Expert answers to "Moving a point along a polynomial curve by..."

High School
Algebra
Algebra I
Quadratic function and equation

Karla Thompson

Answered question

2022-04-14

Moving a point along a polynomial curve by certain arc length

L = \int_{n}^{m} \sqrt{1 + {(b + 2 a x)}^{2}} d x

Answer & Explanation

annie996k88v

Beginner2022-04-15Added 11 answers

Step 1
So, you want to solve for m the equation

4 a L + {sinh}^{- 1} (2 a n + b) + (2 a n + b) \sqrt{1 + {(2 a n + b)}^{2}} =

{\sin h}^{- 1} (2 a m + b) + \sqrt{1 + {(2 a m + b)}^{2}} (2 a m + b)

Let

k = 4 a L + {\sin h}^{- 1} (2 a n + b) + (2 a n + b) \sqrt{1 + {(2 a n + b)}^{2}}

and

x = 2 a m + b

to make

k = {\sin h}^{- 1} (x) + x \sqrt{1 + x^{2}}

A numerical method would be required (Newton method being the simplest) and, as usual, you need a starting guess (as good as possible if you want to reduce the number of iterations).
A reasonable approximation is

{\sin h}^{- 1} (x) + x \sqrt{1 + x^{2}} \sim x^{2} \Rightarrow x_{0} = \sqrt{k}

Let us try with

a = 3, b = 5, n = 7

and

L = 1234

which give

k = 14808 + 47 \sqrt{2210} + {\sin h}^{- 1} (47)

which is

\approx 17022

This would generate the following Newton iterates

(\begin{array}{cc} n & x_{n} \\ 0 & 130.4685531 \\ 1 & 130.4453134 \\ 2 & 130.4453113 \end{array})

corresponding to

m = 20.9075518

With this result, repeating the calculations, we get

L = 1233.9999889

Do you have a similar question?

Recalculate according to your conditions!

New Questions in Algebra I

Didn't find what you were looking for?