Let P2(x) be the second-order Taylor polynomial for cos x centered at x=0 . Suppose that P2(x) is used to approximate cos x for |x| < 0.2. The error in this approximation is the absolute value of the difference between the actual value and the approximation. That is, Error = |P2(x)-cos x|. Use the Taylor series remainder estimate to bound the error in the approximation. Your answer should be a number; that is, you should give a bound for the error which works for all x in the given interval. Hint: Notice that the second- and third-order Taylor polynomials are the same. So you could think of your approximation of cos x as a second-order approximation OR a third-order approximation. Which one gives you a better bound? Error≤ Use the alternating series remainder estimate to bound the error in the approximation. Your answer should be a number; that is, give a bound for the error which works for all x in the given interval. Error≤ In either case, will the actual value of cos x be bigger or smaller than the approximated value, assuming x ≠ 0? bigger or smaller?
Let P2(x) be the second-order Taylor polynomial for cos x centered at x=0 . Suppose that P2(x) is used to approximate cos x for |x| < 0.2.
The error in this approximation is the absolute value of the difference between the actual value and the approximation. That is, Error = |P2(x)-cos x|.
Use the Taylor series remainder estimate to bound the error in the approximation. Your answer should be a number; that is, you should give a bound for the error which works for all x in the given interval.
Hint: Notice that the second- and third-order Taylor polynomials are the same. So you could think of your approximation of cos x as a second-order approximation OR a third-order approximation. Which one gives you a better bound?
Error≤
Use the alternating series remainder estimate to bound the error in the approximation. Your answer should be a number; that is, give a bound for the error which works for all x in the given interval.
Error≤
In either case, will the actual value of cos x be bigger or smaller than the approximated value, assuming x ≠ 0?
bigger or smaller?
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