1. Evaluate
sin(4𝑥) − 2 sin(2𝑥)
𝑥→0
𝑥3
lim
(1)
2. Find the derivative of
𝑦 = √4𝑥2 − 1 + csc−1 2𝑥
3𝑥 − 7
3. Find
∫ 2
𝑑𝑥
𝑥 − 2𝑥 − 15
2
4. Estimate ∫ √𝑥 3 − 1𝑑𝑥
by using trapezoidal rule for 𝑛 = 6.
1
2 sin(3𝑥) − 3 sin(2𝑥)
𝑥→0
𝑥3
1. Evaluate lim
[2]
1
2. Estimate ∫ √1 − 𝑥3 𝑑𝑥
by using trapezoidal rule for 𝑛 = 6.
0
3𝑥 + 7
𝑑𝑥
𝑥 2 + 2𝑥 − 15
4. Find the derivative of
𝑦 = √4𝑥 2 − 1 − sec −1 2𝑥
3. Find
∫
2 sin(2𝑥) − sin(4𝑥)
𝑥→0
𝑥3
1. Evaluate lim
2. Find the derivative of
3. Find
∫
{3}
𝑦 = √9𝑥2 − 1 − sec−1 3𝑥
3𝑥 − 5
𝑑𝑥
𝑥 2 − 𝑥 − 12
1
4. Estimate ∫ √1 − 𝑥 3 𝑑𝑥
by using Simpson′s rule for 𝑛 = 6.
0
3 sin(2𝑥) − 2 sin(3𝑥)
𝑥→0
𝑥3
1. Evaluate lim
2
2. Estimate ∫ √𝑥3 − 1𝑑𝑥
by using Simpson′s rule for 𝑛 = 6.
1
3. Find
∫
3𝑥 + 5
𝑑𝑥
𝑥 2 + 𝑥 − 12
4. Find the derivative of
𝑦 = √9𝑥 2 − 1 + csc −1 3𝑥
〈4〉
sin(4𝑥) − 2 sin(2𝑥)
4 cos 4𝑥 − 4 cos 2𝑥
(1)
=
lim
𝑥→0
𝑥→0
𝑥3
3𝑥 2
−16sin(4𝑥) + 8 sin(2𝑥)
−64 cos 4𝑥 + 16 cos 2𝑥 −64 + 16
= lim
= lim
=
= −8
𝑥→0
𝑥→0
6𝑥
6
6
1. lim
2. Find the derivative of
𝑦 = √4𝑥2 − 1 + csc−1 2𝑥
𝑑𝑦
8𝑥
2
4𝑥
1
4𝑥 2 − 1
√4𝑥 2 − 1
=
−
=
−
=
=
𝑑𝑥 2√4𝑥 2 − 1 2𝑥√4𝑥 2 − 1 √4𝑥 2 − 1 𝑥√4𝑥 2 − 1 𝑥√4𝑥 2 − 1
𝑥
3𝑥 − 7
3. Find
∫ 2
𝑑𝑥
𝑥 − 2𝑥 − 15
3𝑥 − 7
3𝑥 − 7
𝐴
𝐵
=
=
+
𝑥 2 − 2𝑥 − 15 (𝑥 − 5)(𝑥 + 3) (𝑥 − 5) (𝑥 + 3)
𝑥=5
∫
⇨ 𝐴=
3𝑥 − 7
=1
(𝑥 + 3)
𝑎𝑛𝑑
𝑥 = −3
⇨ 𝐵=
3𝑥 − 7
=2
(𝑥 − 5)
3𝑥 − 7
1
2
𝑑𝑥
=
∫
+
(
) 𝑑𝑥 = ln(𝑥 − 5) + 2 ln(𝑥 + 3) + 𝑐
(𝑥 − 5) (𝑥 + 3)
𝑥 2 − 2𝑥 − 15
2
4. Estimate ∫ √𝑥 3 − 1𝑑𝑥
by using trapezoidal rule for 𝑛 = 6.
