Newton's method (intersection points)

In summary, the conversation discusses the use of Newton's Method to approximate all intersection points of two given functions, ln(x) and (x^2)/8 - 2. The process involves equating the two equations, finding the derivative, drawing the graphs, and using the equation Xn+1 = Xn - f(x)/f`(x). The conversation also mentions the importance of using more than 3 decimal places in calculations for a correct answer to 3 decimal places.
  • #1

Homework Statement

Given two functions; y= ln(x) and y=(x^2)/8 - 2

Use Newton's Method to approximate all intersection points of the given functions, each with 3 decimal places.

Homework Equations

Xn+1 = Xn - f(x) / f`(x)

The Attempt at a Solution

Step 1: I equated the two equations, and I got

f(x) = ln(x) - (x^2)/8 +2

Step 2: I found the derivative as f`(x) = 1/x - x/4

Step 3: then I drew the graphs, and I found that they only intersect at a point near 5.4 (because ln(x) doesn't have any graph in the -x direction)

so I started off x1 = 5.4 and then using the equation given above I found the following values:

X2 = 5.435
X3 = 5.436
X4 = 5.435
X5 = 5.436

Can you guys please tell me if I did the solution right and at which point do I stop ... they keep going ... I'm not going same 3 decimal places for any of them ?
I hope someone would answer soon because I have it due tomorrow early morning. Thanks and can you also tell that whatever I did was right and nothing else was to be done!

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  • #2
Yes, you are doing fine with one proviso- if you want the answer correct to 3 decimal places, use more than 3 decimal places, 4 should do, during your calculation. Continue until the 4 decimal places are the same or at least give the same answer rounded to 3 decimal places.

What is Newton's method?

Newton's method, also known as the Newton-Raphson method, is a mathematical algorithm used to find the roots or solutions of a given equation. It is commonly used to find the intersection points of two curves or to approximate the zeros of a function.

How does Newton's method work?

Newton's method works by starting with an initial guess for the root of the equation and then using the tangent line at that point to find a better approximation of the root. This process is repeated until the desired level of accuracy is achieved.

What are the advantages of using Newton's method?

One advantage of using Newton's method is that it can converge to the root of an equation quickly, especially when the initial guess is close to the actual root. It is also a versatile method that can be used for a wide range of equations.

What are the limitations of Newton's method?

Newton's method can fail to converge or produce incorrect results if the initial guess is too far from the actual root or if the function is not well-behaved (e.g. has multiple roots or a vertical tangent). It also requires the calculation of derivatives, which can be computationally expensive.

How is Newton's method used to find intersection points?

To find the intersection points of two curves using Newton's method, the two equations are set equal to each other and rearranged to form a single equation in terms of one variable. This equation is then solved using Newton's method to find the x-values of the intersection points. These x-values can then be substituted into either equation to find the corresponding y-values.

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