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2021 秋高数上期末试题(回忆版)

1.Multiple Choice Questions : (only one correct answer for each of the following questions.)

1-(1)

Let f(x) be a continuous function on [a,a],a>0, then aaf(x)dx=

(A) 0a(f(x)+f(x))dx

(B) 0a(f(x)f(x))dx

(C) 0

(D) 20af(x)dx

1-(2)

If f(x)={2+e1x1+e1x,x0, then at x=0, it is a 0,x=0

(A) jump discontinuity

(B) removable discontinuity

(C) infinite discontinuity

(D) continuous point

1-(3)

If the function f(x) has the third derivative at x=x0, and f(x0)=f(x0)=0,f(3)(x0)>0, then

(A) f(x) has a local minimum at x0

(B) f(x) has a local maximum at x0

(C) f(x) has no local extremum at x0

(D) None of (A),(B) and (C) is correct

1-(4)

If 0xf(t)dt=x42, then 041xf(x)dx=

(A) 8

(B) 16

(C) 128

(D) 256

1-(5)

The number of real roots in (0,1) for 5x20xdt1+t8=0 is

(A) 0

(B) 1

(C) 2

(D) greater than 2

2.Fill in the blanks

(1) If f(x)=(x2+1)(x2+2)(x2+3)(x2+4), then f(6)(0)=

(2) The average value for f(x)=cos4x on [0,π] is

(3) Using Simpson's Rule with n=4 to estimate 241x1dx, the approximation is

(4) If limx(x+axa)x=8, then a=

(5) limn1n(1+cosπn+1+cos2πn++1+cosnπn)=

3

Find the area of the surface generated by revering the curve 4y=x2(1y3) about the y-axis

4

Solve the following first-order linear differential equation

xyy=2xlnx,x>0

5

If the line y=x is tangent to the curve y=logax, find the value of a

6

A isosceles triangle is to be inscribed in a circle of radius R .What is the largest perimeter possible for the isosceles triangle? Please provide the reason

7

Find all values for p such that the improper integral 0exxPdx converges

8.Evaluate the following limits

(1) limx0(1+x)1xex

(2) limx03sinx+x2cos1x(1+cosx)ln(1+x)

9. Evaluate the intergrals

(1) 1eeln2xxdx

(2) 121x3x21dx

(3) 11x6(x5+4)dx

(4) 1(1+x+x2)2dx