Need solution for 4.6 using python. Questions are attached as a screenshot.
Using a step size 0 .. Compare your results with the true value of the derivative. Interpret your results on the (a) Prove that this Maclaurin series expansion is a special case of basis of the remainder term of the Taylor series expansion the Taylor series expansion ((Eq. (4.7)) with 1, = 0 and h= . (b) Use the Taylor series to estimates 48 Use a centered difference approximation of o) to estimate at X for the second derivative of the function examined in Pros. 45. Per X -0.2. Employ the zero-, first, second, and third-order form the evaluation at r-2 using step sizes of h -0.25 and 0.125 . versions and compute the p F for each case. Compare your estimates with the true value of the second deriva .2 The MacLaurin series expansion for cos x is tive. Interpret your result on the basis of the remainder term of the Taylor series expansion. cos The Stefan-Boltzmann law can be employed to estimate the rate of radiation of energy H from a surface, as in Starting with the simplest version, cos I = 1, add terms one at a Aes time to estimate cos(/3), after each new term is added, compute the true and approximate percent relative errors. Use your pocket where H is in watts, A - the surface area (onT), e - the emissivity calculator to determine the true value. Add terms until the absolute that characterizes the emitting properties of the surface (dimension value of the approximate error estimate falls below an error crite less), a a universal constant called the Stefan-Boltzmann con rion conforming to two significant figures. stant (= 5.67 x 10 'W mK), and T = absolute temperature Perform the same computation as in Prob. 4.2, but use the (K). Determine the error of H for a steel plate with a 0.15 m MacLaurin series expansion for the sinx to estimate sincm/3) 0.90, and T - 650 + 20. Compare your results with the exact error. Repeat the computation but with T = 650 + 40. Interpret sin X your results. 4.10 Repeat Prob. 4.9 but for a copper sphere with 4 The Maclaurin series expansion for the arctangent of x is de-radius = 0.15 + 0.01 m, e = 0.90 + 0.05, and T = 550 20. fined for frls Recall that the velocity of the falling parachutist can be com puted by [Eq. 10). (-1)" aretan B (a) Write out the first four terms (n = 0. (b) Starting with the simplest version, arctanx ", add terms one Use a first-order error analysis to e stimate the error of V at 6, if at a time to estimate arctant /6). After each new term is added 9.81 and m - 50 but 12.5 compute the true and approximate percent relative errors. Use 4.12 Repeal Prob. 4. 11 with 8 - 9.81, I = 6, C - 125 15, and your calculator to determine the true value. Add terms until them - 50 + 2. absolute value of the approximate error estimate falls below an 4.13 Evaluate and interpret the condition numbers for error criterion conforming to two significant figures (a) S ) = VAR - 1 for x = 100001 4. Use zero- through third-order Taylor series expansions to (b) fix forx predict f3) for for A - 300 (x) - 25 - 6 + 7 - 88 for = 0.00 using a base point at x in X Compute the true percent relative error (0) FX forx = 1000m e, for each approximation + cos I 4.6 Use zero- through fourth-order Taylor series expansions to pre- 4.14 Employing ideas from See. 4.2, derive the relationships from dict .5) for frIn X using a base point atx Compute the Table 4.3. true percent relative error by for each approximation. Discuss the 415 Prove that Eq. (4.4) is exact for all values of x if meaning of the results. are br + C
