- 1 0 84 0 6 0 4 0 2 L N Tu N 4 3t 8 St 8 3t 4 Tt 8 8 8 4 2 X The Taylor Polynomial For Sin X For Z Near 0 To Order 1 (285.64 KiB) Viewed 39 times
The Taylor polynomial for sin(x) for z near 0 to order x is sin(x) = x- e 1 x + -200 + 0(2) 120 This figure shows a plot of sin(x) [blue) and the above Taylor polynomial [red]. (a) Estimate the largest x for which this lower order approximation: 1 sin(2) 3- 6 is accurate to 1%. Obtain an answer within 20% of the true answer. x^5/120 !!! (b) Estimate the largest x for which sin(2) 3 - 1 6 1 + 120 is accurate to 1%. Obtain an answer within 20% of the true answer. Hint: Make sure your answer makes sense when compared to the plot in the previous question. -X^7/7! (c) This is another plot of sin(x) together with a Taylor polynomial approximation to sin(2) near x = 0. What is the largest n for which (-1)" (2n + 1)! 22n+1 is included in the polynomial plotted? 22 (-1)" 22n+1 for various values of n, taking 2 at a point where the two plots are just starting to diverge. Use a computer if you get bored with your calculator Hint: compute (2n+1)!