a) [3 points] Give the Taylor's series for the function f(x)=cosx about the point 0 , writing explicitly all terms of the Taylor polynomial t4(x) of degree 4 and the remainder R6. Note that there are only 3 non-zero terms in t4(x). Indicate t4(x) and R6. Note that the remainder involves x and an unknown c. Indicate the (smallest) interval where c lies. b) [5 points] Using t4(x) approximate cos1. Indicate the approximate value in 5 significant decimals. Using R6, give an upper bound (as sharp as you can) for the (absolute value of the) error of the approximation to cos 1 . Explain how you got the bound. c) [5 points] Give the Taylor's series for f(x)=cosx about the point 0 , in a form so that the (2n) th term (for a general n ) of the Taylor polynomial t2n(x) of degree 2n and the remainder R2n+2 are shown explicitly. Using R2n+2, give, in terms of n and x, an upper bound (as sharp as you can) for the (absolute value of the) error in the approximation to cos x arising from t2n(x). d) [12 points ] Assume you have an accurate way of calculating the Taylor's series in (c), up to any n. How would you use the Taylor's series in (c) to obtain an efficient and accurate approximation to cosx for (i) x=2 ? (ii) x=4 ? (iii) x=6 ? (iv) some large x( e.g. x=63) ? (v) some negative x ? Generalize for any x and explain. This generalization can be given as pseudo-code of one or more if-then-else statements. What is the maximum number of terms (or what is the maximum n ) that should be used to keep the absolute value of the remainder R2n+2 below 1016 for any x ? Explain. e) [10 points ] In the form of pseudo-code (a for-loop), give a way of calculating the Taylor's polynomial in (c), without using powers (exponentiations) and without using factorials. (You can use additions, subtractions, multiplications and divisions.)