An important application of power series is to represent functions that occur in the natural sciences, in engineering, or in other disciplines of mathematics. Given that a particular power series converges to some function f(x), it is often easier to work with the power series repre- sentation than the function f(x) itself for such tasks as approximating function values, taking derivatives, computing integrals, or discerning the behavior of f(x). You will find the power series of f(x) = 3e-4x (this is the Maclaurin Series) and look at how good of an approximation we can get from it. 1. The first step in finding a power series, as we will discuss, is getting a formula for the nth derivative. Find f'(x), f"(x), f(3) (x), and then come up with a formula for the nth derivative f(n) (x), and explain why this formula is correct. 2. The formula cn= f(n) (0) n! will give you the coefficients for a power series of f. Using this formula, compute Co, C1, C2, C3, and c4. Find a general formula for Cn. 3. Find the first 5 terms of the power series with these coefficients. Also, write down the power series using Σ notation and the formula you had for Cn. 4. For what values of x does this series converge? (Hint: use the ratio test.) Make sure you fully explain your reasoning. 5. The first two terms of the power series give a linear function, let's call this L(x). The first three terms give a quadratic function, and the first four terms give a cubic function, call these Q(x) and C(x) respectively. Plot these three functions, along with f(x), on the interval [0,1]. Use a computer to do this, print it off (make it a big picture), and clearly label the functions. Include this plot with the rest of your work. 6. You have found the power series for f(x). So, L(1), Q(1), and C(1) are all estimates for f(1). Use the Alternating Series Estimation Theorem (Section 11.5) to find bounds for the error of these estimates. 7. How many terms of the power series would you need to use to guarantee that the estimate for f(1) was within 0.05 of the actual value? 8. Plot this new estimate along with f(x) on the interval [0, 1] and compare.