# Newtons method question

#### SyNtHeSiS

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

MHF Hall of Fame
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?

#### SyNtHeSiS

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
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)

#### SyNtHeSiS

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
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.

#### SyNtHeSiS

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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