Math 132 Final Exam Practice Questions 1 Given that a Maclaurin series for sin x is nd the Maclaurin series cid 88 n 0 1 nx2n 1 2n 1 for f x x4 sin x4 A B C D cid 88 n 0 cid 88 n 0 cid 88 n 0 cid 88 n 0 1 n 2n 1x8n 5 2n 1 1 n 8n 4x8n 8 2n 1 1 n 8n 1x8n 5 2n 1 1 n 2n 1x8n 8 2n 1 2 Find the interval of convergence of the power series cid 88 n 1 x 5 n n A I 0 B I 4 6 C I D I 3 Evaluate the integral x2 ln 4x dx x3 ln 4x x3 C 1 3 1 9 cid 90 4 Consider the two parabolas y x2 c2 and y c2 x2 Find the value of the constant c such that the area of the region bounded by the parabolas is 576 5 For the following power series nd the radius and interval of convergence c 6 cid 88 n 0 3n n 1 3 2x 1 n R I cid 21 cid 20 2 6 2 3 1 6 1 6 Represent the function as a power series f x 3x4 1 2x 2 3 cid 88 n 0 1 n2n n 1 xn 4 ln 2 x 2 n cid 88 n 1 1 n 1 2nn 7 Find the Taylor series for f x ln x centered at a 2 Express your answer in summation notation 8 It is known that f x cos x has a Maclaurin series of Determine the Maclaurin series for f x 4x4 cos x5 cid 88 n 0 1 n x2n 2n 4 cid 88 n 0 1 n 2nx10n 4 2n 9 Find the radius and interval of convergence cid 88 n 2 1 n xn 4n ln n cid 88 n 1 3n x 4 n n f x x 9 x2 10 Find the radius and interval of convergence R 4 I 4 4 R I cid 20 13 3 cid 19 11 3 1 3 11 Find the power series representation of the function cid 88 n 0 1 n x2n 1 9n 1 2 12 Find the power series representation of the function and determine the radius of con vergence f x x 1 4x 2 13 Eliminate the parameter to nd a Cartesian equation of the curve x e2t y t 1 14 Find the Cartesian coordinates of the point a 2 5 4 1 1 b 2 3 4 2 2 15 The Cartesian coordinates are given Find two representations of this point in polar coordinates a 2 2 2 2 7 4 2 2 3 4 b 1 3 2 2 3 2 5 3 16 Find the Cartesian equation for the polar curve r 5 cos cid 88 n 0 1 n4n n 1 xn 1 y ln x 1 1 2 cid 18 x cid 19 2 5 2 y2 25 4 r 1 sin 3 cos 17 Find the Polar equation for the curve represented by the following Cartesian equation y 1 3x 3 18 Consider the parametric equations below x sin 2t y cos t 0 t a Draw a graph of the curve de ned by this set of parametric equations Clearly label at least ve ordered pairs x y that have been plotted Draw an arrow from the initial point to indicate which direction the curve is being traced Points 0 1 1 Graph 2 2 0 0 1 2 2 0 1 b Find the equation of the line tangent to the curve at the point cid 16 3 2 1 2 cid 17 c Write an expression for the arc length of the curve as a de nite integral but do not evaluate the integral y x 3 2 1 4 L cid 112 sin t 2 2 cos 2t 2 dt cid 90 0 4 19 Sketch the polar graph of r 3 cos 2 for 0 2 Clearly label at least four ordered pairs r that have been plotted Points 3 0 3 2 6 0 4 3 2 3 3 2 Graph 20 Find the slope of the tangent line to r 3 cos 2 at 4 m 1 21 Find the area of the region enclosed by one loop of the curve r 3 cos 2 9 8 5