ABC University Basic Mathematics Practice Quiz


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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
3.∫0 √1 + 7𝑥𝑑𝑥
4. ∫0
5. ∫0 (2𝑥+1)3 𝑑𝑥
1 𝑒 𝑎𝑟𝑐𝑡𝑎𝑛𝑦
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
1. 𝑎𝑛 = 𝑒 −1/√𝑛
2. 𝑎𝑛 = 1+9𝑛
7. Find the radius of convergence and interval of convergence of the series. (10 分)
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
6. 𝑓(𝑥) = 1+𝑥 , 𝑎 = 2

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Explanation & Answer:
8 Problems


taylor series

indefinite integral

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