{ "format": "recallweave-deck/1", "title": "Newton's method: tangent steps, exact residuals and visible limits", "attribution": "Original teaching text, questions and worked examples prepared for RecallWeave in October 2026. Mathematical reference: Gilbert Strang and Edwin Herman, OpenStax Calculus Volume 1, section 4.9 (Newton's Method), https://openstax.org/books/calculus-volume-1/pages/4-9-newtons-method . Reference checked 2026-10-08. All question wording, calculations and graphics in this contribution are original.", "license": "Original teaching text and examples. This contribution makes no additional reuse-license grant. Linked teaching sources retain their own terms; no textbook prose, exercise or figure is copied.", "concepts": [ "Tangent steps", "Exact residuals", "Failure and repetition", "Interpreting a run" ], "items": [ { "id": "nm-01", "concept": "Tangent steps", "prerequisites": [], "prompt": "For f(x) = 2x − 3, start at x₀ = 4. Where does the tangent meet the x-axis?", "options": [ "3", "5/2", "3/2", "4" ], "answer": 2, "explanation": "The graph is already a straight line. At x₀ = 4, f(x₀) = 5 and f′(x₀) = 2. Its x-intercept is 4 − 5/2 = 3/2. Substituting 3/2 into 2x − 3 gives exactly zero, so one step reaches the root.", "transfer": "Replace the constant −3 with −5. Find the new root and explain why the starting point does not matter for this nonhorizontal straight line." }, { "id": "nm-02", "concept": "Tangent steps", "prerequisites": [], "prompt": "A tangent at xₙ has equation y = f(xₙ) + f′(xₙ)(x − xₙ), with f′(xₙ) ≠ 0. Which rule sets its next x-intercept?", "options": [ "xₙ₊₁ = xₙ − f′(xₙ)", "xₙ₊₁ = xₙ − f(xₙ)/f′(xₙ)", "xₙ₊₁ = xₙ + f(xₙ)/f′(xₙ)", "xₙ₊₁ = f(xₙ)/f′(xₙ)" ], "answer": 1, "explanation": "Set the tangent's y-value to zero and solve: f′(xₙ)(xₙ₊₁ − xₙ) = −f(xₙ). Dividing by the nonzero slope gives a displacement of −f/f′ from the current x. The quotient alone is a correction, not the next coordinate.", "transfer": "If xₙ = 3, f(xₙ) = −6 and f′(xₙ) = 4, compute the signed correction f/f′ and the next coordinate separately." }, { "id": "nm-03", "concept": "Tangent steps", "prerequisites": [], "prompt": "For f(x) = x² − 2, use x₀ = 1. What is the first Newton iterate?", "options": [ "1/2", "0", "2", "3/2" ], "answer": 3, "explanation": "Here f(1) = −1 and f′(1) = 2. The correction f/f′ is −1/2, so x₁ = 1 − (−1/2) = 3/2. A negative function value does not mean the next coordinate must decrease; the slope also matters.", "transfer": "Start the same polynomial at x₀ = 2. Explain why the first step reaches the same coordinate from the other side." }, { "id": "nm-04", "concept": "Exact residuals", "prerequisites": [ "Tangent steps" ], "prompt": "The iterate x = 3/2 is being used for f(x) = x² − 2. What is its exact residual f(x)?", "options": [ "1/4", "0", "1/2", "3/2" ], "answer": 0, "explanation": "The residual is (3/2)² − 2 = 9/4 − 8/4 = 1/4. It measures the function value at the current iterate. It is neither the x-coordinate nor, in general, the horizontal distance to a root.", "transfer": "For x = 17/12, calculate x² − 2 exactly. Compare this function residual with the difference between 17/12 and 3/2." }, { "id": "nm-05", "concept": "Exact residuals", "prerequisites": [ "Tangent steps" ], "prompt": "For f(x) = x², the starting point is x₀ = 0. Both f(x₀) and f′(x₀) are zero. What should the explorer report first?", "options": [ "A cycle of length one with a nonzero residual.", "An exact root has already been reached; no Newton division is needed.", "A nonroot horizontal-tangent failure.", "A new iterate of 0/0." ], "answer": 1, "explanation": "A root is a point where f(x) is zero. That condition is already satisfied, so the process can stop successfully without attempting a division by the derivative. A zero derivative prevents a step away from a nonroot; it does not erase an exact root.", "transfer": "Compare f(x) = x² at x = 0 with f(x) = x² − 2 at x = 0. Which residual changes the stopping decision?" }, { "id": "nm-06", "concept": "Exact residuals", "prerequisites": [ "Tangent steps" ], "prompt": "After three Newton steps for x² − 2 from x₀ = 2, the exact residual is 1/166464. What does that record justify?", "options": [ "The exact root has been reached because a short display might show 0.0000.", "The iteration must diverge because the residual is positive.", "The residual is the exact horizontal error in x.", "The residual is small but nonzero, so this iterate is not an exact root." ], "answer": 3, "explanation": "The rational residual 1/166464 is not zero. A rounded decimal display can hide a nonzero value, while exact arithmetic retains it. A finite run can produce a useful approximation without proving an exact root or guaranteeing all future behavior.", "transfer": "Why can defined Newton steps with rational coefficients and a rational start stay rational, even when the root being approached is irrational?" }, { "id": "nm-07", "concept": "Failure and repetition", "prerequisites": [ "Tangent steps", "Exact residuals" ], "prompt": "For f(x) = x² − 2 at x = 0, f(x) = −2 and f′(x) = 0. What happens to the tangent step?", "options": [ "The horizontal tangent y = −2 has no x-intercept, so the next Newton iterate is undefined.", "The next