Numerical analysis: justify the computation and its limits
Bring ideas together without being told which method to use. Show the reasoning that makes your result valid.
Suggested session length: 180 minutes, as an unvalidated planning estimate. This collection contains 12 tasks; choose a manageable set and return for further sessions. Completing the whole collection may take substantially longer. This is a unit assessment, not a full course or AP examination.
Choose methods before opening guidance. Write the domain, assumptions and intermediate reasoning; a decimal answer or theorem name alone is insufficient.
These public mixed tasks require human mathematical review. Saved work and automated checks do not certify an independent examination, proof validity or semester-course equivalence.
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Change the interval to[0,2]. For x⁴, compute the initial coarse Simpson value, the two-half fine value, correction and exact integral. With absolute tolerance1/100 and a cap of five evaluations, explain the next partition and what is unresolved. Separately let H(x)=F(x/2), where F is the hidden polynomial from the lesson. Give its exact integral and explain the no-scouting failure on the changed interval.
Structure your argument and preserve revisions
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Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
For y′=−(y−t²)+2t, y(0)=1, try h=1/2 with tolerance1/8 using shared-start Euler step doubling. Calculate the coarse value, midpoint, fine value and estimate. Reject if necessary, retry at h=1/4, and record accepted time/state and derivative work. Derive and compute the global rational bound on the accepted path using M=3. If the trial cap is2, can the report claim an endpoint solution or that global error≤1/8? Explain what exact arithmetic and a floating exponential check each establish.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
For h>0 near zero, compare direct evaluation of f(h)=√(1+h)−1 with h/(√(1+h)+1). Derive the relative condition number with respect to h and explain how a well-conditioned problem can have an unstable evaluation. State the range assumptions of the floating-point model.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Bisect x³−5 on [1,2] four times and give a certified interval and midpoint error bound. A colleague instead iterates T(x)=(x+5)/4 on [1,2]. Prove that iteration converges, then explain why it does not solve the original equation.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
For f(x)=(x−2)³ and x₀=3, derive the ordinary Newton error recurrence and compare a multiplicity-corrected update. Explain why the usual simple-root secant order theorem does not apply. Give a bracket safeguard that guarantees interval shrinkage for a continuous sign-changing function.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Interpolate f(x)=x³−x at −1,0,2 in Newton form. Find the exact remainder polynomial and relate it to the derivative remainder theorem. Explain why this successful small example does not justify equally spaced high-degree interpolation for arbitrary smooth data.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
For data (0,1),(1,2),(3,0), derive the natural cubic spline and the least-squares line. Show a thin QR factorization for the line’s design matrix and explain why the two procedures answer different questions.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Approximate ∫₀¹x⁴ dx with composite trapezoids at h=1/2 and Simpson rules at h=1/2 and h=1/4. Use the Simpson difference to construct the usual correction. Construct a smooth nonzero integrand that defeats an adaptive test sampling only these five nodes.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Solve A x=(1,3)ᵀ for A=[[1/100,2],[1,1]]. Explain the multiplier without pivoting, then compare Jacobi convergence before and after swapping the equations. Does a small residual alone imply a small forward error?
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
For A=diag(5,−4,1), analyze power iteration from (1,1,0). Then use inverse iteration with shift 4/5 from the same start: will it find the eigenvalue 1 nearest the shift? Repair the experiment and state the convergence assumptions.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
For y′=−40y, y(0)=1, take h=1/10. Compare one explicit Euler, explicit trapezoidal/Heun and implicit Euler step. Derive the negative-real stability ranges of the first two and explain why second order is not enough to resolve stiffness.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Suppose a one-step method has local defect at most C h^(p+1) and error recurrence eₙ₊₁≤(1+Lh)eₙ+C h^(p+1), e₀=0, with L>0. Derive a global bound through T. Explain a step-doubling estimate and design checks that could reveal a misleading convergence plot.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Review and revisit
Use each rubric to identify a valid argument, a missing condition or a step to revise. For independent evaluation, ask a teacher or reviewer to examine the complete reasoning. Self-review and recorded responses are not verified assessment results.
Coverage and targeted lesson returns
Change the interval to[0,2]. For x⁴, compute the initial coarse Simpson value, the two-half fine value, correction and exact integral. With absolute tolerance1/100 and a cap of five evaluations, explain the next partition and what is unresolved. Separately let H(x)=F(x/2), where F is the hidden polynomial from the lesson. Give its exact integral and explain the no-scouting failure on the changed interval.
For y′=−(y−t²)+2t, y(0)=1, try h=1/2 with tolerance1/8 using shared-start Euler step doubling. Calculate the coarse value, midpoint, fine value and estimate. Reject if necessary, retry at h=1/4, and record accepted time/state and derivative work. Derive and compute the global rational bound on the accepted path using M=3. If the trial cap is2, can the report claim an endpoint solution or that global error≤1/8? Explain what exact arithmetic and a floating exponential check each establish.
For h>0 near zero, compare direct evaluation of f(h)=√(1+h)−1 with h/(√(1+h)+1). Derive the relative condition number with respect to h and explain how a well-conditioned problem can have an unstable evaluation. State the range assumptions of the floating-point model.
Bisect x³−5 on [1,2] four times and give a certified interval and midpoint error bound. A colleague instead iterates T(x)=(x+5)/4 on [1,2]. Prove that iteration converges, then explain why it does not solve the original equation.
For f(x)=(x−2)³ and x₀=3, derive the ordinary Newton error recurrence and compare a multiplicity-corrected update. Explain why the usual simple-root secant order theorem does not apply. Give a bracket safeguard that guarantees interval shrinkage for a continuous sign-changing function.
Interpolate f(x)=x³−x at −1,0,2 in Newton form. Find the exact remainder polynomial and relate it to the derivative remainder theorem. Explain why this successful small example does not justify equally spaced high-degree interpolation for arbitrary smooth data.
For data (0,1),(1,2),(3,0), derive the natural cubic spline and the least-squares line. Show a thin QR factorization for the line’s design matrix and explain why the two procedures answer different questions.
Approximate ∫₀¹x⁴ dx with composite trapezoids at h=1/2 and Simpson rules at h=1/2 and h=1/4. Use the Simpson difference to construct the usual correction. Construct a smooth nonzero integrand that defeats an adaptive test sampling only these five nodes.
Solve A x=(1,3)ᵀ for A=[[1/100,2],[1,1]]. Explain the multiplier without pivoting, then compare Jacobi convergence before and after swapping the equations. Does a small residual alone imply a small forward error?
For A=diag(5,−4,1), analyze power iteration from (1,1,0). Then use inverse iteration with shift 4/5 from the same start: will it find the eigenvalue 1 nearest the shift? Repair the experiment and state the convergence assumptions.
For y′=−40y, y(0)=1, take h=1/10. Compare one explicit Euler, explicit trapezoidal/Heun and implicit Euler step. Derive the negative-real stability ranges of the first two and explain why second order is not enough to resolve stiffness.
Suppose a one-step method has local defect at most C h^(p+1) and error recurrence eₙ₊₁≤(1+Lh)eₙ+C h^(p+1), e₀=0, with L>0. Derive a global bound through T. Explain a step-doubling estimate and design checks that could reveal a misleading convergence plot.