1
𝑏
∫ 𝑓(𝑥)𝑑𝑥 ≅
𝑎
ℎ=
ℎ
[ 𝑦0 + 2𝑦1 + 2𝑦2 + 2𝑦3 + 2𝑦4 + 2𝑦5 + 𝑦6 ]
2
𝑏−𝑎 2−1
1
=
=
𝑛
6
6
𝑥𝑛
1
𝑦𝑛
𝑦0 = 0
Factors
1
Product
0
1 + 1 ⁄6
1 + 2 ⁄6
𝑦1 = 0.7668
2
1.5336
𝑦2 = 1.1706
2
2.3412
1 + 3 ⁄6
1 + 4 ⁄6
𝑦3 = 1.5411
2
3.0822
𝑦4 = 1.9052
2
3.8104
1 + 5 ⁄6
𝑦5 = 2.272
2
4.544
2
𝑦6 = 2.6458
Sum
1
2.6458
17.9572
2
∫ √𝑥 3 − 1𝑑𝑥 ≅
1
1⁄6
17.9572
× 17.9572 =
= 1.4964
2
12
2 sin(3𝑥) − 3 sin(2𝑥)
6 cos 3𝑥 − 6 cos 2𝑥
[2]
lim
𝑥→0
𝑥→0
𝑥3
3𝑥 2
−18sin(3𝑥) + 12 sin(2𝑥)
−54 cos 3𝑥 + 24 cos 2𝑥 −54 + 24
= lim
= lim
=
= −5
𝑥→0
𝑥→0
6𝑥
6
6
1. Evaluate lim
1
2. Estimate ∫ √1 − 𝑥3 𝑑𝑥
by using trapezoidal rule for 𝑛 = 6.
0
𝑏
∫ 𝑓(𝑥)𝑑𝑥 ≅
𝑎
ℎ=
ℎ
[ 𝑦0 + 2𝑦1 + 2𝑦2 + 2𝑦3 + 2𝑦4 + 2𝑦5 + 𝑦6 ]
2
𝑏−𝑎 1−0
1
=
=
𝑛
6
6
𝑥𝑛
0
𝑦𝑛
𝑦0 = 1
Factors
1
Product
1
0 + 1 ⁄6
0 + 2 ⁄6
𝑦1 = 0.9977
2
1.9954
𝑦2 = 0.9813
2
1.9626
0 + 3 ⁄6
0 + 4 ⁄6
𝑦3 = 0.9354
2
1.8708
𝑦4 = 0.8389
2
1.6778
0 + 5 ⁄6
𝑦5 = 0.6491
2
1.2982
1
𝑦6 = 0
Sum
1
0
9.8048
1
∫ √1 − 𝑥 3 𝑑𝑥 ≅
0
1⁄6
9.8048
× 9.8048 =
= 0.8171
2
12
3𝑥 + 7
𝑑𝑥
𝑥 2 + 2𝑥 − 15
3𝑥 − 7
3𝑥 − 7
𝐴
𝐵
=
=
+
𝑥 2 + 2𝑥 − 15 (𝑥 + 5)(𝑥 − 3) (𝑥 + 5) (𝑥 − 3)
3. Find
∫
𝑥 = −5
⇨ 𝐴=
∫
3𝑥 + 7
=1
(𝑥 − 3)
𝑎𝑛𝑑
𝑥=3
⇨ 𝐵=
3𝑥 + 7
=2
(𝑥 + 5)
3𝑥 + 7
1
2
𝑑𝑥
=
∫
+
(
) 𝑑𝑥 = ln(𝑥 + 5) + 2 ln(𝑥 − 3) + 𝑐
(𝑥 + 5) (𝑥 − 3)
𝑥 2 + 2𝑥 − 15
4. Find the derivative of
𝑦 = √4𝑥 2 − 1 − sec −1 2𝑥
𝑑𝑦
8𝑥
2
4𝑥
1
4𝑥 2 − 1
√4𝑥 2 − 1
=
−
=
−
=
=
𝑑𝑥 2√4𝑥 2 − 1 2𝑥√4𝑥 2 − 1 √4𝑥 2 − 1 𝑥√4𝑥 2 − 1 𝑥√4𝑥 2 − 1
𝑥
2 sin(2𝑥) − sin(4𝑥)
4 cos 2𝑥 − 4 cos 4𝑥
{3}
=
lim
𝑥→0
𝑥→0
𝑥3
3𝑥 2
−8 sin(2𝑥) + 16sin(4𝑥)
−16 cos 2𝑥 + 64 cos 4𝑥 −16 + 64
= lim
= lim
=
=8
𝑥→0
𝑥→0
6𝑥
6
6
1. lim
2. Find the derivative of
𝑦 = √9𝑥2 − 1 − sec−1 3𝑥
𝑑𝑦
18𝑥
3
9𝑥
1
9𝑥 2 − 1
√9𝑥 2 − 1
=
−
=
−
=
=
𝑑𝑥 2√9𝑥 2 − 1 3𝑥√9𝑥 2 − 1 √9𝑥 2 − 1 𝑥√9𝑥 2 − 1 𝑥√9𝑥 2 − 1
𝑥
3. Find
𝑥2
3𝑥 − 5
𝑑𝑥
𝑥 2 − 𝑥 − 12
3𝑥 − 5
3𝑥 − 5
𝐴
𝐵
=
=
+
− 𝑥 − 12 (𝑥 − 4)(𝑥 + 3) (𝑥 − 4) (𝑥 + 3)
𝑥=4
∫
∫
⇨ 𝐴=
3𝑥 − 5
=1
(𝑥 + 3)
𝑎𝑛𝑑
𝑥 = −3
⇨ 𝐵=
3𝑥 − 5
=2
(𝑥 − 4)
3𝑥 − 5
1
2
𝑑𝑥
=
∫
+
(
) 𝑑𝑥 = ln(𝑥 − 4) + 2 ln(𝑥 + 3) + 𝑐
(𝑥 − 4) (𝑥 + 3)
𝑥 2 − 𝑥 − 12
1
4. Estimate ∫ √1 − 𝑥 3 𝑑𝑥
by using Simpson′s rule for 𝑛 = 6.
