X 1. Let f(x) = sin a. Suppose that |x| 0.1. Find a reasonable upper-bound...

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X 1. Let f(x) = sin a. Suppose that |x| ≤ 0.1. Find a “reasonable” upper-bound on the error in approxi- mating f(x) by T³½³(x; 0). b. Suppose that |x| ≤0.1. By using "reasonable" bounds, find the smallest n such that you can prove that the error in approximating f(x) by T (x; 0) is less than or equal to 0.0001. c. Suppose that we want T³ ³ (x; 0) to approximate f(x) to within 0.0001. By using "reasonable" bounds, find the largest 8 > 0 such that this accuracy will be achieved provided that |x| ≤ d. 3. Let f(x) = sin(5x). a. Suppose that |x1|≤ 0.1. Find a "reasonable" upper-bound on the error in approx- imating f(x) by T¾³ (x; 1). 4.3. ERROR IN APPROXIMATION 439 b. Suppose that |x1| ≤ 0.1. By using "reasonable" bounds, find the smallest n such that you can prove that the error in approximating f(x) by Tr(x; 1) is less than or equal to 0.0001. c. Suppose that we want T½ ½ (x; 1) to approximate f(x) to within 0.0001. By using "reasonable" bounds, find the largest 5 >0 such that this accuracy will be achieved provided that |x − 1| ≤ 8. X 1. Let f(x) = sin a. Suppose that |x| ≤ 0.1. Find a “reasonable” upper-bound on the error in approxi- mating f(x) by T³½³(x; 0). b. Suppose that |x| ≤0.1. By using "reasonable" bounds, find the smallest n such that you can prove that the error in approximating f(x) by T (x; 0) is less than or equal to 0.0001. c. Suppose that we want T³ ³ (x; 0) to approximate f(x) to within 0.0001. By using "reasonable" bounds, find the largest 8 > 0 such that this accuracy will be achieved provided that |x| ≤ d. 3. Let f(x) = sin(5x). a. Suppose that |x1|≤ 0.1. Find a "reasonable" upper-bound on the error in approx- imating f(x) by T¾³ (x; 1). 4.3. ERROR IN APPROXIMATION 439 b. Suppose that |x1| ≤ 0.1. By using "reasonable" bounds, find the smallest n such that you can prove that the error in approximating f(x) by Tr(x; 1) is less than or equal to 0.0001. c. Suppose that we want T½ ½ (x; 1) to approximate f(x) to within 0.0001. By using "reasonable" bounds, find the largest 5 >0 such that this accuracy will be achieved provided that |x − 1| ≤ 8.