9. (3 points) Find the third order approximation of f(x)=sin(x) at x = 0. Sketch the graph of f(x) in the same coordinate axes as the graph of P3(x). 10. (2 points) Use the expression of P3(x) for f(x)=sin(x) to evaluate the limit lim sin(x) 0+x x One particular use of the Maclaurin series is the polynomial approximation to solutions of some differential equations (equations involving f and its derivatives). To do these kinds of approximations one needs to be skilled in changing the sum index at one's convenience. 11. (3 points) Write the series below with a sum index j starting at 3 and at -2: 00 0 - j=3 )-( ) j=-2 12. (12 points) Consider the differential equation "(x) + f'(x) + f(x) = 0 for all x = R for a function f(x) satisfying (the initial conditions) that f(0) = 1 and f'(0) = 0. Find the third order approximation P3(x) of f(x) at x = 0. Recall that if anx" - bmx = 0 is because n = m and a = am. For some simple differential equations, it is possible to find the exact solution f(x) using its Maclaurin series. While in general these series might not be as informative as the closed form of f(x), they provide some useful information for simple enough initial conditions. 13. (12 points) Consider the differential equation f" (x) = 2f(x) for any x R, satisfying the initial conditions that f(0) = 1 and f'(0) = 0. Find the Maclaurin series of f. 14. (3 points) Does the function have any even or odd symmetry? What if the initial conditions are f(0) = 0 and f'(0) = 1? What if f(0) = 1 and f'(0) = 2? Reflection: It is often useful to put in words what we have learned, since this forces us to consolidate our thoughts and help us reflect on the untold implications of what we have learned. 15. (5 points) Write a paragraph summarizing what you have learned with this assignment and how you could use these lessons in your academic career. Note: Please make sure to state all your references, including the names of the people who helped you in this assignment and specify which exercise they helped with and how.