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.
Understanding Newton's Method & Root Finding
The speed of Newton's method comes from the local shape of a smooth curve. Near an ordinary crossing of the horizontal axis, the curve and its tangent have very similar directions. Each new estimate often has far more correct decimal places than the one before it.
This is called quadratic convergence. For example, an error of one tenth may shrink roughly to one hundredth, then to one ten-thousandth, once the estimates are close enough.
This rapid improvement is not guaranteed at the beginning. The method has a local advantage, meaning it works best after the estimate has entered a suitable region around the solution.
The shape of the graph can make the process behave very differently. At a root where the graph crosses the axis cleanly, the slope is usually usable. At a repeated root, the graph may touch the axis and turn back.
Such roots occur when a factor is repeated, as with x minus three squared. Newton's method still may approach that root, but it often slows down greatly. A nearly flat section is another warning sign.
A small slope makes the tangent line almost horizontal, so its intercept can lie very far from the current point. Students should sketch the graph or inspect a table of values before trusting a sequence of calculator outputs.
A sensible starting estimate comes from information gathered before iteration. A graph can show where the function is near zero and whether there may be several roots. A sign change on an interval is especially useful for locating a crossing, though it does not reveal a root where the curve only touches the axis.
If several roots exist, different starting values can lead Newton's method to different answers. In some cases, estimates can jump back and forth between two values or move into a region with no useful path to a root.
A practical safeguard is to first narrow a sign-change interval with bisection, then use Newton's method from a point inside that interval. This combines reliable bracketing with fast local improvement.
Computers use root finding whenever a model contains an equation that cannot be rearranged neatly. Engineers may solve for a temperature in a heat transfer model, a launch angle in a trajectory calculation, or a current in a circuit with nonlinear components. In school, the same thinking appears when finding intersections of curves, solving exponential or trigonometric equations, and fitting physical measurements to a model.
A calculator can perform the arithmetic, but it cannot decide whether the result makes sense. Record each estimate, the function value there, and the change from the previous estimate.
Use a tolerance that matches the units and purpose of the problem. A root accurate to six decimal places is not automatically meaningful if the measured input data are accurate only to the nearest tenth.