3 𝑥 + cos6 𝑥 then find 𝑑𝑥 .
19. Give an example of a function which is continuous everywhere but not differentiable at a
point.
𝑑𝑦
20. If 𝑥 − 𝑦 = 𝜋 find 𝑑𝑥 .
𝑑𝑦
21. If 𝑦 = 𝑒 cos 𝑥 find 𝑑𝑥 .
22. Check the continuity of the function 𝑓(𝑥) = 2𝑥 + 3 at 𝑥 = 1.
−1
23. Differentiate 𝑒 sin 𝑥 w.r.t. 𝑥.
24. Differentiate sec(tan √𝑥) w.r.t. 𝑥.
25. Differentiate sin(tan−1 𝑒 −𝑥 ) w.r.t. 𝑥.
26. Differentiate cos(sin 𝑥) w.r.t. 𝑥.
27. Differentiate log(log 𝑥) w.r.t. 𝑥.
28. Differentiate √𝑒 √𝑥 w.r.t. 𝑥.
29. Differentiate sin(log 𝑥) w.r.t. 𝑥.
30. Differentiate cos−1 (𝑒 𝑥 ) w.r.t. 𝑥.
31. Differentiate 2√cot(𝑥 2 ) w.r.t. 𝑥.
32. Find the second order derivative of 𝑥 2 + 3𝑥 + 2 with respect to 𝑥.
33. Differentiate √3𝑥 + 2 w.r.t. 𝑥.
2 Mark Questions [Question 15 & Question 16]
dy
1. If 𝑦 + sin 𝑦 = cos 𝑥, Find dx.
𝑑𝑦
𝑦
2. If √𝑥 + √𝑦 = √10, show that 𝑑𝑥 + √𝑥 = 0
dy
3. Find dx, if 2𝑥 + 3𝑦 = sin 𝑥.
𝑑𝑦
4. Find 𝑑𝑥 , if 2𝑥 + 3𝑦 = sin 𝑦
𝑑𝑦
5. If 𝑎𝑥 + 𝑏𝑦 2 = cos 𝑦, find 𝑑𝑥 .
𝑑𝑦
6. If 𝑥𝑦 + 𝑦 2 = tan 𝑥 + 𝑦, find 𝑑𝑥 .
dy
7. Find dx, if 𝑥 2 + 𝑥𝑦 + 𝑦 2 = 100
𝑑𝑦
8. Find 𝑑𝑥 , if sin2 𝑥 + cos2 𝑦 = 𝑘, where 𝑘 is a constant.
𝑑𝑦
2
2
2
9. Find 𝑑𝑥 , if 𝑥 3 + 𝑦 3 = 𝑎3 .
𝑑𝑦
10. If 𝑦 = 𝑥 𝑥 , find 𝑑𝑥 .
𝑑𝑦
11. Find 𝑑𝑥 , if 𝑦 = (log 𝑥)cos 𝑥
1 𝑥
12. Differentiate (𝑥 + 𝑥) w.r.t. 𝑥.
𝑑𝑦
13. If 𝑥 𝑦 = 𝑎 𝑥 , prove that 𝑑𝑥 =
𝑥 log𝑒 𝑎−𝑦
𝑥 log𝑒 𝑥
.
14. Differentiate 𝑥 sin 𝑥 , 𝑥 > 0 w.r.t. 𝑥.
15. Differentiate (sin 𝑥)𝑥 w.r.t. 𝑥.
16. Differentiate (sin 𝑥)cos 𝑥 w.r.t. 𝑥.
𝑑𝑦
17. If 𝑦 = (sin−1 𝑥)𝑥 , then find 𝑑𝑥 .
18. Find the derivative of 𝑥 𝑥 − 2sin 𝑥 w.r.t 𝑥.
dy
1
19. Find dx, if 𝑦 = sec −1 (2𝑥 2 −1), if 0 < 𝑥 <
dy
1−𝑥 2
dy
2𝑥
1
.
√2
20. Find dx, if 𝑦 = cos −1 (1+𝑥 2 ), if 0 < 𝑥 < 1.
21. Find dx, if 𝑦 = sin−1 (1+𝑥 2 )
𝑑𝑦
22. If 𝑦 = cos −1(sin 𝑥) then show that 𝑑𝑥 = −1.
2𝑥
𝑑𝑦
23. If 𝑦 = tan−1 (1−𝑥 2 ), then find 𝑑𝑥 .
3𝑥−𝑥 3
24. If 𝑦 = tan−1 (1−3𝑥 2 ) , |𝑥| <
1
𝑑𝑦
, then find 𝑑𝑥 .
