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Term: 2021 Winter

Name: _________ Time:

Final exam

_______ Total: _________

1. Evaluate the indefinite integral.（30 分）

1. ∫ 𝑐𝑜𝑠(𝜋𝑡/2)𝑑𝑡

2. ∫ 𝑠𝑒𝑐 2 2𝜃𝑑𝜃

1

3.∫0 √1 + 7𝑥𝑑𝑥

3

4. ∫0

1

𝑑𝑥

5𝑥+1

1

𝑥

5. ∫0 (2𝑥+1)3 𝑑𝑥

1 𝑒 𝑎𝑟𝑐𝑡𝑎𝑛𝑦

6.∫−1

𝑑𝑦

1+𝑦 2

2.Find the exact length of the curve. （10 分）

1. 36𝑦 2 = (𝑥 2 − 4)3 , 2 ≤ 𝑥 ≤ 3, 𝑦 ≥ 0

3.Find the exact area of the surface obtained by rotating the curve about the x-axis. （10 分）

1. 𝑦 2 = 𝑥 + 1,0 ≤ 𝑥 ≤ 3

4. Solve the differential equation. （10 分）

1. 𝑥𝑦 ′ + 𝑦 = √𝑥

2. 2𝑥𝑦 ′ + 𝑦 = 2√𝑥

5.Find the exact length of the polar curve. （10 分）

1.𝑟 = 5𝜃 , 0 ≤ 𝜃 ≤ 2𝜋

6.Determine whether the sequence converges or diverges. If it converges, find the limit. （10

分）

1. 𝑎𝑛 = 𝑒 −1/√𝑛

𝑛4

2. 𝑎𝑛 = 1+9𝑛

7. Find the radius of convergence and interval of convergence of the series. （10 分）

𝑥𝑛

1. ∑∞

𝑛=1 𝑛!

𝑛 𝑛

2. ∑∞

𝑛=1 𝑛 𝑥

8. Use the definition of a Taylor series to find the first four nonzero terms of the series for

𝑓(𝑥) centered at the given value of 𝑎. （10 分）

5. 𝑓(𝑥) = 𝑥𝑒 𝑥 , 𝑎 = 0

1

6. 𝑓(𝑥) = 1+𝑥 , 𝑎 = 2

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Explanation & Answer:

8 Problems

Tags:

mathematics

taylor series

indefinite integral

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