Bring ideas together without being told which method to use. Show the reasoning that makes your result valid.
Suggested session length: 240 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.
Attempt these mixed tasks before opening worked guidance. State assumptions and justify all steps; numerical agreement alone is not a proof.
These public problems require human mathematical review. Saved responses and revealed solutions do not establish secure independent examination performance.
Work is untimed. Use the suggested duration for planning, then review errors, return to linked lessons and revise.
Loading saved assessment responses…
0 of 12 tasks have recorded responses. No overall mastery score is assigned.
On the2πcircle compare K_N=1 with the Fejér kernels F_N. Both are nonnegative with normalized mass1. For f(x)=cos(3x), compute convolution with each and show exactly which missing hypothesis prevents K_N from recovering f. A second learner claims S_N f converges uniformly for every continuous f because Fejér means do; identify the logical gap. Finally take a bounded step equal2 on one half-circle and−1 on the other: determine its averaging range, explain why uniform convergence remains impossible, and justify ordinary L² projection convergence without assuming Parseval.
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.
A report uses zero boundary values and fixed initial data sin(2πx), κ=1, T=1/16. It refines N=8 to16 while retaining τ=1/128 and claims that finer spacing must reduce error. Derive both low-mode factors and the N16 highest-mode factor. Explain why a visually mild low-mode run cannot certify stability. Design a same-data stable refinement pair and derive an explicit maximum-error bound, including its derivative constant. Separately add 2⁻¹⁰ sin((N−1)πx): explain why this is a changed-data robustness probe. A run stops after six of eight target steps at an amplitude ceiling; specify which claims and retained evidence remain valid.
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.
Classify uₓₓ+8uₓᵧ+7uᵧᵧ=0. Solve uₜ+4uₓ=3 with u(x,0)=cos x. For Burgers initial data f(ξ)=−sin ξ, determine the first positive gradient-breakdown time and explain why the characteristic formula then ceases to define a global classical solution.
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 Burgers flux F(u)=u²/2 and Riemann states uL=6,uR=2, derive the weak shock speed and verify the entropy inequalities. Reverse the states and give the entropy solution. Derive Rankine–Hugoniot from a moving-interface balance rather than quoting it alone.
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 on (−π,π), derive its Fourier coefficients, explain the endpoint limit of its periodic Fourier series, and obtain Σn≥1 1/n² from Parseval. Contrast Dirichlet and Neumann eigenfunctions for −X″ on (0,L), including the zero mode.
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 uₜ=2uₓₓ+8sin(2x) on 0<x<π, with u(0,t)=1,u(π,t)=3 and initial data 1+2x/π+3sin(2x). Derive the amplitude ODE after boundary lifting. Prove uniqueness by energy for two solutions with the same data, and explain what changes for the whole-line heat kernel.
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.
On the line solve uₜₜ=4uₓₓ for f(x)=x³ and g(x)=x. Compute u(1,1), identify its initial dependence interval, and distinguish pointwise formula verification from finite total-energy claims. Compare a Dirichlet interval sine mode with three-dimensional Huygens propagation.
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.
Find the harmonic extension to the unit disk of boundary data 2+3cos(2θ)−sin(3θ). Use a maximum principle to bound it. Solve −u″=6 on (0,2) with zero endpoint values, then decide whether the same source permits zero Neumann data. Derive the compatibility sign.
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.
With f̂(ξ)=∫e^(−ixξ)f(x)dx, derive the Fourier multipliers for heat uₜ=κuₓₓ and free Schrödinger i uₜ=−uₓₓ. Explain their different L² behavior using Plancherel. Verify distributionally that the derivative of 2H(x−1) is 2δ₁, where δ₁ evaluates a test at x=1, and explain the limits of treating this identity as an ordinary derivative.
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 −u″=12x on (0,1), zero Dirichlet data, derive the weak formulation. Approximate using Vh=span{x(1−x)} and compute the Galerkin coefficient. Compare with the exact solution and explain what coercivity and Galerkin orthogonality do—and do not—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.
Analyze the explicit centered heat update at r=3/5 using the alternating spatial mode. Compare it with r=1/3. For positive-speed upwind advection at ν=2/3, update center −1 and left neighbor 5. Explain why a smaller local truncation error alone does not establish convergence.