√3
𝑑𝑦
25. If 𝑦 = log 7 (log 𝑥), find 𝑑𝑥
26. Prove that greatest integer function defined by 𝑓(𝑥) = [𝑥], 0 < 𝑥 < 3 is not differentiable at 𝑥 =
1.
27. Check the continuity of the function 𝑓 given by 𝑓(𝑥) = 2𝑥 + 3 at 𝑥 = 1.
𝑑𝑦
28. If 𝑥 = 𝑎𝑡 2 , 𝑦 = 2𝑎𝑡 show that 𝑑𝑥 =
1
𝑡
29. Find the derivative of √𝑥 + √𝑦 = 9 at (4, 9).
𝑑𝑦
30. Find 𝑑𝑥 , if 𝑥 = 4𝑡 𝑎𝑛𝑑 𝑦 =
31. Find the derivative of
4
𝑡
(3𝑥 2
5
− 7𝑥 + 3)2 w.r.t. 𝑥.
𝑑𝑦
32. If 𝑦 = sin(log 𝑒 𝑥), then prove that 𝑑𝑥 =
√1−𝑦 2
𝑥
𝑥)
.
33. Find the derivative of cos(log 𝑥 + 𝑒 , 𝑥 > 0
34. Find the derivative of sin(tan−1 𝑒 −𝑥 )
35. Find the derivative of log(cos 𝑒 𝑥 )
3 Mark Questions [Question 28 and Question 29]
1. Differentiate (𝑥 + 3)2 . (𝑥 + 4)3 . (𝑥 + 5)4 with respect to 𝑥.
(𝑥−1)(𝑥−2)
2. Differentiate √(𝑥−3)(𝑥−4) w.r.t. 𝑥.
𝑑𝑦
3. Find 𝑑𝑥 , if 𝑦 𝑥 = 𝑥 𝑦 .
𝑑𝑦
4. Find 𝑑𝑥 , if 𝑥𝑦 = 𝑒 (𝑥−𝑦) .
𝑑𝑦
5. If 𝑦 𝑥 + 𝑥 𝑦 = 𝑎𝑏 , find 𝑑𝑥 .
6. Differentiate (log 𝑥)cos 𝑥 with respect to 𝑥.
7. Differentiate 𝑥 sin 𝑥 + (sin 𝑥)cos 𝑥 w.r.t 𝑥.
𝑑𝑦
8. If 𝑥 = 𝑎𝑡 2 , 𝑦 = 2𝑎𝑡 show that 𝑑𝑥 .
𝑑𝑦
𝜃
9. If 𝑥 = 𝑎(𝜃 + sin 𝜃)& 𝑦 = 𝑎(1 − cos 𝜃) then prove that 𝑑𝑥 = tan 2
𝑑𝑦
10. Find 𝑑𝑥 , if 𝑥 = 2𝑎𝑡 2 𝑎𝑛𝑑 𝑦 = 𝑎𝑡 4
𝑑𝑦
11. Find 𝑑𝑥 , if 𝑥 = 𝑎 cos 𝜃 𝑎𝑛𝑑 𝑦 = 𝑏 cos 𝜃
𝑑𝑦
12. If 𝑥 = sin 𝑡 & 𝑦 = cos 2𝑡 then prove that 𝑑𝑥 = −4 sin 𝑡.
𝑑𝑦
13. Find 𝑑𝑥 , if 𝑥 = 4𝑡 𝑎𝑛𝑑 𝑦 =
4
𝑡
𝑑𝑦
cos 𝜃−2cos 2𝜃
14. If 𝑥 = cos 𝜃 − cos 2𝜃 , 𝑦 = sin 𝜃 − sin 2𝜃 prove that 𝑑𝑥 = − 2 sin 2𝜃−sin 𝜃 .
𝑑𝑦
𝜃
15. If 𝑥 = 𝑎(𝜃 − sin 𝜃)& 𝑦 = 𝑎(1 + cos 𝜃) then prove that 𝑑𝑥 = − cot 2.
𝑑𝑦
𝑡
16. Find 𝑑𝑥 , if 𝑥 = 𝑎 (cos 𝑡 + log tan 2) & 𝑦 = 𝑎 sin 𝑡..
𝑑𝑦
17. Find 𝑑𝑥 , If 𝑥 = 𝑎(cos 𝜃 + 𝜃 sin 𝜃)& 𝑦 = 𝑎(sin 𝜃 − 𝜃 cos 𝜃).
18. If 𝑥 = √𝑎sin
−1 𝑡
−1 𝑡
& 𝑦 = √𝑎cos
𝑑𝑦
𝑦
then prove that 𝑑𝑥 = − 𝑥 .
