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
50.
Want to see the full answer?
Check out a sample textbook solutionChapter 9 Solutions
Calculus: Early Transcendentals (2nd Edition)
- PLEASE ONLY WRITE IN WOLFRAM MATHEMATICA. IT'S ABOUT LAGRANGE INTERPOLATIONarrow_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_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
- Consider 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_forward(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_forwardConsider the function f(x) = 1 3x+1 . We approximate f(x) by the Lagrange interpolating polynomial P₂ (x) at the points xo = 1, x₁ = 1.5 and.x₂ = 2. A bound of the theoretical error of this approximation at x = 1.8 is:arrow_forward
- euler 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_forwardPlease no written by hand solutions Numerical Methodsarrow_forward2. The Lorenz equations originating from models of atmospheric physics are given as follows: dr = 10 (y - 2) dt (2a) %3D dy 28r – y -rz (2b) dt dz ay - 2.6666672 (2c) dt with initial conditions r(0) = y(0) = 2(0) = 5. (a) Evaluate the eigenvalues of the Jacobian matrix at t = 0. Is the problem stiff? Estimate the maximum time step that can be selected to keep the solution stable when the fourth-order Runge-Kutta method is used. (b) Solve the given system to t = 50 using the fourth-order Runge-Kutta method. Set the time step to 0.1. Plot the solution. All three functions (2(t), y(t), z(t)) should be present on one plot. • Set the time step to 10 3 and 10 6. Plot r(t) obtained at the three time steps (the first one is 0.1 from above) on one plot. Describe the behaviour. How does the value of the time step affect the result? Set the time step to 10-6 and use the initial conditions r(0) = y(0) = 5.0 and 2(0) = 5.00001. Plot z(t) obtained at the two different sets of initial conditions on…arrow_forward
- Given 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_forward48arrow_forward• Suppose that we want to find a solution of the equation sin² (2) + 1-2x = 0, on the interval [0, π/2]. Is there a solution of the equation in this interval? How do you know?arrow_forward
- Operations Research : Applications and AlgorithmsComputer ScienceISBN:9780534380588Author:Wayne L. WinstonPublisher:Brooks Cole