3. (12pts) Calculatey??(0)anddxprovided that two variablesxandysatisfy the equationy2?x?ycosxy,y?0.
?Ax2?Bx?2,x?24. (12pts) FindAandBgiven that the derivative off(x)??is 2Bx?A,x?2?continuous for all realx.
5. (12pts)Find the area of the region bounded by curesy?tangent to the graphy?x,y?4and the equation of the
xat the point(1,1). What is the volume of the solid generated by
revolving the region about thex-axis?
6. (12pts)A manufacturing plant has a capacity of 30 articles per week. Experience has shown that narticles per week can be sold at a price of pdollars each wherep?10?0.15nand the cost of producingnarticles is30?3ndollars. How many articles should be made each week to give the largest profit?
x27. (16pts)Sketch the graph of the functiony?2.
x?18. (8pts) Letf(x)be continuous on[0,a], differentiable on(0,a). Iff(a)?0, then for every
positive realRthere is at least one numbercin(0,a)for whichRf(c)?cf?(c)?0
9. (8pts)Is it true or false thatf2(x)is differentiable impliesf(x)exists antiderivative？Justify your answer.
微积分2
江西财经大学05－06学年度第二学期试卷
课程名称：微积分II（C卷） 课程代码：12044 适用对象：05级国际学院本科生 学 时：64
11?) 1 (8pts). Evaluate lim(xx?0e?1x 2 (12pts). Given thatz(x,y)is defined by ztanx?xyz?2xyz, find (1) ?z.
(2) dz.
23?( (3) zU?4,1,1). Where, U=(1,0)
3 (8pts). Compute
?lnxx22dx.
122?2f?2f2?f4(12pts). Calculatex, provided thatf(x,y)?3x3y3. ?2xy?y22?x?y?x?y5 (8pts). Evaluate
12cos?xdxdy. Where, D is the triangle formed by the x?axis, ??2Dy?x,x?1.
6 (8pts). Solve the differential equationy??x?y. x?y?(?1)n9n7 (12pts). Determine the convergence of series ? and find its sum if it converges.
n!n?1 8 (12pts). A manufacturing plant has a product functionf(x,y)?xy, where xandyare
quantities of two distinct inputs respectively. Experience has shown that the prices of two inputs
are 2 and 3 respectively. What is the minimum cost of producing 100 units productivity? 9 (12pts). Find
?????e?xdx
2 10 (8pts). Let that f1(x,y),f2(x,y),g(x,y)are continuous on closed regionD and
g(x,y)?0. Prove 2??gf1f2dxdy????gf12dxdy?DD1?2gf2??dxdy,???0. D
江西财经大学
06－07第二学期期末考试试卷
试卷代码：12044A 授课课时：64
课程名称：微积分II 适用对象：06国际学院本科生
试卷命题人 罗春林 试卷审核人 王平平 1． (5?3?15pts) Fill in the blanks:
(1)
?tanxdx?___________.
?(?1)n (2) ?=____________.
n?0(2n)!lnn?____________.
n??n?1dx?____________. (4) ?12x3 (3) lim (5) dz(1,1)?______________provided that z?3x2?2y2.
2． (5?3?15pts) Choose the best answer for each of the followings:
(1) If the series
?bn?1?nconverges and an?bn(n?1,2,3,?),then
?an?1?n
_________.
(A) must be convergent. (B) must be divergent.
(C) may converges or diverges. (D) The statement (A),(B),(C)
are false.
1x? (2) ???f(x,y)dy??dx?___________. 0?0? (A)
(C)
x?f(x,y)dx??dy (B) ?0???0?111?f(x,y)dx??dy ?0???y?10x1????f(x,y)dxdyf(x,y)dy (D)????dx ?0??x?0??0?? (3) The general solution of the equation y???2y??15y?0is___________. (A) y?Ce5x?Ce?3x (B)y?Ce?5x?Ce3x
(C) y?C1e5x?C2e?3x (D)y?C1e?5x?C2e3x
(4) f(x,y)?2x2?y2?xy?7y has a local minimum at point__________. (A) (?1,?4) (B) (1,?4)
(C) (?1,4) (D) (1,4)
ex?x?1 (5) lim=____________. 2x?02x1 (B) 2 21 (C) (D) 4
4 (A)
x3dx. 3．(8pts) Evaluate ?21?x?2z4．(9pts) Calculate given thatzis defined implicitly as a differentiable function
?x?yof (x,y)by the equationz4?x2z3?y2?xy?2. 5
．
(8pts)
Compute
sinxdxdy??x? with ? bounded by the
curesy?0,y?xandx??.
6．(8pts) Find the interval of convergence of
?(?1)n?1?n(n?1)n. (x?3)4n7． (8pts) Find the general solution ofy??3y?3ex.
8． (12pts) A manufacturer can produce three distinct products in quantities
Q1,Q2,Q3, respectively, and thereby derive a profit
P(Q1,Q2,Q3)?2Q1?4Q2?6Q3. Find the profit-maximizing level of output
22Q1,Q2,Q3, for whichQ12?Q2?Q3?14.
? 9．(10pts) Suppose that an?0,(n?1,2,3?).Show that the convergence of
?an?1n
implies the convergence of counter-example.
?an?1?2n.Is the converse true， prove it or give a
10．(7pts) Suppose that f(x)andg(x)are both continuous on[a,?b]，show that:
bbb22???????af(x)g(x)dx????af(x)dx????ag(x)dx??.
??????2