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MATH 151 TEST 3 SPRING, 2010 5/11/2010Remember to keep your work neat and orderly. Show all of your work. NO WORK = NO CREDIT! Read each question carefully and be sure to answer the question that was asked. Good luck!Name:____________________________________________________________ pts1. Compute the Taylor (Maclaurin) series for f (x )=e−2 x and show that it is the expected series1−2 x+(2 x )22!−(2 x )33 !+. ... Be sure to show the formula for the nth derivative and term! 10−2 x¿f(n)(x)=(−2)ne¿f(n)(0)=(−2)n=(−1)n2n2. Find the interval of convergence for the series∑n=4∞(−1 )n+13 (x +5 )n+26nn2. 11−11 ≤ x ≤ 13. Find the distance between the points (-3, 7, -2) and (-8, 6, 1). 5¿√354. If a =< 3,2,−4 >¿¿ , b =<−5,1,−3>¿¿, and c =<−3,5,1>¿¿ find each of the following.a.2 a⋅bb. a×b4,7= -2 =<-2, 29, 13>c. Any non-zero vector orthogonal to c .d. |−2 a|4,5= <1, 0, 3>¿2√295. Compute the 4th-degree Taylor polynomial about x=π6 for f (x )=sin x . Note -sinπ6=12;cosπ6=√32. This is different than the known series about x=0 !10T4(x)=12+√32(x−π6)−12(12 !)(x−π6)2−√32(13!)(x−π6)3+12(14 !)(x−π6)46. What is the equation of the sphere centered at the point (-4, 0, 3) with a radius of 5? 4(x+4)2+ y2+(z −3)2=257. What is the maximum possible error if we approximate ∑n=2∞(−1)nn by ∑n=236(−1)nn?4= abs val of 37th term = 1/378. State whether each of the following series is absolutely convergent (AC), conditionally convergent (CC), or divergent (D). Justify your answers!a.∑n=2∞(−1)nn52n−1b. ∑n=2∞(−1)n+1(3−2n+3 n2)n2+5 n−76,5AC (ratio test, limit= ½) Div : limit = 3 ≠ 0c. ∑n=2∞(4n+1−79n−27)d. ∑n=2∞(−1)n(√n+3√n )n+4√n5,5AC (geom., r = 4/9) CC compare to ∑(−1)n√n9. Given that 11−3 x=1+3 x +9 x2+27 x3+. .. for |3 x|<1 and11+2 x=1−2 x+4 x2−8 x3+.. . for |2 x|<1, find the 3rd-degree Maclaurin polynomial forf (x )=11−3 x⋅11+2 x by multiplying the two series together.10T3(x)=1+x+7 x2+3 x310.Sketch the vector represented by 3 u−2


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SCSU MAT 151 - MATH 151 TEST 3

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