MATH 126 Final Review
3. Find the Taylor series for f(x) centered at the given value of a. [Assume that f has a power series expansion. Do not show that Rn(x) → 0.] f(x) = lnx , á=a
((-1)^n+1(x-a)^n)/(n) * (1/a)^n
1. Find the Maclaurin series for f(x) using the definition of a Maclaurin series. [Assume that f has a power series expansion. Do not show that Rn(x) → 0.] f(x) = sin(πx/a) R =
(-1)^n * (π^2n+1)/(2n+1)! * (1/a)^2n+1 ; R=infinity
2. Find the Maclaurin series for f(x) using the definition of a Maclaurin series. [Assume that f has a power series expansion. Do not show that Rn(x) → 0.] f(x) = a(1-x)^-2 R =
a(n+1)x^n ; R=1
13. Find an equation of the sphere that passes through the origin and whose center is (a, b, c).
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14. Find a unit vector that has the same direction as ai − bj + ck.
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15. Find the angle between the vectors. (First find an exact expression and then approximate to the nearest degree.) a = <x, x, x> , b = <x, x, x> *Exact *Aproximate
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16. Determine whether the lines L1 and L2 are parallel, skew, or intersecting. Random numbers L1: x = 6 + 4t, y = 8 − 2t, z = 2 + 6t L2: x = 3 + 12s, y = 9 − 6s, z = 12 + 15s *MC : parallel, skew, intersecting *If they intersect, find the point of intersection. (If an answer does not exist, enter DNE.)
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17. Find parametric equations and symmetric equations for the line. (Use the parameter t.) The line through (4, −3, 5) and parallel to the line x + 5 = y/2 = z − 5 *(x,y,z)= *Mc symmetric eq
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18. Consider the points below. P(2, 0, 2), Q(−2, 1, 3), R(6, 2, 4) (a) Find a nonzero vector orthogonal to the plane through the points P, Q, and R. (b) Find the area of the triangle PQR.
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19. Find an equation of the plane. The plane through the origin and the points (2, −4, 6) and (8, 1, 1)
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20. Consider the following. x = sin(t), y = csc(t), 0 < t < π/2 (a) Eliminate the parameter to find a Cartesian equation of the curve. (b) Sketch the curve and indicate with an arrow the direction in which the curve is traced as the parameter increases.
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21. Find a Cartesian equation for the curve. r = a sin(θ) Identify the curve.
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22. Find the slope of the tangent line to the given polar curve at the point specified by the value of θ. r = cos 2θ, θ = π/4
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23. Consider the following. x = a + t^2, y = t^2 + t^3 (a) Find dy/dx and d^2y/dx^2. dy/dx = d^2y/dx^2 = (b) For which values of t is the curve concave upward? (Enter your answer using interval notation.)
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24. Find the area of the region that is bounded by the given curve and lies in the specified sector. r = e−θ/a, π/2 ≤ θ ≤ π
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25. Evaluate the limit. lim x → 1 : x^a − 1/ x^b − 1
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26. Find the limit. Use l'Hospital's Rule if appropriate. If there is a more elementary method, consider using it. lim x→∞ (e^x + x)^a/x
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29. Find the exact length of the curve. x = e^t + e^−t, y = 5 − 2t, 0 ≤ t ≤ a
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30. Evaluate the integral. (Use C for the constant of integration.) p^a ln p dp
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31. Evaluate the integral. (Use C for the constant of integration.) ln(ax + 1) dx
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32. Evaluate the integral. 0 to π/2 a cos^2(θ )dθ
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33. Evaluate the integral. (Use C for the constant of integration.) dx/sqrt (x^2 + a)
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34. Evaluate the integral. (Remember to use absolute values where appropriate. Use C for the constant of integration.) a dx / (x − 1)(x^2 + b)
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35. Evaluate the integral. (Remember to use absolute values where appropriate. Use C for the constant of integration.) (ax^2 + 2x − a)/(x^3 − x) dx
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36. Determine whether the integral is convergent or divergent. 2 to ∞ e^−ap dp *Mc: convergent, divergent *If it is convergent, evaluate it. (If the quantity diverges, enter DIVERGES.)
