[Solved] MATH 205 Final Examination

10] 1. (a) Sketch a graph of the functionf(x) = (4 x2 2 x 02 2|x 2| 0 < x 4on the interval 2 x 4 and calculate the definite integral R42f(x) dxin terms of area (do not antidifferentiate).(b) Use the Fundamental Theorem of Calculus to calculate the derivative ofF(x) =x2Rxesin( t) dt,and determine whether F is increasing or decreasing at x = 1.[6] 2. Find F(x) such that F0(x) = x2 + 2xx2 + 4and F(0) = 0.[11] 3. Find the following indefinite integrals:(a)Z13 xx2 x 6dx (b)Zx3/2ln2(x) dx[12] 4. Evaluate the following definite integrals (give the exact answers):(a)Z102x4x + 1dx (b)Z214 x2 dx .[8] 5. Evaluate the given improper integral or show that it diverges:(a)Zedxx ln(x2)(b)Z10dx(1 x)3/4MATH 205 Final Examination[18] 6. (a) Sketch the curves y = x (3 x2) and y = x, and find the area enclosed bythe two curves. (HINT: find first the points of intersection of the curves.)(b) Sketch the region enclosed by y = cos(2x) and the x-axis on the interval[0,2], and find the volume of revolution of this region about the axis y = 1.(c) Find the average value of the function f(x) = sec4(x) on the interval4,4

.[9] 7. Find the limit of the sequence {an} as or prove that it does not exist:(a) an =en n33n(b) an =(1)nn1 + 4n2(c) an = ln(n + 2n2) ln(2n + n2).[12] 8. Determine whether the series is divergent or convergent, and if convergent,then whether absolutely or conditionally convergent:(a)Xn=1n2/31 + 2n(b)Xn=1(1)n+1 sin 1n(c)Xn=1(1)n+1 2n+1n![6] 9. Find (a) the radius and (b) the interval of convergence of the series Pn=1(x + 2)3nn2 8n[8] 10. (a) Derive the Maclaurin series of f(x) = x2e3x.(HINT: start with the series for ez and then let z = 3x).(b) Find the values of x for which the following series convergesXn=0(x2 + 1)n2n+1and, for these values of x, find the sum of the series as a function of x.Bonus question [5]. Find the values of p (if any) for which the seriesPn=51n ln n(ln (ln n))p is convergent