By Richard L Burden; J Douglas Faires

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Suppose f is to be evaluated at x0 in (a, b), but instead of computing the actual value f (x0 ), the approximate value, f˜(x0 ), is the actual value of f at x0 + , that is, f˜(x0 ) = f (x0 + ). a. 8 to estimate the absolute error |f (x0 ) − f˜(x0 )| and the relative error |f (x0 ) − f˜(x0 )|/|f (x0 )|, assuming f (x0 ) = 0. b. If = 5 × 10−6 and x0 = 1, ﬁnd bounds for the absolute and relative errors for i. f (x) = ex ii. f (x) = sin x c. Repeat part (b) with = (5 × 10−6 )x0 and x0 = 10. c. 26. 27.

B. 5). c. Repeat part (a) using x0 = 1. d. Repeat part (b) using the polynomial from part (c). √ Find Taylor polynomial P3 (x) for the function f (x) = x + 1 about x0 = 0. 5 using P3 (x), and ﬁnd the actual errors. Find the second Taylor polynomial P2 (x) for the function f (x) = ex cos x about x0 = 0. a. 5). 5)| using the error formula, and compare it to the actual error. b. Find a bound for the error |f (x) − P2 (x)| in using P2 (x) to approximate f (x) on the interval [0, 1]. 1 1 c. Approximate 0 f (x) dx using 0 P2 (x) dx.

08206. 15 K = 17◦ C. 08206) In the laboratory, it was found that T was 15◦ C under these conditions, and when the pressure was doubled and the volume halved, T was 19◦ C. Assume that the data are rounded values accurate to the places given, and show that both laboratory ﬁgures are within the bounds of accuracy for the ideal gas law. 3 Algorithms and Convergence The use of an algorithm is as old as formal mathematics, but the name derives from the Arabic mathematician Muhammad ibn-Mˆsâ al-Khwarârizmî (c.