Quantitative Methods Answers

use Euler’s method to obtain a four-decimal approximation of the indicated value. First use h = 0.1 and then use h = 0.05. Find an explicit solution for each initial-value problem

Use Lagrange polynomial estimate (2) for the given data.

x -2 -1 0 4

f(x) -2 4 1 8

Find the values of the first and second derivatives of y = 4x2+2x-1 for

x=1.25 from the following table using Lagrange’s interpolation

formula.

x : 1 1.5 2 2.5

y : 5 11 19 29

Find the root of the equation x^3+x^2-100=0 using iteration method

Is there any relation between the degree of the polynomial and the number of data points in polynomial curve fitting? If yes, write the relation. If no, write a brief (no more than two lines) explanation.

Consider three-points Gaussian quadrature. The three points on the x-axis and the associated weights are:

x₁ =0, x₂ = -"\\sqrt{3\/5}" , x₃ = "\\sqrt{3\/5}" ; w₁ = 8/9, w₂ = w₃ = 5/9

Apply this 3-points Gaussian quadrature to the polynomial

P(x) = 2x⁵-x⁴+x²-1

Evaluate integral of a = 0 and b =π\2 sin(t)dt by applying Simpsons rule with four equal intervals

a.     The following is a list of hospitals in Kenya.

ID NO

Name

Address

01

St. Luke Hospital

1140, Kisumu

02

Matter Hospital

1256, Nakuru

03

KNH

1000, Nairobi

04

Thika Level 5

1416, Thika

05

Consolata General Hospital

3120, Mombasa

06

Eldoret Referral

1127, Eldoret

07

Children Psychiatric Hospital

2123, Kisumu

i.                   A sample of three hospitals is to randomly selected. The random numbers are 09, 02, 05, 10, 00, 07. Which hospitals are included in the sample?                                                                                                                                 (3 marks)

ii.                 A sample is to contain every second location. We select 01 as the starting point. Which hospitals will be included in the sample?                  (3 marks)

Let f(x) =x²-a. Show that the Newton-raphson method leads to the recurrent

Use Runge-Kutta’s method of order two to determine approximate solutions for y(0;1) and y(0;2)

y′ = f(t,y) = ycos(t) y(0) = 2