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.
Diagonalize Uₜ+A Uₓ=0 for A=[[0,3],[3,0]], and derive its energy flux. Derive the field equation from action density (uₜ²−4uₓ²−9u² )/2 with compactly supported variations. Explain why variable coefficient systems need additional energy terms.
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
On the2πcircle compare K_N=1 with the Fejér kernels F_N. Both are nonnegative with normalized mass1. For f(x)=cos(3x), compute convolution with each and show exactly which missing hypothesis prevents K_N from recovering f. A second learner claims S_N f converges uniformly for every continuous f because Fejér means do; identify the logical gap. Finally take a bounded step equal2 on one half-circle and−1 on the other: determine its averaging range, explain why uniform convergence remains impossible, and justify ordinary L² projection convergence without assuming Parseval.
A report uses zero boundary values and fixed initial data sin(2πx), κ=1, T=1/16. It refines N=8 to16 while retaining τ=1/128 and claims that finer spacing must reduce error. Derive both low-mode factors and the N16 highest-mode factor. Explain why a visually mild low-mode run cannot certify stability. Design a same-data stable refinement pair and derive an explicit maximum-error bound, including its derivative constant. Separately add 2⁻¹⁰ sin((N−1)πx): explain why this is a changed-data robustness probe. A run stops after six of eight target steps at an amplitude ceiling; specify which claims and retained evidence remain valid.
Classify uₓₓ+8uₓᵧ+7uᵧᵧ=0. Solve uₜ+4uₓ=3 with u(x,0)=cos x. For Burgers initial data f(ξ)=−sin ξ, determine the first positive gradient-breakdown time and explain why the characteristic formula then ceases to define a global classical solution.
For Burgers flux F(u)=u²/2 and Riemann states uL=6,uR=2, derive the weak shock speed and verify the entropy inequalities. Reverse the states and give the entropy solution. Derive Rankine–Hugoniot from a moving-interface balance rather than quoting it alone.
For f(x)=x on (−π,π), derive its Fourier coefficients, explain the endpoint limit of its periodic Fourier series, and obtain Σn≥1 1/n² from Parseval. Contrast Dirichlet and Neumann eigenfunctions for −X″ on (0,L), including the zero mode.
Solve uₜ=2uₓₓ+8sin(2x) on 0<x<π, with u(0,t)=1,u(π,t)=3 and initial data 1+2x/π+3sin(2x). Derive the amplitude ODE after boundary lifting. Prove uniqueness by energy for two solutions with the same data, and explain what changes for the whole-line heat kernel.
On the line solve uₜₜ=4uₓₓ for f(x)=x³ and g(x)=x. Compute u(1,1), identify its initial dependence interval, and distinguish pointwise formula verification from finite total-energy claims. Compare a Dirichlet interval sine mode with three-dimensional Huygens propagation.
Find the harmonic extension to the unit disk of boundary data 2+3cos(2θ)−sin(3θ). Use a maximum principle to bound it. Solve −u″=6 on (0,2) with zero endpoint values, then decide whether the same source permits zero Neumann data. Derive the compatibility sign.
With f̂(ξ)=∫e^(−ixξ)f(x)dx, derive the Fourier multipliers for heat uₜ=κuₓₓ and free Schrödinger i uₜ=−uₓₓ. Explain their different L² behavior using Plancherel. Verify distributionally that the derivative of 2H(x−1) is 2δ₁, where δ₁ evaluates a test at x=1, and explain the limits of treating this identity as an ordinary derivative.
For −u″=12x on (0,1), zero Dirichlet data, derive the weak formulation. Approximate using Vh=span{x(1−x)} and compute the Galerkin coefficient. Compare with the exact solution and explain what coercivity and Galerkin orthogonality do—and do not—establish.
Analyze the explicit centered heat update at r=3/5 using the alternating spatial mode. Compare it with r=1/3. For positive-speed upwind advection at ν=2/3, update center −1 and left neighbor 5. Explain why a smaller local truncation error alone does not establish convergence.
Diagonalize Uₜ+A Uₓ=0 for A=[[0,3],[3,0]], and derive its energy flux. Derive the field equation from action density (uₜ²−4uₓ²−9u² )/2 with compactly supported variations. Explain why variable coefficient systems need additional energy terms.
