江西财经大学国际学院
2019 级微积分期末复习试卷（一）
——By 刘京
from 微积分助教组
1. (15ps) Full in the blank of each statement in the following five equalities such
that the equality holds. Then write the corresponding answer on the answer book
by the title number. You can be gained 3 points for the right thing on per blank.
𝑥
1. lim
√1+𝑥−1−2
𝑥→0
2
𝑒 𝑥 −1
1
2. lim (√𝑛2
𝑛→∞
=
1
+𝑛
+ √𝑛2
1
+ √𝑛2
+2𝑛
+3𝑛
1
+ ⋯ + √𝑛2
+𝑛2
)=
3. 𝑓(𝑥) = 𝑥 3 + 2𝑥 − 4, 𝑔(𝑥) = 𝑓[𝑓(𝑥)], 𝑔′ (0) =
4. ∫
𝑥𝑙𝑛(𝑥+√1+𝑥 2 )
(1+𝑥 2 )2
=
𝑑𝑥
5. 𝑦 = 𝑥 3 + 3𝑥 + 1, 𝑑𝑦|
=
𝑦=1
2. (15pts) Choose the one that the statement is right from four choices marked A,
B, C, and D, with which each statement of the following five statements. Then write
the corresponding answer on the answer book by the title number. You can be
gained 3 points for the right choice of per the statement.
6. Assume that 𝑔(𝑥) have second derivative when 𝑥 = 0 , and 𝑔(0) = 𝑔′ (0) =
𝑔(𝑥)
,𝑥 ≠ 0
0. If 𝑓(𝑥) = { 𝑥
, then when 𝑥 = 0, 𝑓(𝑥) is ( )
0 , 𝑥=0
A. discontinuous
B. continuous, but not differentiable
C. differential, but the derivative is discontinuous
D. differential, and the derivative is continuous
1
7. Assume that 𝑓(𝑥) have one derivative when 𝑥𝜖[0,4], and 𝑓′(𝑥) ≥ 4, 𝑓(2) ≥ 0,
1
and the following must have 𝑓(𝑥) ≥ 4 is ( )
A. [0,1]
B. [1,2]
𝑥1
C. [2,3]
D. [3,4]
8. 𝑓(𝑥) = ∫0 𝑥 (𝑡 2 − 𝑡)𝑑𝑡(𝑥 > 0), then the minimum value is ( )
6
B. −1
A. − 13
1
C. 0
D. − 2
9. Assume that 𝑓(𝑥) is continuous when 𝑥 = 0, then the wrong statement of
following is ( )
A. If lim
𝑓(𝑥)
is exists, then 𝑓(0) = 0
𝑥→0 𝑥
𝑓(𝑥)+𝑓(−𝑥)
B.
If lim
C.
If lim 𝑥
D.
If lim
is exists, then 𝑓(0) = 0
𝑥
𝑥→0
𝑓(𝑥)
is exists, then 𝑓 ′ (0) is exists
𝑥→0
𝑓(𝑥)−𝑓(−𝑥)
𝑥
𝑥→0
is exists, then 𝑓 ′ (0) is exists
10. When 𝑥 → 0+ , the right following is ( )
A. √1 + 𝑥 − 1~𝑥
B. ln(1 + 𝑥) − 𝑥~𝑥
C. cos(𝑠𝑖𝑛𝑥) − 1~𝑥
D. 𝑥 𝑥 − 1~𝑥
3. Give the solution of everyone of the following problems and write operation
process and answer on the answer book by the title number. You can be gained
mark of per problem by the right process and answer.
6
6
11. Find
a) lim √𝑥 6 + 𝑥 5 − √𝑥 6 − 𝑥 5
𝑥→∞
𝑑𝑥
b) ∫ 2+𝑐𝑜𝑠𝑥
提出x
三角替代
12. If 𝑓(𝑥) = 𝑒 10𝑥 𝑥(𝑥 + 1)(𝑥 + 2) ⋯ (𝑥 + 10), then find the value of 𝑓′(0)
极限的定义式
13. Discuss the real root of the equation 2𝑥 3 − 9𝑥 2 + 12𝑥 − 𝑎 = 0
14. Assume that 𝑓(𝑥) is continuous on [a,b], differentiable on (a,b). and 𝑓(𝑎) =
𝑓(𝑏) = 0, proof that:
a) ∃𝜃𝜖(𝑎, 𝑏), 𝑓(𝜃) + 𝜃𝑓 ′ (𝜃) = 0
b) ∃𝜇𝜖(𝑎, 𝑏), 𝜇𝑓(𝜇) + 𝑓 ′ (𝜇) = 0
𝑑2
𝑥
𝑠𝑖𝑛𝑡
15. Find 𝑑𝑥 2 ∫0 ( ∫1
√1 + 𝑢4 𝑑𝑢)𝑑𝑡
𝑔(𝑥)−𝑒 −𝑥
16. Assume that 𝑓(𝑥) = {
𝑥
0,
, 𝑥≠0
, and 𝑓(𝑥) has second derivative which
𝑥=0
is continuous, and 𝑔(0) = 1, 𝑔′ (0) = −1
a) Find 𝑓′(𝑥)
b) Discuss the continuity of 𝑓 ′ (𝑥) 𝑜𝑛 (−∞, +∞)
17. Assume that 𝑦 = 𝑓(𝑥) has inverse function 𝑔(𝑥) , and 𝑓(𝑎) = 3, 𝑓 ′ (𝑎) =
1, 𝑓 ′′ (𝑎) = 2. Find 𝑔′′(3)