Part 1: Fill in the blanks.
1. Let x0=1, xn=ln(1+xn−1)(n≥1). Find limn→+∞nxn.2. Find I=∫02π1+tanxcosxdx.3. Let L:{2x−4y+z=03x−y−2z=9 and plane π:4x−y+z=1. Find the equation of projection of straight line L on the plane π.4. Find ∑n=1+∞arctan4n2+4n+12.5. Solve (x+1)dxdy+1=2e−y where y(0)=0.Part 2: Proof-based Questions
6. Let f(x)=−21(1+e1)+∫−11∣x−t∣e−t2dt. Prove that f(x) has only two real roots in the interval (−1,1).7. Let f(x,y) be a function that exists a continuous second-order partial differentiation in closed region D={(x,y)∣x2+y2≤1} such that ∂x2∂2f+∂y2∂2f=x2+y2. Find limr→0+(tanr−sinr)2∫∫x2+y2≤r2(x∂x∂f+y∂y∂f)dxdy.8. It is given that every orientable smooth closed surface S in half space in R3{(x,y,z)∈R3∣x>0} exists ∫∫Sxf′(x)dydz+y(xf(x)−f′(x))dzdx−xz(sinx+f′(x))dxdy=0, where f is twice continuously differentiable on the interval (0,+∞) and limx→0+f(x)=limx→0+f′(x)=0. Find f(x).9. Let f(x)=∫0x(1−u⌊u⌋)du. Discuss the convergence and divergence of ∫1+∞xpef(x)cos(x2−x21)dx, where p is positive number.10. Let an be a positive sequence that is monotonically decreasing and tends to 0 and f(x)=∑n=1∞annxn. Prove that if the series ∑n=1∞an diverges, then integral ∫1+∞x2lnf(x)dx diverges too.