# differentiation and Integration

hello,,, please I need the right answer for these questions with all steps….I hope that you make each step clear because I want to study by these answers.
practice_questions_in_unit_1_and_unit_3.docx

practice_questions_in_limits_and_derivatives.docx

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Practice questions in differentiation, Integration and Variable separable method:
ð??
1. Show that: ð??ð?¥(8ð?ð??ð? ð??ð?(ð?¥))=â??8ð?ð??ð? ð??ð?(ð?¥)cot(x)
2.
Find the second derivative of the following:
2
ð?¦=ln (ð?¥ 3 ð?? 5ð?¥ )
ð??ð?¦
ð?¥
3. Find ð??ð?¥ implicitly, if ð?? ð?¥ð?¦ â?? ð?¦ = 5ð?¥ 4 â?? 2â??ð?¥ ð?¦
2ð??
ð??
4. Evaluate:â?«0 (sec(5ð?¥) tan(5ð?¥) cosâ¡(3ð?¥) â?? )ð??ð?¥
4
5. Evaluate the following integral:
â¡â¡â¡â?«(
7â¡lnâ¡(3ð?¥)
ð?¥5
)ð??ð?¥
â¡
3â¡â¡â¡
ð?¥
3
â¡â¡â¡â¡â¡â¡6.â¡â¡â¡ð¸ð?£ð??ð??ð?¢ð??ð?¡ð??: â?«2 (ð?¥ 2 â?? 3 â¡)ð??ð?¥
â??ð?¥
7.
Find the following integrals:
a.â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â?« sinâ¡(1 â?? ð?¥)(2 â?? cosâ¡(1 â?? ð?¥))4 ð??ð?¥
ð??â¡â¡â¡
â?«0 ð?¥â¡ð? ð??ð? 2 (3ð?¥)ð??ð?¥
b.
ð??
8. Show thatð??ð?¥ (cot(ð?¥ 3 )) = â??3ð?¥ 2 ð?ð??ð?¡ð?¥â¡ð?ð??ð? ð??ð?ð?¥ (Hint:â¡ð?ð??ð?¡ð?¥ =
ð??ð?¦
9. Find ð??ð?¥ implicitly if â¡10ð?¥ð?¦ 2 â?? 4ð? ð??ð??ð?¥ =
10ð?¦
ð?¥2
ð?ð??ð? ð?¥
ð? ð??ð??ð?¥
1
â¡â¡ð??ð??â¡ ð? ð??ð??ð?¥)
â??6
10. Evaluate the following integrals:
3
3
5
â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡ð??. â?«(5ð?¥ â?? ð?¥ 2 â¡ â??ð?¥ 2ð?¥ â?? 6)ð??ð?¥
ð??
b.â?«0 (sin(2ð?¡) â?? 7ð?? â??3ð?¡ ð?¡ 5 )ð??ð?¡
8. Find the general solution of each differential equation.
ð??ð?¦
a.â¡â¡â¡â¡ð??ð?¥ = 2ð?¦ð?¥ ð?¦ð?¥ 2
ð??ð?¦
b.â¡â¡ð??ð?¥ = 12ð?¥ 3 ð?¦, ð?¦(0) = 2
ð??ð?¦
1
c. .â¡â¡ð??ð?¥ = ð? ð??ð? 2 (ð?¦) , ð?¦(0) = 1
ð??
9. Show thatð??ð?¥ (cot(ð?¥ 3 )) = â??3ð?¥ 2 ð?ð??ð?¡ð?¥â¡ð?ð??ð? ð??ð?ð?¥ (Hint:â¡ð?ð??ð?¡ð?¥ =
ð??ð?¦
10. Find ð??ð?¥ implicitly if â¡10ð?¥ð?¦ 2 â?? 4ð? ð??ð??ð?¥ =
10ð?¦
ð?¥2
â??6
ð?ð??ð? ð?¥
ð? ð??ð??ð?¥
1
â¡â¡ð??ð??â¡ ð? ð??ð??ð?¥)
11. Evaluate the following integrals:
3
5
3
â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡ð??. â?«(5ð?¥ â?? ð?¥ 2 â¡ â??ð?¥ 2ð?¥ â?? 6)ð??ð?¥
ð??
b.â?«0 (sin(2ð?¡) â?? 7ð?? â??3ð?¡ ð?¡ 5 )ð??ð?¡
12. Solve the ordinary differential equation:
ð?¥ð?¦â¡ð??ð?¥â?? (ð?¥ 2 1)ð??ð?¦â¡â¡= 0.
13. Find the order and degree of the differential equation:
ð??2 ð?¦
ð??ð?¦
ð??ð?¦
ð?¥ð?¦ ð??ð?¥ 2 â?? 3(ð??ð?¥ )2 ð?¦ ð??ð?¥ = 0
14. Find ð?¦ â?²â?² , if ð?¦ = lnâ¡(2ð?¥ 3 â?? cosâ¡(ð?¥))
15. Find ð?¦ â?² , implicitly if
3
ð?¥3
