Suppose you want to estimate cos 3 (where 3 means 3 radians). You can do this is two ways: (i) using the Maclaurin series for cos x (aka Taylor series at 0); (ii) using the Taylor series for cos x at π. Note that 3 is closer to π than to 0, so you can expect that the second method will work better. For each of the methods (i) and (ii), use the Remainder Estimate to find the degree of the Taylor polynomial that will guarantee that approximation is to within 0.001. (That is, the error is less than 0.001.)
1. Suppose you want to estimate cos 3 (where 3 means 3 radians). You can do this is two ways: (i) using the Maclaurin series for cos x (aka Taylor series at 0); (ii) using the Taylor series for cos x at π. Note that 3 is closer to π than to 0, so you can expect that the second method will work better. For each of the methods (i) and (ii), use the Remainder Estimate to find the degree of the Taylor polynomial that will guarantee that approximation is to within 0.001. (That is, the error is less than 0.001.) You do not need to compute the actual approximations unless you really want to.
2. Represent the function cos(√ t) as power series centered at 0 and use this power series to find a power series representation of the indefinite integral R cos(√ t) dt.
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