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xxxx大学

第一学期期末考试试卷

课程名称：Calculus (I) Problem 1(3×5=15pts.)Fill in the blank of each unfinished statement such that it is right. Mark your answer on the answer sheet.

(1) lim(x?x???x2?2x)? .

x?0x?0(2) If lim[f(x)?g(x)]?2 and lim[f(x)?g(x)]?1, then limf(x)g(x)? . x?0(3)

?1?1(1?x2?sinx)dx? . 1?x41?ifx?0?xsin(4) Let f(x)??. Then f?(x)? . x?ifx?0?0(5)

?1?xdx? . 1?xProblem 2(3×5=15pts.)For each blank in the following statement, choose the best answer from the choice given below. Mark your choice on the answer sheet.

(6) If f(x) is differentiable at x?0 such that f(0)?0 and f?(0)?1, then

limx?0f(2x)?f(3x)=________.

xdf(ef(x))?________. dxA.0 B.1 C.3 D.5

(7) Let f(?) be a differentiable function. Then

A. ef(x)f?(x)f?(ef(x)) B. ef?(x)f?(x)f?(ef(x)) C. f?(x)f?(ef(x)) D. ef?(x)f?(ef(x))

(8) lim?1?17?x?0?x??? ________. 7??171xA. e B. e C. e7 D. e?7

(9) If f and g are continuous and f(x)?g(x) for a?x?b, then

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A. f?(x)?g?(x) for a?x?b B. C.

?baf(x)dx??g(x)dx

ab?f(x)dx??g(x)dx D. limx?cf(x)?1 for a?c?b g(x)(10) Suppose that f(x)is continuous on[a,b], and x0is a fixed number in[a,b], then

dx0f(t)dt= _________. dx?a A. f(x0) B. f?(x0) C. 0 D. ?f(x0)

Please show all your work on answer sheet, unless instructed otherwise. Partial credit will be given only for work shown. The point value of each problem is written next to the problem – use your time wisely.

Problem 3(8pts.) Find the equation of the tangent line to the curve

y3?xy2?cos(xy)?2

at the point (0,1).

Problem 4(8pts.) Evaluate lim(x?011?x). sinxe?1?sinx?Problem 5(8pts.) Let f(x)??x??ax?b4ifx?0ifx?0

Determine a and b such that f is differentiable everywhere. Problem 6(8pts.) Find

?0xdx. 1?2xProblem 7(8pts.) Let R be the region bounded by the curve y?x2, and the line

y?2x.

(a)Evaluate the area of the region R.

(b)Find the volume of the solid generated by revolving the R about the y-axis. Problem 8(8pts.) Determine the production level that will maximize the profit for a company with cost and demand functions

C(x)?0.1x2?4x?60,p(x)?16?0.2x

Problem 9(8pts.) Sketch the graph of f(x)?x2?Problem 10(8pts.)

(a) (3pts.)Use the Mean Value Theorem to prove that

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1. xlim(sinx?2?sinx)?0

x??(b) (5pts.) Let f(x) be continuous on [a,b] and suppose that f??(x) exists for all x in (a,b). Prove that if there are three values of x in [a,b] for which f(x)?0 then there is at least one value of x in (a,b) such that f??(x)?0.

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