Foundations — Newton-Raphson method for root finding

Before you can walk with the parent note, you must own every symbol it uses without apology. This page lists each one from absolute zero — plain meaning, the picture it lives in, and why the method cannot work without it.

1. The two axes: $x$ and $y$

• Plain words: $x$ = "how far across", $y$ = "how high up".
• Picture: graph paper with a horizontal line ($x$) and a vertical line ($y$) crossing at the origin $(0,0)$.
• Why the topic needs it: every curve, tangent, and root below lives on this $x$–$y$ plane, so we fix the two labels once, right at the start, before any of them appear.

2. The function $f$ and the equation $f(x)=0$

• Plain words: "$x$ goes in, a height $y$ comes out."
• Picture: a wiggly curve drawn over the $x$-axis; its up–down position at each $x$ is the value $f(x)$ read off the $y$-axis.
• Why the topic needs it: the whole game is "solve $f(x)=0$". Without a function there is nothing to zero out.

Why "$=0$" and not "$=5$"? Any equation can be rearranged so one side is zero. Solving $g(x)=5$ is the same as solving $f(x)=g(x)-5=0$. So "find where the curve meets the axis" is the universal shape of every equation. That is why the method targets zero specifically.

3. The guess $x_n$ and the subscript $n$

• Plain words: $x_0,x_1,x_2,\dots$ is a list of guesses, each hopefully better than the last.
• Picture: a row of dots marching along the $x$-axis, creeping toward the root.
• Why the topic needs it: Newton-Raphson is a repeated process. We need names to talk about "the current guess $x_n$" versus "the next guess $x_{n+1}$". The counter lets one formula describe every step at once.

4. Height: $f(x_n)$ — how far from zero we are

• Picture: a vertical line from the guess $x_n$ on the axis up (or down) to the curve. That segment's signed length is $f(x_n)$.
• Why the topic needs it: it is the thing we want to drive to zero, and it becomes the top of the fraction $f/f^{\prime }$ in the formula.

5. Slope, the limit, and the derivative $f^{\prime }(x_n)$

This is the tool the parent note leans on hardest, so we build it with full care.

• Picture: a right triangle riding on the line — horizontal leg $\Delta x$, vertical leg $\Delta y$. A steep line has a tall triangle; a flat line has a short one.

But a curve is not straight — its steepness changes from place to place. So what is the slope at a single point?

• Plain words: "keep making $\Delta x$ tinier and watch what number the answer approaches."
• Picture: the second point on the curve sliding toward the first; the chord between them tilting toward one settled tilt.

• Plain words: the derivative answers "how steep is the curve at exactly this spot?"
• Why the topic needs it: Newton-Raphson replaces the curve by the tangent line, and a line is fixed once you know one point plus its slope. $f^{\prime }(x_n)$ is that slope. It becomes the bottom of the fraction $f/f^{\prime }$.

6. The tangent line — the star of the method

• Picture: a ruler laid against the curve touching at one point, matching the curve's lean.
• Why the topic needs it: near the touch point the tangent and the curve are almost identical (this is Tangent line and linear approximation). So the tangent's own zero is a great stand-in for the curve's zero — and a line's zero is easy to find exactly.

7. Finding where a line hits zero — and the Newton update

Taking the tangent $y-f(x_n)=f^{\prime }(x_n)(x-x_n)$ and putting $y=0$: $0-f(x_n)=f^{\prime }(x_n)(x_{n+1}-x_n)\Rightarrow x_{n+1}-x_n=-\frac{f(x_n)}{f^{\prime }(x_n)}.$

Now add $x_n$ to both sides to get the guess by itself — this is the standard Newton-Raphson update the parent note uses: $\boxed{x_{n+1}=x_n-\frac{f(x_n)}{f^{\prime }(x_n)}}$

• Plain words: the ratio $\frac{f(x_n)}{f^{\prime }(x_n)}$ is the horizontal distance the tangent travels to drop from height $f(x_n)$ down to the axis. Height divided by steepness = how far sideways to reach zero. The new guess is "old guess minus that jump".
• Why the minus sign? If the curve is above the axis ($f>0$) and rising ($f^{\prime }>0$), you must step left (backwards) to reach zero — the minus does that. Every sign combination works out the same way; test them:

8. Tolerance $\epsilon$ and the stop rule

• Picture: a thin band of width $\epsilon$ hugging the root; once your dot lands inside the band, you quit.
• Why the topic needs it: the true root usually has infinite decimals, so we can never land exactly on it. $\epsilon$ tells the process when "practically there" counts as done.

9. The true root $r$, and $O((\text{error}{)}^2)$

• Picture: a fixed green mark on the axis (that's $r$); the distance from your marching dot $x_n$ to that mark is the error.

• Plain words: if your current error is $0.01$, the next error is around $(0.01{)}^2=0.0001$ — the number of correct digits roughly doubles each step.
• Why the topic needs it: this is the payoff line of the whole method, explained fully via Taylor series. Here you only need to read the symbol, now that $r$ has a name.

10. When is the method trustworthy? (convergence conditions)

• Why the topic needs it: these are the fine print behind every "it converges" claim. The parent note's misbehaving Example 3 breaks condition 3; its double-root warning is condition 2; its "guard against tiny $f^{\prime }$" advice protects condition 2 and the flat-tangent case from Section 7.

Prerequisite map

Equipment checklist

Test yourself — cover the right side and answer aloud.

Connections

• Tangent line and linear approximation — the tangent is the method's whole engine.
• Derivatives — definition and rules — how the slope $f^{\prime }$ is actually computed.
• Taylor series — where quadratic convergence and the $O(\cdot )$ symbol come from.
• Fixed-point iteration — the "repeat a guess" pattern generalised.
• Bisection method — the safe fallback when $f^{\prime }\approx 0$.
• Roots of polynomials — a common place these functions come from.