MATH 455 Instructions Homework 3 Written assignments are graded based on quality of work and fully worked out solutions Points will be deducted for incomplete reasoning and disorganized work even if your answers are correct 1 Chebyshev Interpolation a List the Chebyshev interpolation nodes x0 xn 1 in the given interval i 1 1 n 6 ii 2 2 n 4 iii 4 12 n 6 iv 0 3 0 7 n 5 a the form for f cid 48 cid 48 x0 form b Provide the upper bound for x x0 x x1 x x2 x xn 1 on the intervals and nodes used in part c Use Chebyshev interpolation to nd a 5th degree interpolating polynomial P5 x on 1 1 for f x ex Use the interpolation error formula to nd a worse case estimate for the error ex P5 x valid for all x in 1 1 d Answer the same question as in part c but for the interval 0 6 1 0 2 Numerical Di erentiation a Apply the method discussed in class to derive the 2nd order backward nite di erence formula for f cid 48 cid 48 x0 of Af x0 Bf x0 h Cf x0 2h Df x0 3h f cid 48 cid 48 x0 O h2 Speci cally determine the values of A B C D b Use your result in part a and substitute h for h to determine the 2nd order forward nite di erence formula c Apply the method discussed in class to derive the 4th order centered nite di erence formula for f cid 48 x0 of the Af x0 2h Bf x0 h Cf x0 Df x0 h Ef x0 2h f cid 48 x0 O h4 Speci cally determine the values of A B C D and E d Assuming a round o error of magnitude 10 16 determine the optimal step size h associated with the O h4 centered nite di erence formula for f cid 48 x0 given by f x0 2h 8f x0 h 8f x0 h f x0 2h f cid 48 x0 O h4 12h Use desmos to graph the total error E the error due to round o Er and the error due to truncation ET 1