Trapezoidal rule Puns

Here :

https://www.macmillanlearning.com/studentresources/highschool/mathematics/rogawskiapet2e/additional_proofs/error_bounds_proof_for_numerical_integration.pdf

At page 5, he suddenly starts to use a function G.

Where does come that new function of "G(x) = f (x) − S(x) − q(x − x j−1)(x − x j ) " ?

What's the relationship of G(x) to f(x) and S(x) ?

import sympy as sym
from sympy.abc import x
def trapizodial_rule(b,a,n):
delta_x=(b-a)/(n)

    print(delta_x)
    a=a-delta_x
    subinterval=[]
    import math
    integrand= (math.e**x)*(x**3)+(1/(4*x**0.5))

    while a + delta_x <= b:
    a = delta_x + a
    subinterval.append(a)
    
    list1=[]
    for n in range(0,n):
    f1=integrand.subs(x,subinterval[n])
    list1.append(f1)
    
    list2=[]
    for n in range(1,len(list1)-1):
    f2=2*list1[n]
    list2.append(f2)
    
    middleterm = sum(list2)
    outerterm = list1[0] + list1[n+1]
    traprule=((delta_x/2)*(middleterm + outerterm))
    print(traprule)

the correct output is around 16 million but I got around 9 million I have another program that is almost exactly like this but the indexes for some of the lists are a little different because the subinterval list could have an even or an odd number of items. the other program gives the correct approximation for the integrand x^3 on the interval [2,4]. I'm up for any suggestions because the goal is to have any integrand to be approximated not just x^3

Is it okay to directly place the long summations in pm solver? need help from mods

I made a program in python for the trapezoidal rule and want to make a graph that shows the approximation error, approximation, and true value. The problem is that I don’t know the terminology for what that type of graph would be called. I was going to use matlibplot but I don’t know what the appropriate graph to use. I think it would be cool to make a graph but I need help. Any suggestions would be greatly appreciated.

How would I be able to do the trapezoidal rule for example in the integral of x^2+1/x+1 -tanx/x+2 from -4 to 5 with 3 sub intervals? It would have an undefined value when at the -1 value.

Hello,

I've got the trapezoidal rule where func is a function declared in the program.

// multi application trapezoidial rule
double trap(double a, double b, double (* func)(double), int segments){
    double integral  = func(a) + func(b);
    double h = (b-a)/n;

    for (int i = 1; i < n; i++){
        integral += 2 * func(a + i*h);
    }
    integral *= (b-a) / (2*n);
    return integral;
}

How can I convert this if I am given the following table of answers instead of a function:

I've stored this table in a struct of x and y.

https://imgur.com/a/BH8upKd

%Trapezoidal

f =@(x) 1-exp(-x);
I= trap(f(x),0, 4);
function I=trap(func,a,b,n,varargin)
if nargin<3
    error('at least 3 inputs required');
end
if ~(b>a)
    error('upper bound must be greater than lower')
end
if nargin<4||isempty(n)
    n=100;
end
    x=a;
    h=(b-a)/n;
    s=func(a,varargin{:});
    for i=1 : n-1
        x=x+h;
        s=s+2*func(x,varargin{:});
    end
    s=s+func(b,varargin{:});
    I=(b-a)*s/(2*n);
end

I keep getting this error

Array indices must be positive integers or logical values.

Error in sym/subsref (line 900)
            R_tilde = builtin('subsref',L_tilde,Idx);

Error in ICA_9>trap (line 24)
    s=func(a,varargin{:});

Error in ICA_9 (line 7)
I= trap(f(x),0, 4);

I honestly don't know what's wrong it can someone please point me to the right direction?

What does the k variable represent in the trapezoidal rule?

https://preview.redd.it/f1rr9rspjk451.png?width=511&format=png&auto=webp&s=9349834db098074f45655c891847b500ec3369cd

Thanks!

https://www.youtube.com/watch?v=rt7UfceiLtA

So I’m trying to do a question using the trapezoidal rule (question 2)

https://imgur.com/a/uP41PaT

I haven’t done a question that has dx in the numerator before (dx is usually written to the side like in question 1). I was wondering if we would just treat this as if the numerator is 1 or if not, how would I solve?

Update: I got it right.

I had a question like this on my test a few classes ago and was unsure how to do it. Thanks!

https://www.youtube.com/watch?v=rt7UfceiLtA