The Taylor polynomial for sin(x) for a near 0 to order 5 is 1 1 sin(x)...

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The Taylor polynomial for sin(x) for a near 0 to order 5 is 1 1 sin(x) = x - + -x +0(x7) 6 120 The figure below shows a plot of sin(x) [blue] and the above Taylor polynomial [red]. 0.8 0.6 0.4 0.2 1 00|77 4 3 8 2 5 3 7 8 4 8 (a) Estimate the largest x for which this lower order approximation: is accurate to 1%. Obtain an answer within 5% of the true answer. 1 sin(x) = x - 6 23 (b) Estimate the largest x for which 1 1 sin(x) + -x5 6 120 is accurate to 1%. Obtain an answer within 5% of the true answer. Hint: Make sure your answer makes sense when compared to the plot in the previous question. (c) Below is another plot of sin(x) together with a Taylor polynomial approximation to sin(x) near x = 0. 0.5 A A -0.5- Ela What is the largest n for which is included in the polynomial plotted above? (-1)n (2n+1)! -x2n+1 (-1)" Hint: compute (2n+1)! computer if you get bored with your calculator. x+1 for various values of n, taking x at a point where the two plots are just starting to diverge. Use a The Taylor polynomial for sin(x) for a near 0 to order 5 is 1 1 sin(x) = x - + -x +0(x7) 6 120 The figure below shows a plot of sin(x) [blue] and the above Taylor polynomial [red]. 0.8 0.6 0.4 0.2 1 00|77 4 3 8 2 5 3 7 8 4 8 (a) Estimate the largest x for which this lower order approximation: is accurate to 1%. Obtain an answer within 5% of the true answer. 1 sin(x) = x - 6 23 (b) Estimate the largest x for which 1 1 sin(x) + -x5 6 120 is accurate to 1%. Obtain an answer within 5% of the true answer. Hint: Make sure your answer makes sense when compared to the plot in the previous question. (c) Below is another plot of sin(x) together with a Taylor polynomial approximation to sin(x) near x = 0. 0.5 A A -0.5- Ela What is the largest n for which is included in the polynomial plotted above? (-1)n (2n+1)! -x2n+1 (-1)" Hint: compute (2n+1)! computer if you get bored with your calculator. x+1 for various values of n, taking x at a point where the two plots are just starting to diverge. Use a