Calculus: An Integrated Approach to Functions and their Rates of Change, Preliminary Edition Part 96. A major complaint of professors teaching calculus is that students don't have the appropriate background to work through the calculus course successfully. This text is targeted directly at this underprepared audience. This is a single-variable (2-semester) calculus text that incorporates a conceptual re-introduction to key precalculus ideas throughout the exposition as appropriate. This is the ideal resource for those schools dealing with poorly prepared students or for schools introducing a slower paced, integrated precalculus/calculus course | Approximating a Function by a Polynomial 931 Alternating signs -1 and -1 1 can be used to indicate alternating signs. Which is needed to do the job is determined by the notational system you happen to have chosen. The simplest way of determining which you need is by trial and error. Try -1 and check it with a particular fc-value. If it doesn t work switch to -1 1. EXAMPLE SOLUTION Approximate 34 using the appropriate second degree Taylor polynomial. 1 Let x x 5. We must center the Taylor polynomial at a point near 34 at which the values of and can be readily computed. An off-the-cuff approximation of 34 is 34 32 2 we know that 34 is a bit more than 2. Center the Taylor polynomial at x 32. 32 P2 x 32 32 x - 32 x - 32 2 x x1 32 2 f x 1 x-5 32 A-L. 11 5 5 324 5 24 80 -4 9 -4 1 -1 -1 x x-5 32 ---J ----- 25 25 29 25 27 3200 Therefore 11 P2 x 2 80 x - 32 - 6400 x - 32 2- 734 - 2 34 2 2 - 2 ------ 80 6400 40 1600 This agrees with the actual value of 3 to four decimal places. If you study closely the numerical data in this section you can start to get a sense of the magnitude of the error involved in a Taylor polynomial approximation. The size of the error can be estimated by graphing x - P x using a calculator or computer. In the next section we will state Taylor s Theorem which will provide not only a method of estimating errors independent of a calculator but also an invaluable theoretical tool. PROBLEMS FOR SECTION For Problems 1 through 7 do the following. a Compute the fourth degree Taylor polynomial for x at x 0. b On the same set of axes graph x P1 x P2 x P3 x and P4 x . c Use P1 x P2 x P3 x and P4 x to approximate and . Compare these approximations to those given by a calculator. 1. x e x 2. x ln 1 x 3. x tan-1 x 932 CHAPTER 30 Series 4. x 1 x 4 5. x V1 x 6. x 2x4 3x2 x 1 7. x 1 x 2 8. Below is a graph of x . For each quadratic given explain why the quadratic could not be the second degree Taylor polynomial for x at x 0. a 2 3x 1 x2
