Newton's method is a numerical technique for estimating roots of equations that are difficult or impossible to solve exactly. This cheat sheet helps students connect algebra, graphs, derivatives, and iterative approximation. It is useful for applied math problems where a solution must be found to a desired level of accuracy.
Students need it to choose a starting value, apply the update formula, and judge whether an answer is reliable.
The central idea is to replace a curve near a point with its tangent line and use the tangent line's x-intercept as a better estimate of the root. The main formula is x_new = x - f(x) / f'(x), repeated until the values stop changing much. Root finding also includes checking sign changes, interpreting convergence, and avoiding cases where the derivative is zero or the method moves away from the root.
Good numerical work always includes a reasonableness check using the original function.
Key Facts
- A root of f(x) is a value r such that f(r) = 0.
- Newton's method uses the iteration x_(n+1) = x_n - f(x_n) / f'(x_n).
- The tangent line at x_n is y = f(x_n) + f'(x_n)(x - x_n), and setting y = 0 gives the Newton update.
- A common stopping rule is stop when |x_(n+1) - x_n| < tolerance or when |f(x_n)| < tolerance.
- If f(a) and f(b) have opposite signs, then a continuous function f has at least one root in the interval (a, b).
- Newton's method usually converges quickly when the starting guess is close to a simple root and f'(x) is not near 0.
- Newton's method can fail or jump far away when f'(x_n) = 0, f'(x_n) is very small, or the starting guess is poorly chosen.
- After estimating a root r, substitute it into the original function and check that f(r) is close to 0.
Vocabulary
- Root
- A root is an input value that makes a function equal to zero.
- Iteration
- An iteration is one repeated step of a numerical method used to improve an approximation.
- Newton's Method
- Newton's method is a root-finding algorithm that uses tangent lines and derivatives to improve a starting guess.
- Derivative
- A derivative gives the instantaneous rate of change or slope of a function at a point.
- Convergence
- Convergence means that a sequence of approximations gets closer to a final value.
- Tolerance
- A tolerance is the maximum acceptable error used to decide when an approximation is accurate enough.
Common Mistakes to Avoid
- Using x_(n+1) = x_n + f(x_n) / f'(x_n) instead of subtracting is wrong because the tangent line's x-intercept comes from x_n - f(x_n) / f'(x_n).
- Forgetting to recompute f(x_n) and f'(x_n) at each new estimate is wrong because Newton's method depends on the current tangent line, not the first one.
- Starting where f'(x_n) = 0 or very close to 0 is unsafe because the division can be undefined or produce a huge jump.
- Stopping after one iteration without checking error is wrong because the first estimate may still be far from the true root.
- Reporting a root without testing it in the original equation is risky because a rounded or nonconvergent approximation may not actually make f(x) close to 0.
Practice Questions
- 1 Use Newton's method for f(x) = x^2 - 10 with starting value x_0 = 3. Find x_1 and x_2.
- 2 Use Newton's method for f(x) = x^3 - 2x - 5 with starting value x_0 = 2. Find x_1 to three decimal places.
- 3 For f(x) = cos(x) - x, write the Newton update formula and use x_0 = 1 to find x_1.
- 4 Explain why Newton's method may fail or behave unpredictably if the starting value is chosen at a point where the tangent line is nearly horizontal.