Please show work on all steps taken to get to the answer.

1. Using the Maclaurin series for flix} = sin 2;, derive the Maclaurin series for gEx] = Isin 2x. Hint: It is not necessary to do any differentiation to do this problem. 2. By multiplying the appropriate Taylor series about c = (J, compute the first four terms of the Taylor series about c = {J for x] = e'Jr cos x. Hint: It is not necessary to do any differentiation to do this problem. 3. Compute the Taylor series of flint} = about c = l. Hint: The first two terms don't fit the pattern of the remaining terms, so you will need to write those two terms out before writing the rest of the series compactly as an infinite series. This one is a bit messier than most of the examples we've seen up to this point. '1 4. 1Write the value of f 212 air as an infinite series. 0 5. Determine the Taylor series for f (x) =- centered at c = 3. Hint: Do this by computing the derivatives of f and construction the coefficients of the Taylor Series. 6. Use the Maclaurin series you know for f(x) = sin x to find the Maclaurin series for g(x) = x sin(x2). Hint: It is not necessary to do any differentiation to do this problem. 7. Write the value of sin x2 dx as an infinite series. Hint: There is a similar example in one of the videos.Taylor Series and Maclaurin Series There was a subtle point in the last lesson that you may not have noticed. Questions we posed as Find a power series for f(x) = ... . The indefinite article a suggests that there may be several different power serie that add up to the given function y = f(x). Well, as it turns out, that is not that case. The truth is that for given function y = f(c), there are only two o series centered at some number c that add up to f: there is either one such power series, or there are none at all. Given a func annot possibly be more than one power series abou that will add up to f(x) for x's in an interval about c. As shown in the video https://www.youtube.com/watch?v=GUtLtRDox3c (he uses a where we use c and visa versa!) if f(x) = _oan(x - c)" for x's in an interval about c, then the coefficients an must be given by the formula = f(") (@) n = 0,1,2,3, . .. . In other words, if (and that is a really big IF!) there is a power series that adds up to f(x) for x's near c, then that series must be (@ (x - C)". 1 The Most Common Error Concerning The Power Series Derived For a Function It is important to realize that there is no promise that _ (x (x - c)" adds up to f(x) for the x's i any interval about c. The only sure thing is that if any power series adds up to f it must be this one 2 Taylor Series and Maclaurin Series 2.1 Vocabulary Now for a little vocabulary. The power series SC (x -C)" is called the Taylor series for f about c. The special case when c = 0 (which is the most important situa- tion), the power series is called the Maclaurin series, but most people would also be comfortable calling i the Taylor series for f about 0. In fact, if you just said, consider the Taylor series for f, most people would jus assume you meant the Taylor series about 0. The coefficients that appear in the Taylor Series of f are call the Taylor coefficients for f at x = c. 2.2 A Short List of Taylor Series You are Expected to Know There is a half dozen functions which pop up so often you should just know the Taylor series: 1. 1 - x 2X (|x]