To answer the question, you must include the following steps. • Use an appropriate Taylor series expansion for the given function. Choose the center of the expansion to be close to the given point. Using an appropriate multiple of 7/2 will suffice: the required precision can then be achieved by Taylor polynomials with fairly low degree. Using a closer multiple of 7/4 or 1/6 will sometimes lead to closer approximations with fewer terms, but the Taylor series at those centers will be more complicated. Find the Taylor series of the required function with your chosen center. You can use any method. • Use the Taylor Remainder Estimate to find the degree n of the Taylor polynomial T, that will provide approximation to within 0.000001. Explain this step carefully. • Compute the value of the Taylor polynomial T, at the given point to find the approximation. (Use the degree n that you found in the previous step). • Use a calculator to compute the actual value of the function at the given point. Check that your Taylor polynomial calculation indeed gives an approximation with the required precision. Note: work in radians! All Taylor series expansions use radians. You will have to use quantities like in your formulas. Make sure to take enough digits in the decimal expansion of r for the required precision to work. A1. Estimate sin(70°).
To answer the question, you must include the following steps.
• Use an appropriate Taylor series expansion for the given function. Choose the center of the expansion to be close to the given point. Using an appropriate multiple of π/2 will suffice: the required precision can then be achieved by Taylor polynomials with fairly low degree. Using a closer multiple of π/4 or π/6 will sometimes lead to closer approximations with fewer terms, but the Taylor series at those centers will be more complicated.
• Find the Taylor series of the required function with your chosen center. You can use any method.
• Use the Taylor Remainder Estimate to find the degree n of the Taylor polynomial Tn that will provide approximation to within 0.000001. Explain this step carefully.
• Compute the value of the Taylor polynomial Tn at the given point to find the approximation. (Use the degree n that you found in the previous step).
• Use a calculator to compute the actual value of the function at the given point. Check that your Taylor polynomial calculation indeed gives an approximation with the required precision. Note: work in radians! All Taylor series expansions use radians. You will have to use quantities like π 10 in your formulas. Make sure to take enough digits in the decimal expansion of π for the required precision to work.
Estimate sin(70◦ )
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