Using the third order Taylor’s series method, find the solution of the initial value problem:

at x = 0.1 taking h = 0.1

 From the following data, calculate the population in the year 1985:

 If 

f(x) = e^ax  , 

show that

Estimate the eigen values of the matrix:

using the Gerschgorin bounds. Also, draw the rough sketch of the region in which the eigen values lie.

Find the first term of the series whose second and subsequent terms are ,0,3,8 − 0,1 using difference table.

Using synthetic division method, check whether α = 3 is a root of the polynomial equation:

x⁴ +  x³ - 13x² - x + 12 = 0

Solve the system of equations:

x + 2y + z = 3

3x − 2y − z4 = −2

2x + 3y − z = −6

using Gauss-Elimination method.

 Given the data: 

f(3) = 168,

f(7) = 120

and f(9) = 72 

If  f(k) is estimated as 138 using the Newton’s form of the interpolating polynomial, then find the value of .k

Using Runge-Kutta fourth order method with h = 0.1, find an approximation value of y(0.1) for the initial value problem:

b) Evaluate the integral:

using composite trapezoidal rule with h = ,2.0 compare with the exact value.

) Use the Euler’s method to solve numerically the initial value problem: 

with h = 2.0 on the interval [0, 1].

Using the data sin (0.1) = 0.09983 and sin (0.2) = 0.19867, find an approximate value of sin (0.15) by Lagrange’s interpolation. Also obtain a bound on the truncation error.

 Use secant method to determine the root of the equation cos x - xe^x = 0. Take the initial approximation as x₀ = 0, x₁ = 1 and perform two iterations of the method.

 Find the inverse of the matrix:

using LU decomposition method with u₁₁ = u₂₂ = u₃₃ = 1.

Calculate the nth divided difference of 1/x, on the nodal points X₀ , X₁ , ......., X_n .

Using Newton-Rapshson method, find an iterative formula to compute the reciprocal of a natural number N.

Which of the following statements are true and which are false? Give a short proof or a counterexample in support of your answer:

i) The equation x³ 4x 16 = 0 has a root in the interval [3, 4].

ii) The order of convergence of the secant method is 0.62.

iii) For the system of linear equations:

5x + y + 2z = 34

4y − 3z = 12

10x − 2y + z = −4

the matrix is diagonally dominant.

iv) The numerical method:

is relatively stable.

 v) The method:

converges to 1.5 for any choice of initial approximation.

 The sugar level of 6 patients were recorded before and after taking a new drug:

I can definitely do that for you! Here's the table:

Using t-test and a significance level of 5% can you determine if the new drug causes significant reduction in sugar level. You must write Ho and H₁ clearly and explain your results.

The following table shows the data on mortality rate of plants on use of a new fertilizer. Use chi-square or any other test to determine, if the fertilizer has any effect on enhancing the living rate.

explain your result

A company has the following production and sales data.

Sure, here is the table you provided:

(a) Construct a scatter plot (diagram) for the given data using a spreadsheet package.

(b) Find the best linear regression line, assuming that sales in an independent variable and production is a dependent variable. Explain your answer.

Consider the following data of sales of wheat by a farm house in a week:

Sure, here is the text from the image in a table format:

Find the moving averages of length 4 and 5. Plot these moving averages using spreadsheet.