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Using Taylor polynomials of ln (x + 1), calculate an approximation of ln (2) with an error less than 0.1
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- Is there a number x such that ln x=2 ? If so, whatis that number? Verify the result.Original Question: ) Use the Taylor polynomial of ln(x) with degree 3, centered at 1 to approximate ln 1.1, then bound the error. Find an upper bound for |Rn(x)| at the x value that yields the requested approximationa. Prove that ƒ(x) = x - ln x is increasing for x> 1. b. Using part (a), show that ln x < x if x > 1.
- Determine the fourth Taylor polynomial of f(x) = ln(1 - x)at x = 0, and use it to estimate ln(0.9).Say that the function f(x) is x for -π < x < 0 and zero for 0 < x < π. Find the a0 coefficient for the Fourier Series.Using the remainder formula for the Taylor polynomial approximation, estimate the error in your approximation to sin N°.
- Compute the Fourier series of the indicated functions for x ∈(−L, L):f (x) = x^2Compute the Fourier series of the indicated functions for x ∈(−L, L):f (x) = e^xUse the second Taylor polynomial of f(x) = ln x at x = 1 toestimate ln 0.8.Approximate root 7.1 with a Taylor polynomial of degree 2 centered at x=9. (Enter either an exact answer or at least 6 decimal places.)
- What is the fourier series of the piecewise: f(x) = { x^3+2, -L ≤ x ≤ L f(x+2L), other partsSolve the equation f(x) = x2 + ln x - 3 = 0 using the Newton–Raphson metho. Detail the steps of iterative calculations. Discuss the results. (Convergence to 8 decimal places).Show that f is continuous on (−∞, ∞). f(x) = 1 − x2 if x ≤ 1 ln(x) if x > 1 On the interval (−∞, 1), f is ---Select--- a polynomial an exponential a root a logarithmic a rational function; therefore f is continuous on (−∞, 1). On the interval (1, ∞), f is ---Select--- a polynomial an exponential a root a logarithmic a rational function; therefore f is continuous on (1, ∞). At x = 1, lim x→1− f(x) = lim x→1− = , and lim x→1+ f(x) = lim x→1+ = , so lim x→1 f(x) = . Also, f(1) = . Thus, f is continuous at x = 1. We conclude that f is continuous on (−∞, ∞).