â?? 2ð?¥â??ð?¥ 5 â?? ln(ð?¥ ð?¦) = ð?? â??6 cosâ¡(ð?¦ 2 )
16. Evaluate: â?« 4ð?¥ 2 cos(ð?¥ 3 2) ð??ð?¥
17. Solve the initial value problem: ð?¦ â?² =
2 6â¡lnâ¡(ð?¥)
18. Evaluate:â?«1
ð?¥3
â¡ð??ð?¥
ð?¥(ð?? 3ð?¥ â??5)â¡
5ð?¦ 2
â¡, ð?¦(0) = 1
Practice questions in Limits and derivatives
1. Evaluate the following limit:
ð?¥ 4 â??625
lim( ð?¥ 5 â??3125) lim(
ð?¥â??5
ð?¥ 2 ð?¥â??2
ð?¥â??1
2. Find the value of a, if limâ¡(
ð?¥ 6 â??ð??6
ð?¥â??ð??
ð?¥â??ð??
ð?¥ 2 â??1
)
) =192
3. Evaluate the following limit:
.limâ¡(
5ð?¥ sin(â??3ð?¥)
)
5ð?¥ 2 ð?¥
ð?¥â??0
â??ð?¥?â??â??2ð?¥
]
ð?¥ 2 â??2ð?¥
4. Evaluate:â¡lim [
ð?¥â??2
3 sin(7ð?¥)
3
5. Given that lim (â¡ð??â¡ sin(4ð?¥)) = 4, find the value of n.
ð?¥â??0
â??2ð?¥â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡ð??ð??â¡ð?¥ < 0 6. For â¡ð??(ð?¥) = {5 â?? ð?¥â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡ð??ð??â¡0 â?¤ ð?¥ â?¤ 3 , evaluate the following: (ð?¥ â?? 3)2 â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡ð??ð??â¡ð?¥ > 3
(i) lim ð??(ð?¥)
ð?¥â??3
(ii) ð??(3)
5ð?¥ â?? 1,â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡ð??ð??â¡ð?¥ < â??2 7. Evaluate lim ð??(ð?¥), where f is defined by f(x) = {â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡5â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡ð??ð?? â?? 2 < ð?¥ < 2 ð?¥â??â??2 3ð?¥ 1,â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡ð??ð??â¡ð?¥ > 2
8. Evaluate:
limâ¡(
3sinâ¡(2ð?¥)
ð?¥
3
ð?¥â??0 7ð?¥â¡cosâ¡( )
â??
â??2 cos(2ð?¥)
3ð?¥
3
â?? 2ð?¥)
9. Evaluate the following limits:
ð?¥ 2 â?? ð?¥ â?? 2 ð?¥ 2 â?? 2ð?¥
limâ¡( 2
â??
)
ð?¥ â??4
2ð?¥ â?? 4
ð?¥â??2
10. Evaluate the following limits
(ð?¥ 4 â??1296)
a. lim [ (ð?¥ 3
ð?¥â??2
â??216)
]
ð?¥ 2 ð?¥â??2
b. lim( ð?¥ 2 â??3ð?¥)â¡
ð?¥â??1
11. By using the definition of derivative, find ð?? â?² (ð?¥), if
a.â¡ð??(ð?¥) = 2ð?¥ 2 â?? 7ð?¥
b. ð??(ð?¥) = â??4ð?¥ 3
c.â¡ð??(ð?¥) = â??â??2ð?¥ 7
12. Find ð?¦ â?² , if ð?¦ = (2ð?¥ â?? 3)(ð?¥ 4 2ð?¥) 4ð?? 2ð?¥
13. Find ð?¦ â?² , if ð?¦ =
14. Find
15. Find
ð??ð?¦
(5ð?¥â??2)(ð?? ð?¥ )â??(2sinâ¡(ð?¥)
, if ð?¦ =
ð??ð?¥
ð??ð?¦
ð??ð?¥
(ð?¥)2
ð?¥ 5 cos(2ð?¥â??3)â??5ð?? 8â??7ð?¥
ð?¥ 3 â??3ð?¥
, if ð?¦ = (4ð?¥ â?? 3) ln(6ð?¥) â?? ð?? 2ð?¥ (4ð?¥ 2 â?? 9)
.
â??ð?¥â??â??2ð?¥
]
ð?¥ 2 â??2ð?¥
ð?¥â??2
4. Evaluate:â¡lim [
â??ð??ð?? ð?ð??
5. Find: ð¥ð¢ð¦ [ ð?ð??ð? â??ð?ð?? ]
ð??â??ð??
6. Evaluate: lim [
ð?¥â??3
ð?¥ 3 â??27
ð?¥â??3
ð?¥ 2 ð?¥â??2
2â??ð?¥
] â¡â¡ lim( ð?¥ 2 â??3ð?¥) â?? â¡ lim( â??3ð?¥)
ð?¥â??1
ð?¥â??0

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