# Find the 3th taylor polynomial of f x tan 1 x

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26. Find the 3th Taylor polynomial of
27. Find the Taylor series of11-2x2using geometric series.Xn=0(2x2)n=Xn=02nx2n28. Does the seriesn=1(-1)nlnnconverge absolutely, converge conditionally, or diverge?
29. Determine if the following statements are true or false:(a) The sequencean=1+cos(πn) converges to 0.Fanconverge.(b) Ifan|converges, then{xR||x-1|<3}is-2.Xanconverge.|an|converges, then{xR||x-1|<3}is-2.Xanconverge.
(c) The least upper bound of the set{xR||x-1|<3}is-2.
(d) The set{xR| -1<cos(x)<1}does not have a least upper bound.
(e) The greatest lower bound of{1.1,1.11,1.111,1.1111, ...}is 1.1.
(f) The set{1.1,1.11,1.111,1.1111, ...}does not have a greatest upper bound.Fanconverges.(g) If limn→∞an=0, thenXn=1rnconverges to11-rwhen|r|<1.FXn=1(-1)nnconverges.Xanconverges.
(h) The geometric seriesXn=1rnconverges to11-rwhen|r|<1.FXn=1(-1)nnconverges.(i) The alternating harmonic seriesXn=1(-1)nnconverges.
(j) A series can converge both absolutely and conditionally.
(k) The values ofXn=11dxx2are the equal.
(l) Ifc,0, thenn=1c=c+c+c+· · ·diverges.
30. Convert the polar coordinates to rectangular coordinates.(a) [r, θ]=[-2, π/4](-2,-2)(b) [r, θ]=[1/2,7π/6]-34,-14(c) [r, θ]=[-3,13π/4]322,32231. Convert the rectangular coordinates to polar coordinates.Choose the angleθso thatθ[0,2π].
32. Write the equationx2+y2+2x=px2+y2in polar coordinates.
33. Write the polar equationr=31-2 sinθin rectangular coordinates.
34. Sketch the following polar curves on the polar grid.(a)r=θ2, 0θ2π(b)r=1+cosθ
(c)r=1-2 cosθ35. Find the area outside the curver=2 sin(2θ) but inside the curver=3.
36. Find the area of one of the petals in the graph of the petal curver=sin(3θ).
37. Find the area of the region bounded by the cardioidr=2+2 cosθ.
38. Suppose the path of an object on the liney=2xis given in timetby the equationsx(t)=2-cos(t) andy=4-2 cos(t),t[0,4π]. 1. Check that this path is actually on theliney=2x. 2. Describe the path that the object takes in the plane.
39. Find all the points (x,y) where the tangent line to the curve given byx(t)=2t3+3t2-12tandy(t)=t2-2t+1,t(-∞,) is vertical.
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