Quantitative Methods Answers

a.Calculate forward and backward difference approximations of 𝑂(ℎ) and 𝑂(ℎ2) and

central difference approximations 𝑂(ℎ2) and (𝑂(ℎ4) for the first derivative of 𝑦 =

sin 𝑥 at 𝑥 = 𝜋

4

and the step size is 𝜋

12

b.Estimate the percentage relative error, 𝜀𝑡 for each approximation. 

find the numerical value of y'(10) for y = sinx given that:

sin 0 = 0.000 , sin 10 = 0.1736 , sin 20 = 0.3420 , sin 30 = 0.5000 , sin 40 =0.6428

Using n=6 integrate the function in the interval [1, 2] 

(i) Complete the table

 x        1       
f(x) 

(ii) Use trapezoidal rule.

(iii) Simpsons rule to evaluate the integral.

(i) Using 5 points x=0, 1,2,3,4 complete the table

 x     0        1       2        3        4 
f(x) 

(ii) Find "the f\u2019(0.5)."

(iii) Find the exact value if

"f\u2019(x)=(-a)\/100 e^(-ax\/100) cos\u2061\u30163ax-3a\u3017 e^(-ax\/100) sin\u20613ax"

y is a function of x satisfying the equation xy″ + ay′ + (x – b) y = 0, where a and b are integers. Find the values of constants a and b if y is given by the following table:

x: 0.8 1 1.2 1.4 1.6 1.8 2 2.2

y: 1.73036 1.95532 2.19756 2.45693 2.73309 3.02549 2.3333 3.65563

Prove that

d/dx (Yx) = 1/h (Yx+h – Yx–h) –1/2h (Yx+2h – Yx–2h) +1/3h (Yx+3h – Yx–3h) – ......... ...

Write an algorithm to compute the value of a function using Lagrange’s interpolation.

Find a Lagrange’s interpolating polynomial for the data given below:

x0 = 1, x1 = 2.5, x2 = 4 and x3 = 5.5

f(x0) = 4, f(x1) = 7.5, f(x2) = 13 and f(x3) = 17.5

Also, find the value of f(5).

Determine by Lagrange’s formula, the percentage number of criminals under 35 years:

Age % number of criminals

under 25 years 52

under 30 years 67.3

under 40 years 84.1

under 50 years 94.4