How does $$ y-f\left(x_0\right)=f^{\prime}\left(x_0\right)\cdot\left(x-x_0\right)\rarr x_1=x_0-\frac{f\left(x_0\right)}{f^{\prime}\left(x_0\right)} $$

The equation 
\[
y - f(x_0) = f'(x_0) \cdot (x - x_0)
\]
is the **tangent line** to the curve $ y = f(x) $ at the point $ (x_0, f(x_0)) $. We use this tangent line as an approximation to the curve to iteratively solve for the root of $ f(x) = 0 $, which is the foundation of **Newton's Method**.

### Derivation of Newton's Method:
1. The tangent line approximation of $ f(x) $ around $ x_0 $ can be expressed as:
   \[
   y = f(x_0) + f'(x_0) \cdot (x - x_0).
   \]
2. To find the root of $ f(x) = 0 $, we set $ y = 0 $ and solve for $ x $:
   \[
   0 = f(x_0) + f'(x_0) \cdot (x - x_0).
   \]
3. Rearranging to isolate $ x $, we get:
   \[
   x = x_0 - \frac{f(x_0)}{f'(x_0)}.
   \]
4. This new value $ x $ is designated as $ x_1 $, the next approximation for the root of $ f(x) = 0 $. Thus, we have:
   \[
   x_1 = x_0 - \frac{f(x_0)}{f'(x_0)}.
   \]

In summary, we start at an initial guess $ x_0 $ and use the tangent line to obtain a new approximation $ x_1 $. This process is repeated iteratively to bring the approximations closer to the actual root.

Would you like more details or have any questions?

Here are 5 questions to explore further:
1. Why does Newton's Method converge faster than other root-finding methods?
2. How does the choice of $ x_0 $ affect the convergence of Newton's Method?
3. What happens if $ f'(x_0) = 0 $ during Newton's Method?
4. Can Newton's Method fail to converge, and if so, under what conditions?
5. How can we modify Newton's Method to improve its convergence for certain functions?

**Tip:** Always check if $ f'(x) $ is close to zero at any iteration in Newton's Method; a small derivative can lead to large steps and instability.

Explore how Newton's Method uses the tangent line approximation to iteratively find roots of functions, starting from an initial guess and refining with the formula x_1 = x_0 - f(x_0)/f'(x_0). This method is foundational in calculus for approximating solutions to f(x) = 0, involving calculus concepts such as derivatives and tangent lines.

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