Quantitative Methods Answers

Find the minimum number of intervals required to evaluate int (e^(-x^2 +1))dx with an accuracy of 0.5 × 10^(-4) , by using Trapezoidal rule.

Calculate the fourth degree Taylor polynomial about x0 = 0.5 for the function f(x) = sin^(-1) √x

Using the classical fourth order Runge-Kutta method, find the approximate value of y(0.6) for the initial value problem dy/dx = sin xy ; y(0) =1 with the step size h =0.2

Perform iterations of Newton-Raphson method to approximate a root of the equation f(x) = x^4 -x^3 +x -1 =0 , until the roots at successive iterations are closer than 10^(-5). How many iterations do you need for this much accuracy.

bureau of labor statistics want to use statistics to determine whether there is a difference between 2 countries on some measure of labor or between time periods of one country.hourly compensation in US dollars for manufacturing production workers in country A in 2015 and 2019 were $25.96 and $28.29 respectively.Hourly compensation for country B were $19.26 in 1995 and $23.89 in 2004.actual hours worked in country A in 2015 were 36.1and 35.5 in 2019.country B 37.1 in2015 and 35.9 in 2019. was the hourly labor cost higher in country A in 2019 than country B in 2019?

Perform 6 iteration to find zero of the equation: x+3=e^x using bisection algorithm. How many steps of the algorithm are needed to compute toot of the equation?

Civil and transportation engineers must often estimate the future traffic flow on roads and bridges to plan for maintenance or possible future expansion. The following data gives the number of vehicles (in millions) crossing a bridge each year for 10 years. Fit a cubic polynomial to the data and use the fit to estimate the flow in year 2000.
Year 1990 1991 1992 1993 1994 Vehicle flow (millions) 2.1 3.4 4.5 5.3 6.2
Year 1990 1991 1992 1993 1994 Vehicle flow (millions) 6.6 6.8 7 7.4 7.8

The interpolating polynomial of degree \\(\\leq n\\)with the nodes\\(x_{0}, x_{1},\\cdots, x_{n}\\) can be written as

9.One of these describes the Lagrange’s interpolating polynomial P(x)
a.\\(p(x) = L_{1}(x)f_{0} + L_{2}(x)f_{1} + L_{3}(x)f_{2} + L_{4}(x)f_{3}\\)
b.\\(P(x) = L_{0}(x)f_{0} + L_{1}(x)f_{1} + L_{2}(x)f_{2} + L_{3}(x)f_{3}\\)
c.\\(P(x) = L_{0}(x)f_{3} + L_{1}(x)f_{2} + L_{2}(x)f_{1} + L_{3}(x)f_{0}\\)
d.\\(P(x) = L_{0}(x)f_{3} + L_{1}(x)f_{2} + L_{2}(x)f_{1} + L_{3}(x)f_{0}\\)

10.One of these is a method of solving system of linear equations:
a.Square Method
b.Inverse method
c.Lagrange Method
d.Transformation Method

7.Iterative methods of the solutions of systems of equations are ________________
a.finite
b.sequential
c.infinite
d.non-sequential

8.Stirling\'s formula for interpolation is given by
a.\\(P_n (x) = f (x_0) + \\frac{s}{2}[df_\\frac{1}{2} + d_\\frac{-1}{2}] + \\frac{s^2}{2}d^2f_0\\)
b.\\(P_n (x) = f (x_0) + \\frac{s}{2}[df_\\frac{1}{2} + d_\\frac{-1}{2}] + \\frac{s^2}{2!}d^2f_0\\)
c.\\(P_n (x) = f (x_0) + \\frac{s - 1}{2}[df_1+ d_\\frac{-1}{2}] + \\frac{s^2}{2!}d^2f_0\\)
d.\\(P_n (x) = f (x_0) + \\frac{s}{2}[df_\\frac{1}{2} + d_\\frac{-1}{2}] + \\frac{s^2}{2!}df_0\\)