1. ( points) Compute (a: lye42' dm 1 . 2. ( points) Compute / 9 (393\"2 1)3 - x/E (3; 0 . . 5:1: 3. ( pomts) Compute the integral / m dm. 1 points) Consider the function f (as) = (1 + g) . a) Using any method, nd the rst four non-zero terms of the Maclaurin series for f (at) b) Write the Maclaurin series from (a) using summation notation. c) Determine the interval of convergence of the Maclaurin series. a) Draw a picture and shade the region R. - ( ( ( ( 5. ( points) Let R be the region bounded by the curves 3; = 6'5"", 3; = 0, a: = 0, and .7; = ln4. ( (b) Calculate the volume of the solid obtained by rotating R about the line 3; = 2. - ( points) Determine if the following series converge or diverge. Show all of your work. (a) 2 cos (3k) 2 7. points) (a) Convert the polar point (3, g) to rectangular coordinates. ( (b) Convert the rectangular point (3, 4) to polar coordinates. (c) Find the area of the region R in the scy-plane of polar points (5", 6) that satisfy the inequalities 2 S r S 3 and 0 S 6 3 7r. [Hint: Sketch R and you should not need calculus to nd the area] 8. ( points) Consider the graph of y = :33. (a) Write an integral expression that calculates the length along this curve from a: = 0 to a: = 1. (b) Using Simpson's method with 4 subintervals, approximate the integral you wrote in part (a). If 1 you did not nd an integral in (a), then approximate / x/zr + :33 da: 0 9. ( points) Consider the parametric curve given by 13(6) = ln(1 + t) and y(t) = t2 for 0 S t g 1. (a) Find the plane curve equation, 3; = f (:8), corresponding to this parametric curve. ( b) Sketch the parametric curve in the Qty-plane, indicating orientation, and making sure to carefully mark the beginning and ending points. 1 (c) Find the equation of the tangent line at t = E. 2 2 10. ( pomts) Let A = [5 _1 1 . Find the eigenvalues of A and an associated eigenvalue of each. 11. ( points) Use Gauss-Jordan elimination to solve the system 2:c+y=16 3m+2y=17 1. Consider the following data a: 011511.52 \"$025620 2 Use Simpson's method with four subintervals to estimate f x) dun. o 2. What is the third order Taylor polynomial of {328:6 centered at 0? 3. If the third order Taylor polynomial of f($) centered at 0 is p3($)=2$%1:2+4$3 then what is f\"(0)? 4. Find the rst four nonzero terms of the Maclaurin series of tan(m + 7r/4). 1 5. Use a known Maclaurin series to write m as a series. 6. Graph the set of polar points {(T', 6) : *1