Numerical Analysis (1) Let f(x)= ex2 + 4x³ . Find the third Taylor polynomial of f about xo = 2. Then use this polynomial to approximate f(2.03), f'(2.03)
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- Number of terms What is the minimum order of the Taylor polynomial required to approximate the following quantities with an absolute error no greater than 10-3? (The answer depends on your choice of a center.) cos (-0.25)Approximating sin x Let ƒ(x) = sin x, and let pn and qn be nth-order Taylor polynomials for ƒ centered at 0 and p, respectively.a. Find p5 and q5.b. Graph ƒ, p5, and q5 on the interval [-π, 2π]. On what interval is p5 a better approximation to ƒ than q5? On what interval is q5 a better approximation to ƒ than p5?c. Complete the following table showing the errors in the approximations given by p5 and q5 at selected points.d. At which points in the table is p5 a better approximation to ƒ than q5? At which points do p5 and q5 give equal approximations to ƒ? Explain your observations.Number of terms What is the minimum order of the Taylor polynomial required to approximate the following quantities with an absolute error no greater than 10-3? (The answer depends on your choice of a center.) ln 0.85
- Number of terms What is the minimum order of the Taylor polynomial required to approximate the following quantities with an absolute error no greater than 10-3? (The answer depends on your choice of a center.) e-0.5Two parts to this question. Find the second-degree Taylor polynomial for f(x)=√x f(x)=x centered at x=4 p2(x)= Use p2(x) to approximate squareroot(4.1)A ruptured pipe in a North Sea oil rig produces a circular oil slick that is y meters thick at a distance x meters from the rupture. Turbulence makes it difficult to directly measure the thickness of the slick at the source (where x = 0), but for x > 0, it is found thaty =0.5(x² + 3x)/x³ + x² + 4xAssuming the oil slick is continuously distributed, howthick would you expect it to be at the source?
- integral the x^2 e^-xDynamic profit function is P(t)= 2 - (t - 5) x ln (t + 1), here t is measured in years, and P is measured in hundreds of euros. a) use marginal analysis to estimate how fast company's profit was growing initially b) use Taylor formula to write down square approximation of the given profit function around t=2. Round coefficient of the Taylor polynomial to 3 decimals c) use results from step b) to estimate total company's profits between first and fourth years of operation d) estimate average company's profit between first and fourth years of operation e) use initial function to estimate total company's profits between first and fourth years of operation. Compare results with step c) the question is not graded, as the exam was yesterday, I want to check my answersNumerical Method Evaluate x2/3 from x0=1.2 and xn=2.4 of order 3 using Simpson's 3/8 R.
- Consider the function f(x) = 3√x. (a.) Approximate f(x) with T2(x), the second degree Taylor polynomial, centered ata= 1. You do not need to expand/simplify the polynomial. b.) Use Taylor’s Inequality, to estimate the accuracy of your approximationwhenxis within the interval 0.5≤ x≤ 1.5. Round the maximum error|Rn(x)|to 3 decimal places.using a graphing calculator or computer algebra system. Plot f (x) = cos(x2) sin x for 0 ≤ x ≤ 2π. Then illustrate local linearity at x = 3.8 by choosing appropriate viewing rectanglesIntegration by Rationalizing Substitution