Assist me please

Paros

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I have been battling with these exercise please help
1.using the nodal points x0=0,x1=1/3 and x2=1 construct lagrange interpolation polynomial for the function f(x) =sin(π x/2). Hence evaluate f(3/4).

And
2.compute the roots of the equations x-sin18x=0.75 at x0=5 using Newton Raphson iterative formulae. Hint stop after six iterations.
 

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Subhotosh Khan

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I have been battling with these exercise please help
1.using the nodal points x0=0,x1=1/3 and x2=1 construct lagrange interpolation polynomial for the function f(x) =sin(π x/2). Hence evaluate f(3/4).

And
2.compute the roots of the equations x-sin18x=0.75 at x0=5 using Newton Raphson iterative formulae. Hint stop after six iterations.
12500
What have you tried? Where are you exactly stuck?
 

Otis

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Paros

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The issue I have with number one is am nt sure of what i solved and the formulae am using to find f(3/4) not sure too..
And quetion 2 is I cant simplify fx into f prime. Please work it out let me learn..
 

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HallsofIvy

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For \(\displaystyle f(x)= sin(\pi x/2)\), \(\displaystyle f(0)= sin(0)= 0\), \(\displaystyle f(1/3)= sin(\pi/6)= \frac{1}{2}\), and \(\displaystyle f(1/2)= sin(\pi/4)= \frac{\sqrt{2}}{2}\). The Lagrange Polynomial then is \(\displaystyle 0\frac{(x- \pi/6)(x- \pi/4)}{(0-\pi/6)(0- \pi/4)}+ \frac{1}{2}\frac{(x- 0)(x- \pi/4)}{(\pi/6- 0)(\pi/6- \pi/4)}+ \frac{\sqrt{2}}{2}\frac{(x-0)(x-\pi/6)}{(\pi/4- 0)(\pi/4- \pi/6)}\).
 

Paros

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Sorry i taught its going to be in a tabular form
 

Otis

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The issue I have with … quetion 2 is I cant [find the derivative] ... Please work it out let me learn.
Typing sin18x could mean sin(18)∙x or sin(18x). Below, I've guessed it's supposed to be sin(18x).

💡 \(\;\) Trig functions are functions, so it's best to use 'function notation' (i.e., type parentheses around inputs).

f(x) = x - sin(18x) - 0.75

We differentiate function f term by term:

The first term is x, and the derivative of x is 1

The second term is -sin(18x) -- that's a composite function, so we use the Chain Rule to get -18∙cos(18x)

The third term is a constant, and the derivative of any constant is 0

😎
 

Subhotosh Khan

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Paros

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Thanks Otis I think I now understand a little on that questions on the Newton raphson.. But the divided difference is not yet clear to me.
A table like the normal table we always draw in the forward n backward differences.
 
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