0
𝑏
∫ 𝑓(𝑥)𝑑𝑥 ≅
𝑎
ℎ=
ℎ
[ 𝑦0 + 4𝑦1 + 2𝑦2 + 4𝑦3 + 2𝑦4 + 4𝑦5 + 𝑦6 ]
3
𝑏−𝑎 1−0
1
=
=
𝑛
6
6
𝑥𝑛
0
𝑦𝑛
𝑦0 = 1
Factors
1
Product
1
0 + 1 ⁄6
0 + 2 ⁄6
𝑦1 = 0.9977
4
3.9908
𝑦2 = 0.9813
2
1.9626
0 + 3 ⁄6
0 + 4 ⁄6
𝑦3 = 0.9354
4
3.7416
𝑦4 = 0.8389
2
1.6778
0 + 5 ⁄6
𝑦5 = 0.6491
4
2.5964
1
𝑦6 = 0
Sum
1
0
14.9692
1
∫ √1 − 𝑥 3 𝑑𝑥 ≅
0
1⁄6
14.9692
× 14.9692 =
= 0.8316
3
18
3 sin(2𝑥) − 2 sin(3𝑥)
6 cos 2𝑥 − 6 cos 3𝑥
〈4〉
=
lim
𝑥→0
𝑥→0
𝑥3
3𝑥 2
−12sin(2𝑥) + 18 sin(3𝑥)
−24 cos 2𝑥 + 54 cos 3𝑥 −24 + 54
= lim
= lim
=
=5
𝑥→0
𝑥→0
6𝑥
6
6
1. lim
2
by using Simpson′ s rule for 𝑛 = 6.
2. Estimate ∫ √𝑥3 − 1𝑑𝑥
1
𝑏
∫ 𝑓(𝑥)𝑑𝑥 ≅
𝑎
ℎ=
ℎ
[ 𝑦0 + 4𝑦1 + 2𝑦2 + 4𝑦3 + 2𝑦4 + 4𝑦5 + 𝑦6 ]
2
𝑏−𝑎 2−1
1
=
=
𝑛
6
6
𝑥𝑛
1
𝑦𝑛
𝑦0 = 0
Factors
1
Product
0
1 + 1 ⁄6
1 + 2 ⁄6
𝑦1 = 0.7668
4
3.0672
𝑦2 = 1.1706
2
2.3412
1 + 3 ⁄6
1 + 4 ⁄6
𝑦3 = 1.5411
4
6.1644
𝑦4 = 1.9052
2
3.8104
1 + 5 ⁄6
𝑦5 = 2.272
4
9.088
2
𝑦6 = 2.6458
Sum
1
2.6458
27.117
2
∫ √𝑥 3 − 1𝑑𝑥 ≅
1
3. Find
∫
1⁄6
27.117
× 27.117 =
= 1.5065
3
18
3𝑥 + 5
𝑑𝑥
𝑥 2 + 𝑥 − 12
3𝑥 + 5
3𝑥 + 5
𝐴
𝐵
=
=
+
𝑥 2 + 𝑥 − 12 (𝑥 + 4)(𝑥 − 3) (𝑥 + 4) (𝑥 − 3)
𝑥 = −4
∫
⇨ 𝐴=
3𝑥 + 5
=1
(𝑥 − 3)
𝑎𝑛𝑑
𝑥=3
⇨ 𝐵=
3𝑥 + 5
=2
(𝑥 + 4)
3𝑥 + 5
1
2
𝑑𝑥
=
∫
+
(
) 𝑑𝑥 = ln(𝑥 + 4) + 2 ln(𝑥 − 3) + 𝑐
(𝑥 + 4) (𝑥 − 3)
𝑥 2 + 𝑥 − 12
4. Find the derivative of
𝑦 = √9𝑥 2 − 1 + csc −1 3𝑥
𝑑𝑦
18𝑥
3
9𝑥
1
9𝑥 2 − 1
√9𝑥 2 − 1
=
−
=
−
=
=
𝑑𝑥 2√9𝑥 2 − 1 3𝑥√9𝑥 2 − 1 √9𝑥 2 − 1 𝑥√9𝑥 2 − 1 𝑥√9𝑥 2 − 1
𝑥