Question

I just need help with C and D. I tried (0,1) and (-1,0) for C and 1/6! for D

(4 points) Consider the error in using the approximation sin:) on the interval (-1,1]. (a) Reasoning informally, on what interval is this approximation an overestimate? (0,1) An underestimate? [-1.0) (For each, give your answer as an interval or list of intervals, e.g., to specify the intervals -0.25 <2<0.5 and 0.75<2<1, enter [-0.25, 0.75), (0.75,1) Enter none if there are no such intervals.) (b) Use the Error Bound for Taylor Polynomials to find a good smallest bound for the error in approximating sin(2) with 2 on this interval: error bound = 1/6 Now, consider the error in using the approximation sin(2) x+ on the same interval. (c) Reasoning informally, on what interval is this approximation an overestimate? nono An underestimate? none (For each, give your answer as an interval or list of intervals, e.g., to specify the intervals -0.25 <<0.5 and 0.75 <3 < 1, enter (-0.25, 0.75), (0.75, 1) Enter none if there are no such intervals.) on this interval: (d) Use the Error Bound for Taylor Polynomials to find a good smallest bound for the error in approximating sin(C) with 2+ error bound = 0.0001985

Answer 1

(c) overestimated [-1,0) and underestimated from (0,1] since sin1=0.84 while x-x^3 at x=1 is 0.8333 so underestimated in (0,1] similarly sin(-1)=-0.84 while x-x^3 at x=-1 is -0.8333 so 0verestimated [-1,0).

(d) smallest error bound= 1/5!

Since sinx=x+0x^2+(x^3)/6 +0x^4 . So R4(x)=1/5!