Working with binomial series Use properties of power series, substitution, and factoring to find the first four nonzero terms of the Maclaurin series for the following functions. Give the interval of convergence for the new series (Theorem 9.4 is useful). Use the Maclaurin series
48.
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Calculus: Early Transcendentals (2nd Edition)
- (a) Derive the Gaussian quadrature formula of the form f f(x) dx = a f(h) + bf (2h) + cf (3h). Find the order of convergence of this formula. (b) Apply the formula you found on part (a) to Xi+k = x₁ + fik f(t, x(t))dt, to find an approximate formula for the numerical solution of the IVP (t) = f(t, x(t)), x(to) = xo. (c) Use the IVP (t) = 10 x(t) + 11t-5t²-1, x(0) = 0 to test the validity of the formula on the interval [0, 2] with h=0.25. [Hint: the exact solution is x(t) = (2) ft.arrow_forwardLet p=0.54617 and q=0.54601. Use four-digit to express p−q. Find the relative error of the previous part. If α and β are the roots of an equation, illustrate with an example howloss of significance can be avoided if two numbers are very close to each other.arrow_forward5. Using the following Truth Table, (a) Find out the maxterm expansion for the function F A В C F 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 And then simplify the maxterm expansion to get a minimum POS. (Use simplification theorems)arrow_forward
- 48arrow_forwardeuler and Chinese reaminder theorem x=[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14] x61=7 mod15 find x in rangearrow_forward6. (Stein's lemma) Suppose that X is normally distributed with mean u and variance o2. If g is a continuously differentiable function such that E{g(X)(X- µ)} and E{dg(X)/dx} both exist, prove that E{g(X)(X – u)} = o²E{dg(X)/dx}.arrow_forward
- 9. The solution to the rocurrence equation: T(2') - 3T(2)1, T(1)-1 is:arrow_forwardConsider a gas in a piston-cylinder device in which the temperature is held constant. As the volume of the device was changed, the pressure was mecas- ured. The volume and pressure values are reported in the following table: Volume, m Pressure, kPa, when I= 300 K 2494 1247 831 4 623 5 499 416 (a) Usc lincar interpolation to estimate the pressure when the volume is 3.8 m. (b) Usc cubic splinc interpolation to cstimate the pressure when the vol- ume is 3.8 m. (c) Usc lincar interpolation to cstimate the volume if the pressure is meas- ured to be 1000 kPa. (d) Usc cubic splinc interpolation to cstimate the volume if the pressure is mcasured to be 1000 kPa. 4.arrow_forwardGiven the following elliptic curve y2 ≡ x3 + 3x + 7 mod 13, answer the following: Find the addition of points (3,2) and (8,6) on this curve.arrow_forward
- Operations Research : Applications and AlgorithmsComputer ScienceISBN:9780534380588Author:Wayne L. WinstonPublisher:Brooks Cole