Math 231E. Fall 2013. Midterm 3 Practice Problems.Problem 1. Find the partial fractions expansions of the following functions. Do polynomialdivision if necessary to reduce it to the required form.a. f(x) =2x + 1x2− 4b. f(x) =x3− 1x3− 3x + 2c. f(x) =3x + 1(x + 1)(x2+ 1)d. f(x) =5x + 3x2(x2+ 1)Problem 2. Give the form of the partial fractions expansion for the following functions. Ifnecessary do polynomial division to reduce it to the required form.a. f(x) =x10+ 5x − 11(x4− 16)(x2+ 4)b. f(x) =x2+ x − 1(x3− 3x + 2)x2c. f(x) =3x + 1(x + 1)2(x2+ 1)3d. f(x) =5x + 3x(x2+ 1)2(x2+ 3)3Problem 3. Evaluate the following indefinite integrals;a.Z√x2− 1xdxb.Zx2√1 − x2dxc.Zx2(1 + x2)2dxd.Zx3√1 + x2dxe.Z1√1 + x2dxf.Zex√1 − e2xdxProblem 4. Indicate if each of the integrals below is proper or improper. If the integral isimproper indicate which point or points is responsible.a.Z10sin(x)x3/2dxb.Z10xx2− 2dxc.Z∞0dxx2− πd.Z∞1e11−x−xdxProblem 5. Decide if the following integrals converge or divergea.Z∞1x − 1x3+ 5b.Z∞0x271 + exdxc.Z∞πsin2(x)x2d.Z101√1 − xdxe.Z201x2− 4dxf.Z∞11 + cos2(x)x32dxProblem 6. State the comparison test for integrals, including all hypotheses.Problem 7. Compute the volume and the surface area of the solid of revolution obtainedby rotating the parabola y =x22around the y−axis.Problem 8. Given the implicitly defined curve ye−y= x set up (but do not evaluate)the integral respresenting the surface area of the solid of revolution formed by rotating theportion of this curve between x = 0 and x = e−1about the x-axis.Problem 9. Evaluate the following integrals.a.Zsin3(x) cos2(x)dxb.Zsec2(x) tan2(x)dxc.Zsec3(x)dxd.Zsin2(x)dxProblem 10. Find the limits of the following sequences, or explain why the limits do notexista. an=n2− 2n + 1n2b. an= n sin(πn/2)c. an= n2013(.99)nd. an=(2n)!(n!)2Problem 11. Evaluate the convergence of the following seriesa.∞Xn=11n2b.∞Xn=11n ln(n)c.∞Xn=1(1/2)nd.∞Xn=1n2(1/2)nProblem 12. State the monotone convergence theorem for sequences.Problem 13. Suppose that f(x) is continuous on [0, 2] and satisfies f(0) = f(2). Show thatthere exists a point η ∈ [1, 2] such that f(η) = f(η − 1).Problem 14. Let a0, a1, a2. . . anbe real numbers with the property thata0+a12+a23+ . . .ann + 1= 0.Show that there is a point x ∈ (0, 1) such that a0+ a1x + . . . anxn= 0.Problem 15. Evaluate the integralZ∞01x4+ 1dxHint: Either use the fact that x4+ 1 = (x2+ 1)2−2x2to factor the denominator, or usethe trick from HW10.6.Problem 16. Let f be a continuous function from [0, 1] to [0, 1]. (In other words as x variesover [0, 1] the range of f is [0, 1].) Show that f(x) has a fixed point: a point x such thatf(x) = x.Problem 17. A curve y = f(x) hs the following property. If one computes the volume ofthe solid of revolution obtained by rotating the portion of the curve between x = 0 and x = babout the x-axis, then the volume is b2. Find the curve.Answer the above if the revolution is done about the y-axis.Problem 18. Prove with calculus that eπ>
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