Let f(x) = cos(x). Approximate the integral I = f(x) dx in three different ways, each...

Transcribed Image Text:

Let f(x) = cos(x). Approximate the integral I = f(x) dx in three different ways, each with n = 4: (a) a Right Riemann Sum; (b) the Trapezoidal Rule; and (c) Simpson's Rule. Give each answer correct to six decimal places. (a) R4 = (b) T4 = (c) S4= It can be shown that for each i in [0, 1], f" (x)| 4 and f(4) (x)| 43. Combine these facts with the error-control inequalities discussed in class to complete the following guarantees concerning the absolute errors in the approximations from parts (b) and (c). Give each answer correct to six decimal places. (d) |I-T| vork2/2022W2_MATH_101A_ALL_2022W2/WW05/16/?effective User=16LSBJUN2C06 (e) |I - S4 Using the facts given for parts (d) and (e), and the error formulas, find the smallest integer n for which we can guarantee that the approximation T is within 10-5 of the exact value I. Repeat for Sn. (f) For Tn: n= WeBWork: 2022W2 MATH 101A ALL 2022W2 : WW05: 16 (g) For Sn: n = Let f(x) = cos(x). Approximate the integral I = f(x) dx in three different ways, each with n = 4: (a) a Right Riemann Sum; (b) the Trapezoidal Rule; and (c) Simpson's Rule. Give each answer correct to six decimal places. (a) R4 = (b) T4 = (c) S4= It can be shown that for each i in [0, 1], f" (x)| 4 and f(4) (x)| 43. Combine these facts with the error-control inequalities discussed in class to complete the following guarantees concerning the absolute errors in the approximations from parts (b) and (c). Give each answer correct to six decimal places. (d) |I-T| vork2/2022W2_MATH_101A_ALL_2022W2/WW05/16/?effective User=16LSBJUN2C06 (e) |I - S4 Using the facts given for parts (d) and (e), and the error formulas, find the smallest integer n for which we can guarantee that the approximation T is within 10-5 of the exact value I. Repeat for Sn. (f) For Tn: n= WeBWork: 2022W2 MATH 101A ALL 2022W2 : WW05: 16 (g) For Sn: n =