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37. Determine whether the integral is convergent or divergent. 6 to 8 a/(x − 6)^3 dx *Mc: convergent, divergent *If it is convergent, evaluate it. (If the quantity diverges, enter DIVERGES.)
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38. Consider the solid obtained by rotating the region bounded by the given curves about the x-axis. y = 1/x, x = 1, x = a, y = 0 Find the volume V of this solid.
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39. Use the method of cylindrical shells to find the volume V of the solid obtained by rotating the region bounded by the given curves about the x-axis. y = x^3, y = a, x = 0 *V = *Graph
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40. Determine whether the sequence converges or diverges. If it converges, find the limit. (If an answer does not exist, enter DNE.) an = n + b / an + b
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41. Determine whether the geometric series is convergent or divergent. 1 to ∞Sigma (−a)^n − 1 / b^n *Mc: convergent, divergent *If it is convergent, find its sum. (If the quantity diverges, enter DIVERGES.)
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42. Use the Integral Test to determine whether the series is convergent or divergent. 1 to ∞Sigma a/n^b Evaluate the following integral. 1 to ∞ a / x^b dx Since the integral [is/is not] finite, the series is [convergent/divergent]
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43. Test the series for convergence or divergence. 1 to ∞Sigma (-1)^n [(an − b)/ cn + b] Evaluate the following limit. (If the quantity diverges, enter DIVERGES.) lim n → ∞ (−1)^n [(an − b) / cn + b] Since lim n → ∞ (−1)^n [(an − b) / cn + b] , [exists and equals 0, does not exist, exists and does not equal 0] , [the series is convergent, the series is divergent]
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44. Determine whether the series is convergent or divergent. n=1 to ∞Sigma cos^2 (n)/ (n^a + b) *Mc: convergent of divergent
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45. Determine whether the series is absolutely convergent, conditionally convergent, or divergent. n=0 to ∞Sigma (−a)^n/n! Mc: absolutely convergent conditionally convergent divergent
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46. Determine whether the series is convergent or divergent. n=1 to ∞Sigma nthsqrt (a) Mc: convergent divergent If it is convergent, find its sum. (If the quantity diverges, enter DIVERGES.)
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47. Find the radius of convergence and interval of convergence of the series. n=1 Sigma [(-a)^nx^n] / 3rd sqrt n R= I= ([ blank )]
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48. Find the radius of convergence and interval of convergence of the series. n=2 Sigma [(-10^nx^n+a] / n+1 R= I= ([ blank ])
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49. Find the radius of convergence and interval of convergence of the series. f(x) = a/(b-x) Determine the interval of convergence. (Enter your answer using interval notation.)
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7. Use the binomial series to expand the function as a power series. a(1-x/b)^2/3 (multiple choice) R =
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8. Use the binomial series to expand the function as a power series. f(x) = a sqrt (1+x/b) (multiple choice) R =
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9. Find the Taylor polynomial T3(x) for the function f centered at the number a. f(x) = 1/x , á=4 *Blank *Graph
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28. Find the exact length of the curve. x = y^a/b + 1/ay^c , 1 ≤ y ≤ 2
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10. Find the Taylor polynomial T3(x) for the function f centered at the number a. f(x) = ln(x)/x , á=1 *Blank *Graph
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11. Find the Taylor polynomial T3(x) for the function f centered at the number a. f(x) = xe^-ax , á=0 *Blank *Graph
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12. Find a vector a with representation given by the directed line segment AB. A(f, g), B(x, y) *Blank *Graph
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27. Find the exact length of the curve. y = x^3/3 + 1/4x , 1 ≤ x ≤ a
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4. Find the Taylor series for f(x) centered at the given value of a. [Assume that f has a power series expansion. Do not show that Rn(x) → 0.] f(x) = acos(x) , á = bπ
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5. Find the Taylor series for f(x) centered at the given value of a. [Assume that f has a power series expansion. Do not show that Rn(x) → 0.] f(x) = ax^-2 , á = 1
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6. Use the binomial series to expand the function as a power series. a/(b+x)^3 R =
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