Find the root of an equation, knowing that a solution is located between two points.

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Ben_clm
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#1
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#1
The question is:

3.

The equation

f(x)= arcsin(2x) + 2x - 1.5

can be rearranged into the iterative form xn+1 = a sin(bx + c) when f(x) = 0

(a) Find the constant a, b and c of this expression.

The equation f(x) = 0 has a solution located between x = 0.3 and x = 0.4

(b) Hence, or otherwise, find the root of the equation f(x) = 0 correct to 3 decimal places. Give your answers in radians.

I found that a = 1/2, b = -2 and c = 1.5,
But I am unsure about what I need to do for part b. Could someone please give me a clue about the method I need to use?
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Pangol
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#2
Report 4 weeks ago
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(Original post by Ben_clm)
The question is:

3.

The equation

f(x)= arcsin(2x) + 2x - 1.5

can be rearranged into the iterative form xn+1 = a sin(bx + c) when f(x) = 0

(a) Find the constant a, b and c of this expression.

The equation f(x) = 0 has a solution located between x = 0.3 and x = 0.4

(b) Hence, or otherwise, find the root of the equation f(x) = 0 correct to 3 decimal places. Give your answers in radians.

I found that a = 1/2, b = -2 and c = 1.5,
But I am unsure about what I need to do for part b. Could someone please give me a clue about the method I need to use?
Your part (a) is fine.

This is an iterative methods question. Since you know that there is a root between 0.3 and 0.4, start with a value in this interval (taking x1 = 0.35 is the obvious choice), and plug this into xn+1 = 1/2 sin(-2xn + 1.5) to get x2, x3, etc. It should be clear when things have settled down in the first three decimal places.

This is the standard approach to iterative questions - have you met this before?
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Ben_clm
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#3
Report Thread starter 4 weeks ago
#3
Yes I did to iteration method before usually but they make it more obvious, but now that you mention it seems obvious 😅. Thank you very much for your help!
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