𝑑𝑦
3
𝑦
19. If 𝑥 = 𝑎 cos3 𝜃 & 𝑦 = 𝑎 sin3 𝜃 then prove that 𝑑𝑥 = − √𝑥
3𝑥−𝑥 3
20. If 𝑦 = tan−1 (1−3𝑥 2 ), −
1
√3
<𝑥<
1
𝑑𝑦
, find of 𝑑𝑥 .
√3
√1+𝑥 2 −1
21. If 𝑦 = tan−1 (
𝑥
2𝑥+1
𝑑𝑦
1
), prove that 𝑑𝑥 = 2(1+𝑥 2 )
𝑑𝑦
22. If 𝑦 = sin−1 (1+4𝑥 ) find 𝑑𝑥 .
sin 𝑥
𝑑𝑦
1
23. If 𝑦 = tan−1 (1+cos 𝑥), then prove that 𝑑𝑥 = 2
24. Differentiate sin2 𝑥 with respect to 𝑒 cos 𝑥 .
𝑑𝑦
1
25. If 𝑥√1 + 𝑦 + 𝑦√1 + 𝑥 = 0, 0 < 𝑥 < 1& 𝑥 ≠ 𝑦, prove that 𝑑𝑥 = − 1+𝑥 2.
𝑑𝑦
26. If cos 𝑦 = 𝑥 cos(𝑎 + 𝑦) , cos 𝑎 ≠ ±1, prove that 𝑑𝑥 =
cos2 (𝑎+𝑦)
sin 𝑎
.
27. Prove that if the function is differentiable at a point 𝑐, then it is also continuous at that point.
28. Prove that the function 𝑓 given by 𝑓(𝑥) = |𝑥 − 1|, 𝑥 ∈ 𝑅 is not differentiable at 𝑥 = 1.
29. Prove that the greatest integer function defined by 𝑓(𝑥) = [𝑥], 0 < 𝑥 < 3 is not differentiable at 𝑥 =
1.
4 Mark Questions [Question 49 (b) or 50(b)]
1. Find the relationship between 𝑎 & 𝑏 so that the function 𝑓 defined by
𝑎𝑥 + 1, 𝑖𝑓 𝑥 ≤ 3
𝑓(𝑥) = {
is continuous at 𝑥 = 3.
𝑏𝑥 + 3, 𝑖𝑓 𝑥 > 3
𝜆(𝑥 2 − 2𝑥), 𝑖𝑓 𝑥 ≤ 0
2. For what value of 𝜆, is the function defined by 𝑓(𝑥) = {
is continuous at 𝑥 =
4𝑥 + 1, 𝑖𝑓 𝑥 > 0
0.
𝑘 cos 𝑥
3. Find the value of 𝑘, if 𝑓(𝑥) = {
𝜋−2𝑥
, 𝑥≠
3, 𝑥 =
𝜋
2
𝜋
2
𝜋
is continuous at 𝑥 = 2 .
June – 2019
𝑘𝑥 2 , 𝑥 ≤ 2
4. Find the value of 𝑘, if 𝑓(𝑥) = {
is continuous at 𝑥 = 2.
3, 𝑥 > 2
1−cos 2𝑥
, 𝑥≠0
5. Find the value of 𝑘, if 𝑓(𝑥) = { 1−cos 𝑥
is continuous at 𝑥 = 0.
𝑘, 𝑥 = 0
𝑘𝑥 + 1, 𝑖𝑓 𝑥 ≤ 5
6. Find the value of 𝑘, if 𝑓(𝑥) = {
is continuous at 𝑥 = 5.
Mar – 2019
3𝑥 − 5, 𝑖𝑓 𝑥 > 5
𝑘𝑥 + 1, 𝑖𝑓 𝑥 ≤ 𝜋
7. Find the value of 𝑘, if 𝑓(𝑥) = {
is continuous at 𝑥 = 𝜋.
cos 𝑥 , 𝑖𝑓 𝑥 > 𝜋
8. Find the points of discontinuity of the function 𝑓(𝑥) = 𝑥 − [𝑥], where [𝑥] indicates the
greatest integer not greater than 𝑥. Also write the set of values of 𝑥, where the function is
continuous.
𝑥 3 − 3, 𝑖𝑓 𝑥 ≥ 2
9. Find all points of discontinuity of 𝑓 defined by 𝑓(𝑥) = { 2
.
𝑥 + 1, 𝑖𝑓 𝑥 < 2
10. Define a continuity of a function at a point. Find all points of discontinuity of 𝑓 defined by
𝑓(𝑥) = |𝑥| − |𝑥 + 1|.
11. Find the values of 𝑎 𝑎𝑛𝑑 𝑏 such that the function defined by
5, 𝑖𝑓 𝑥 ≤ 2
𝑓(𝑥) = {𝑎𝑥 + 𝑏, 𝑖𝑓 2 < 𝑥 < 10 is continuous function.