iterate is zero because the slope is zero.", "The method has found a root.", "The method should silently choose a nearby starting point." ], "answer": 0, "explanation": "The tangent is the constant line y = −2. It never meets y = 0, and the quotient f/f′ would divide a nonzero value by zero. The explorer records this stop rather than inventing a replacement point.", "transfer": "Change only the starting point to 1/1000. Compute the next coordinate and explain why a very small nonzero slope can create a large jump." }, { "id": "nm-08", "concept": "Failure and repetition", "prerequisites": [ "Tangent steps", "Exact residuals" ], "prompt": "For f(x) = x³ − 2x + 2, Newton steps from 0 give 0 → 1 → 0. The residuals are 2 → 1 → 2. What has been established?", "options": [ "The root is halfway between 0 and 1.", "The iteration has converged because it returned to its start.", "An exact nonroot cycle of length two repeats under the same deterministic rule.", "One more identical step is guaranteed to find a root." ], "answer": 2, "explanation": "At 0, f = 2 and f′ = −2, so the next point is 1. At 1, f = 1 and f′ = 1, so the next point is 0. Returning to the same exact nonroot point repeats the same future steps. This proves a cycle for this start, not that the polynomial has no real root.", "transfer": "Start the cycle example at a different allowed point. Explain why a different finite trace does not invalidate the exact 0 → 1 → 0 calculation." }, { "id": "nm-09", "concept": "Failure and repetition", "prerequisites": [ "Tangent steps", "Exact residuals" ], "prompt": "For f(x) = (x − 1)² and x ≠ 1, Newton's rule simplifies to xₙ₊₁ = (xₙ + 1)/2. Starting at 3, what happens to the distance from the root on each step?", "options": [ "It is halved: 2, 1, 1/2, 1/4, … .", "It is squared: 2, 4, 16, … .", "It becomes exactly zero after the first step.", "It remains fixed because the root is repeated." ], "answer": 0, "explanation": "Subtract 1 from the update: xₙ₊₁ − 1 = (xₙ − 1)/2. From x₀ = 3, the iterates are 3, 2, 3/2, 5/4, … . The error decreases steadily but remains nonzero at every finite step from this start. A repeated root can approach differently from a simple root.", "transfer": "Use the explorer's repeated-root example and compare the exact residual sequence with the exact distance-to-1 sequence. Why does the residual shrink by a factor of four?" }, { "id": "nm-10", "concept": "Interpreting a run", "prerequisites": [ "Tangent steps", "Exact residuals", "Failure and repetition" ], "prompt": "Replace f(x) with g(x) = −3f(x), keeping the same start. Whenever the derivative is nonzero, what happens to the Newton x-intercepts?", "options": [ "Their signs reverse.", "Their magnitudes are multiplied by three.", "They stay the same because g/g′ = f/f′.", "Every step becomes undefined." ], "answer": 2, "explanation": "Both the function value and derivative are multiplied by −3, so their ratio is unchanged: (−3f)/(−3f′) = f/f′. The graph and residual values change, but the tangent x-intercepts follow the same recurrence. This reasoning requires a nonzero scale factor.", "transfer": "Compare x² − 2 with −3x² + 6 in the explorer. Which exact fields change and which remain identical?" }, { "id": "nm-11", "concept": "Interpreting a run", "prerequisites": [ "Tangent steps", "Exact residuals", "Failure and repetition" ], "prompt": "For f(x) = x² − 1, a start of −3 gives the first iterate −5/3. Which interpretation is justified?", "options": [ "The next iterate must be +1 because +1 is a root.", "This first step stays on the negative side; a starting point influences the path, and one step alone is not a convergence proof.", "The polynomial has only one real root.", "The method always chooses the root nearest zero." ], "answer": 1, "explanation": "The first correction is 8/(−6) = −4/3, giving −3 + 4/3 = −5/3. The polynomial has roots at −1 and +1. Newton's method follows the local tangent from the chosen start; it does not carry a separate instruction naming a preferred root. A finite observation should not be overstated.", "transfer": "For a nonzero x, simplify this polynomial's update to (x² + 1)/(2x). What does that expression say about the sign of the next iterate?" }, { "id": "nm-12", "concept": "Interpreting a run", "prerequisites": [ "Tangent steps", "Exact residuals", "Failure and repetition" ], "prompt": "The explorer stops because the next complete exact step would exceed its arithmetic bound. What does that result mean?", "options": [ "The polynomial has no root.", "The iterates have been proved to diverge.", "The displayed approximate coordinate should be accepted as an exact root.", "The bounded computation has stopped; the last complete exact steps remain valid, and no convergence or divergence is inferred." ], "answer": 3, "explanation": "A resource limit is a limit of this computation. It is separate from the mathematical facts that establish an exact root, a zero-slope obstruction, or an exact repeated cycle. The explorer keeps completed records and does not invent an unfinished next step or relabel the limit as a theorem.", "transfer": "Explain how a chosen step-limit stop differs from an exact cycle stop. What evidence would you need to make a stronger mathematical claim?" } ] }