Newtons method question

Apr 2010
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Suppose the line y = 5x - 4 is tangent to the curve y = f(x) when x = 3. If Newton's method is used to locate a root of the equation f(x) = 0 and the initial approximation is x1 = 3, find the second approximation.

I am not sure how to find f(x). I tried finding the anti-derivative of the tangent line and used that as my f(x), but that didnt work.
 

Chris L T521

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May 2008
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Suppose the line y = 5x - 4 is tangent to the curve y = f(x) when x = 3. If Newton's method is used to locate a root of the equation f(x) = 0 and the initial approximation is x1 = 3, find the second approximation.

I am not sure how to find f(x). I tried finding the anti-derivative of the tangent line and used that as my f(x), but that didnt work.
You don't need to know what \(\displaystyle f(x)\) is! Recall that by Newton-Raphson Method, \(\displaystyle x_{n+1}=x_n-\frac{f(x_n)}{f^{\prime}(x_n)}\).

You're given \(\displaystyle x_1=3\implies x_2=x_1-\frac{f(x_1)}{f^{\prime}(x_1)}=3-\frac{f(3)}{f^{\prime}(3)}\). Note that \(\displaystyle f(3)\) is the value of the tangent line at 3 (observe that the function and the tangent line have the same value at 3) and \(\displaystyle f^{\prime}(3)\) is the slope of the given tangent line.

Can you finish the problem now?
 
Apr 2010
156
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So basically you indirectly using the value of f(x) and f'(x), by using the tangent equation, since it shares the same common point at x = 3?
 

Chris L T521

MHF Hall of Fame
May 2008
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So basically you indirectly using the value of f(x) and f'(x), by using the tangent equation, since it shares the same common point at x = 3?
Exactly!!! (Nod)
 
Apr 2010
156
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For a similar question (anti-derivatives) saying: Given the graph of f passes through the point (1, 6) and that the slope of its tangent line at (x, f(x)) is 2x + 1, find f(2).

In order to solve it, you had to find the anti-derivative of the tangent line. Howcome in the newtons method question this doesnt work, if it would work in this question?
 

HallsofIvy

MHF Helper
Apr 2005
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These are completely different problems. Here you were specifically asked to find f(2). You have to find f(x), the anti-derivative, to do that.

With Newton's method you approximate a function by its tangent line and follow that tangent line to y= 0. At each step you don't want to find where f(x)= 0, just where the tangent line crosses y= 0.
 
Apr 2010
156
0
These are completely different problems. Here you were specifically asked to find f(2). You have to find f(x), the anti-derivative, to do that.

With Newton's method you approximate a function by its tangent line and follow that tangent line to y= 0. At each step you don't want to find where f(x)= 0, just where the tangent line crosses y= 0.
Oh ok, but howcome theres no subbing in of y = 0 for the tangent line in the newton method formula?
 
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