21, 𝑖𝑓 𝑥 ≥ 10
|𝑥| + 3 𝑖𝑓 𝑥 ≤ −3
𝑖𝑓 − 3 < 𝑥 < 3
12. Discuss the continuity of the function 𝑓(𝑥) = { −2𝑥,
6𝑥 + 2, 𝑖𝑓 𝑥 ≥ 3
𝑥, 𝑥 ≥ 0
13. Verify whether the function 𝑓(𝑥) = { 2
is continuous function or not.
𝑥 , 𝑥<0
𝑥
, 𝑖𝑓 𝑥 < 0
14. Find all points of discontinuity of 𝑓 where 𝑓 is defined by 𝑓(𝑥) = { |𝑥|
.
−1, 𝑖𝑓 𝑥 ≤ 0
5 Mark Questions
𝑑2 𝑦
1. If 𝑦 = 5 cos 𝑥 − 3 sin 𝑥 then show that 𝑑𝑥 2 + 𝑦 = 0
𝑑2 𝑦
2. If 𝑦 = 𝐴 sin 𝑥 + 𝐵 cos 𝑥 then show that 𝑑𝑥 2 + 𝑦 = 0
𝑑2 𝑦
𝑑𝑦
3. If 𝑦 = 𝐴𝑒 𝑚𝑥 + 𝐵𝑒 𝑛𝑥 then show that 𝑑𝑥 2 − (𝑚 + 𝑛) 𝑑𝑥 + 𝑚𝑛𝑦 = 0
𝑑2 𝑦
𝑑𝑦
4. If 𝑦 = 3𝑒 2𝑥 + 2𝑒 3𝑥 then show that 𝑑𝑥 2 − 5 𝑑𝑥 + 6𝑦 = 0
𝑑2 𝑦
5. If 𝑦 = 500𝑒 7𝑥 + 600𝑒 −7𝑥 then show that 𝑑𝑥 2 = 49𝑦
𝑑2 𝑦
𝑑𝑦
6. If 𝑦 = sin−1 𝑥, then prove that (1 − 𝑥 2 ) 𝑑𝑥 2 − 𝑥 𝑑𝑥 = 0
𝑑2 𝑦
𝑑𝑦
7. If 𝑦 = tan−1 𝑥, then prove that (1 + 𝑥 2 ) 𝑑𝑥 2 + 2𝑥 𝑑𝑥 = 0
8. If 𝑦 = 5 cos(log 𝑥) + 7 sin(log 𝑥) then show that 𝑥 2 𝑦2 + 𝑥𝑦1 + 𝑦 = 0
9. If 𝑦 = 3 cos(log 𝑥) + 4 sin(log 𝑥) then show that 𝑥 2 𝑦2 + 𝑥𝑦1 + 𝑦 = 0
𝑑2 𝑦
𝑑𝑦
10. If 𝑦 = (sin−1 𝑥)2 , then prove that (1 − 𝑥 2 ) 𝑑𝑥 2 − 𝑥 𝑑𝑥 = 2
11. If 𝑦 = (tan−1 𝑥)2 , then prove that (𝑥 2 + 1)2 𝑦2 + 2𝑥(𝑥 2 + 1)𝑦1 = 2
12. If 𝑦 = cos−1 𝑥, find
𝑑2 𝑦
𝑑𝑥 2
interms of 𝑦 alone.
13. If 𝑒 𝑦 (𝑥 + 1) = 1 then show that
14. If 𝑦 = 𝑒
(1 −
𝑎 cos−1 𝑥
𝑑2 𝑦
𝑥 2 ) 𝑑𝑥 2
𝑑2 𝑦
𝑑𝑥 2
𝑑𝑦 2
= (𝑑𝑥 )
−1 𝑥
, − 1 ≤ 𝑥 ≤ 1, show that𝑦 = 𝑒 𝑎 cos
, − 1 ≤ 𝑥 ≤ 1,
show that
𝑑𝑦
− 𝑥 𝑑𝑥 − 𝑎2 𝑦 = 0
𝑑2 𝑦
15. If 𝑥 = 𝑎(cos 𝑡 + 𝑡 sin 𝑡) 𝑎𝑛𝑑 𝑦 = 𝑎(sin 𝑡 − 𝑡 cos 𝑡), find 𝑑𝑥 2 .
16. If (𝑥 − 𝑎)2 + (𝑦 − 𝑏)2 = 𝑐 2 , 𝑐 > 0, prove that
of 𝑎 & 𝑏.
3
𝑑𝑦 2 2
[1+( ) ]
𝑑𝑥
𝑑2 𝑦
𝑑𝑥2
is a constant
independent