Create a MATLAB script to find the first and second derivati

Create a MATLAB script to find the first and second derivative of given function using Forward, Backward, central and Taylor numerical schemes. Test your code using the following functions: f(x) = xe^x + 3x^2 + 2x - 1 and find f\'(3) and f\"(3) for with h = 0.1, 0.01 and 0.001 Approximate y\'(1) and y\"(1) using the following table

Solution

function []=forward_First(x,h)

y0=x*exp(x)+3*x^2+2*x-1;

y1=(x+h)*exp(x+h)+3*(x+h)^2+2*(x+h)-1;

y_forward=(y1-y0)/h;

y0=x*exp(x)+3*x^2+2*x-1;

y1=(x-h)*exp(x-h)+3*(x-h)^2+2*(x-h)-1;

y_backward=(y0-y1)/h;

y0=(x+h)*exp(x+h)+3*(x+h)^2+2*(x+h)-1;

y1=(x-h)*exp(x-h)+3*(x-h)^2+2*(x-h)-1;

y_central=(y0-y1)/(2*h);

fprintf(\'\ Derivate of function f at 3 with h= %0.4f is \ \ \\t using Forward Difference Expansion : %0.4f \ \ \\t using BackwardDifference Expansion : %0.4f \ \ \\t using Central Difference Expansion : %0.4f\ \',h,y_forward,y_backward,y_central);

end

OUTPUT:

Derivate of function f at 3 with h= 0.1000 is

    using Forward Difference Expansion : 105.8704

    using BackwardDifference Expansion : 95.2159

    using Central Difference Expansion : 100.5431

----

forward_First(3,0.01)

Derivate of function f at 3 with h= 0.0100 is

    using Forward Difference Expansion : 100.8763

    using BackwardDifference Expansion : 99.8120

    using Central Difference Expansion : 100.3442

----

Derivate of function f at 3 with h= 0.0010 is

    using Forward Difference Expansion : 100.3954

    using BackwardDifference Expansion : 100.2890

    using Central Difference Expansion : 100.3422

-----------------Second derivative------

function []=forward_second(x,h)

y0=x*exp(x)+3*x^2+2*x-1;

y1=(x+h)*exp(x+h)+3*(x+h)^2+2*(x+h)-1;

y2=(x+2*h)*exp(x+2*h)+3*(x+2*h)^2+2*(x+2*h)-1;

y_forward=(y2-2*y1+y0)/h^2;

y0=x*exp(x)+3*x^2+2*x-1;

y1=(x-h)*exp(x-h)+3*(x-h)^2+2*(x-h)-1;

y1=(x-2*h)*exp(x-2*h)+3*(x-2*h)^2+2*(x-2*h)-1;

y_backward=(y0-2*y1+y2)/h^2;

y2=(x+h)*exp(x+h)+3*(x+h)^2+2*(x+h)-1;

y1=(x-h)*exp(x-h)+3*(x-h)^2+2*(x-h)-1;

y0=x*exp(x)+3*x^2+2*x-1;

y_central=(y2-2*y0+y1)/(h^2);

fprintf(\'\ Derivate of function f at 3 with h= %0.4f is \ \ \\t using Forward Difference Expansion : %0.4f \ \ \\t using BackwardDifference Expansion : %0.4f \ \ \\t using Central Difference Expansion : %0.4f\ \',h,y_forward,y_backward,y_central);

end

OUTPUT:

>> forward_second(3,0.001)

Derivate of function f at 3 with h= 0.0010 is

    using Forward Difference Expansion : 106.5483

    using BackwardDifference Expansion : 601840.5127

    using Central Difference Expansion : 106.4277

(b)

First derivative

>> forward_second(1,0.001)

Derivate of function f at 3 with h= 0.0010 is

    using Forward Difference Expansion : 14.1657

    using BackwardDifference Expansion : 80591.1157

    using Central Difference Expansion : 14.1548
>>

Second derivative

>> forward_second(1,0.001)

Derivate of function f at 3 with h= 0.0010 is

    using Forward Difference Expansion : 14.1657

    using Central Difference Expansion : 14.1548
>>