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Question: a zosz2z 31 0 2031 and 1213 b...

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(a) zosz-2z? + 3-1 = 0, ..20.31 and [1.2.1.3]. (b) 2r cos(2r) (x 2)2-, 2,3 and 3.4. (c) (1-2)s_ In z = 0, [L2] and |e4). 13. Find the third Taylor polynomial Fle) for the function f(r) = vr about ro = 0. Approximate v675 and v/1.5 using PG), and find the actual errors. 14, Let g(z)-1 -r2-|s+4). Pro e that for zo € po,21 the se- . . converges to the unique fixed point quence 1h +1-g(n), k = 0, 1, 2 15. Obtain a bound for the error in estimating the function f(x) sin(/2 0,1 by the Lagrange interpolation polynomial agreeing with f at In Lagrange interpolation. the error tenn E.(z) has the form For n-2,o+h and o+2h, prove that if To SS then 2h Hint. Use the substitutions t = z-ri, t + h = r-zo, and t-h =z-z2. and the function e(t) -t2 on the interval-h stSh. Set e (t) o and solve for t in terms of h 16. A car traveling along a straight road is clocked at a number of points. The data from the observations are given in the following table, where the time is in seconds, the distance is in feet, and the speed is in feet per second. Time 0358 13 Distance0 225 383 623 983 72 780 74 (a) Use a Hermit polynomial to predict the position of the car and its speed when t 10 s (b) Use the derivative of the Hermit polynomial to determine whether the car ever exceeds a 55 mi/h speed limit on the road. If so, what is the first time the car exceeds this speed? (c) What is the predicted maximum speed for the car? 17. (a) Use the Hermit interpolat ing method to construct an approximate polynomial for the following data 0.0 0.223633622.1691753 1.0 0.65809197 1 2.0466965 (b) Use the polynomial constructed in (a) and the given value of z to approximate f(x) and calculate the absolute error
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