I. Multiple-choice questions (5 small questions in this big question, 2 points for each small question, *** 10)

Of the four options listed in each question, only one meets the requirements of the topic. Please fill in the code in brackets after the title. Wrong selection, multiple selection or no selection will not be scored.

1. If f(x) is odd function and there is always f(x+3)-f(x- 1)=0 for any real number x, then f(2)= ().

A.- 1 B.0 C. 1 D.2

2. Limit = ()

Arabic -3 Arabic -2 Arabic-1 Arabic -3

3. If the curve y=f(x) has a tangent at x=x0, then the derivative f' (x0) ().

A equals 0 B. C. without D. does not necessarily exist.

4. Let the function y=(sinx4)2, then the derivative = ().

A.4x3cos(2x4) B.4x3sin(2x4)

C.2x3cos(2x4) D.2x3sin(2x4)

5. If f' (x2) = (x >; 0), then f(x)= ()

A.2x+C B. +C C.2 +C D.x2+C

2. Fill in the blanks (this big question * * 10 small question, 3 points for each small question, 30 points for * * *).

Please fill in the correct answers in the blanks of each question. You don't score if you fill it wrong or not.

6. If f(x+ 1)=x2-3x+2, then f () = _ _ _ _ _ _

7. The sum of infinite series is _ _ _ _ _ _ _.

8. If the function f(x)= and f(x0)= 1, then the derivative f' (x0) = _ _ _ _ _.

9. If the derivative f' (x0) = 10, the limit is _ _ _ _ _ _ _.

10. The monotonic decreasing interval of the function f(x)= is _ _ _ _ _ _ _.

1 1. The minimum value of the function f(x)=x4-4x+3 in the interval [0,2] is _ _ _ _ _ _ _ _.

12. The differential equation y÷+x(y')3+sin y = 0 has the order of _ _ _ _ _ _ _.

13. definite integral _ _ _ _ _ _ _ _.

14. Derivative _ _ _ _ _ _ _.

15. Let the function z=, then the partial derivative _ _ _ _ _ _ _.

Third, the calculation problem (a) (this big question ***5 small questions, each small question 5 points, ***25 points)

16. let y=y(x) be an implicit function determined by the equation ex-ey=sin(xy) and find the differential dy.

17. Find the limit.

18. Find the concave-convex interval and inflection point of curve y=x2ln x.

19. Calculate infinite generalized integral. 20. Let the function z= and find the second-order partial derivative.

Four, calculation problem (2) (this big problem ***3 small questions, 7 points for each small question, ***2 1 point)

2 1. Let an original function of f(x) be, and find the indefinite integral xf'(x)dx.

22. Find the area a of the plane figure surrounded by the curve y=ln x and its tangent at point (e, 1) and the X axis.

23. Calculate the double integral, where d is the area enclosed by the curve y=x2- 1 and the straight line y=0 and x=2.

Five, the application problem (this topic 9 points)

24. Let the cost function of a factory producing q tons of products be C(q)=4q2- 12q+ 100, and the demand function of this product be q=30-.5p, where p is the price of the product.

(1) Find the revenue function r (q) of the product; (2) Find the profit function L (q) of the product; (3) How many tons can this product produce and get the maximum profit? What is the maximum profit?

The problem of proving intransitive verbs (5 points for this big problem)

25. It is proved that the equation x3-4x2+ 1=0 has at least one real root in the interval (0, 1).