Approximate solutions with explicit Euler steps

Continue modelling task 1 with a numerical method

Use the logistic model below on the interval from 0 to 2. Compare step sizes of 1, one half and one quarter at the same endpoint. Use old-point updates, record all rows and compare with the exact candidate solution. Check whether the population stays between 0 and 80.

What is the first Euler value with a step size of 1?

What is the Euler value after two steps of size 1?

Technology: execute the complete source

function euler(f, x0, y0, h, N) { if (typeof f !== 'function' || ![x0,y0,h].every(Number.isFinite) || h <= 0 || !Number.isInteger(N) || N < 0 || N > 100000) throw new Error('Invalid forward grid'); let x = x0, y = y0; const rows = [[x,y]]; for (let n = 0; n < N; n++) { const slope = f(x,y); const nextY = y + h*slope; const nextX = x0 + (n+1)*h; if (![slope,nextX,nextY].every(Number.isFinite)) throw new Error('Non-finite step'); x = nextX; y = nextY; rows.push([x,y]); } return rows; } const f = (t,p) => 0.4*p*(1-p/80); const exact = t => 80/(1+3*Math.exp(-0.4*t)); const runs = [1,0.5,0.25].map(h => { const rows = euler(f,0,20,h,2/h); if (rows.some(([t,p]) => t<0 || t>2 || p<0 || p>80)) throw new Error('Outside the declared model region'); const endpoint = rows.at(-1)[1]; return {h,N:2/h,rows,endpoint,error:endpoint-exact(2)}; }); const constantCheck = euler(()=>3,0,2,0.25,4).at(-1)[1]; if (constantCheck !== 5) throw new Error('Constant-rate check failed'); console.log(JSON.stringify({constantCheck,exactEndpoint:exact(2),runs},null,2));

Technology: reproduce and explain the refinement, ungraded

Run the displayed JavaScript program in a compatible runtime with Array.at. Retain its JSON rows and exact endpoint, identify the old-point update, explain the constant-rate check, and compare the three signed endpoint errors. Record the runtime and rounding. Add a failure test and explain what the finite-region check can and cannot establish

Model: audit the computation independently

The constant-rate endpoint must be 5. The h=1 rows are (0,20),(1,26),(2,33.02). The exact endpoint is 80/(1+3e^(−0.8)), about 34.072. The finer endpoint errors approach zero in the supplied runs. Keep full internal precision and round only the table. A negative h is rejected by the shared function; a non-finite slope is rejected during stepping. The region check examines stored nodes, not every point of the exact solution. Agreement or plausible nodes alone does not validate the biological assumptions or establish a general error theorem

What comparison isolates the displayed endpoint approximation error?

Technology-enabled exam practice: make a defensible decision

Write a short recommendation for an endpoint estimate accurate to 0.01 thousand cells at t=2. The three displayed errors all exceed 0.01 in magnitude, so none meets that absolute-error target against the known exact solution. Continue refinement or use the exact expression; record the actual error before claiming the target. Without an exact reference, a refinement difference would need additional assumptions or an error estimate

Does the coarse estimate exceed the exact endpoint? Enter 1 for yes or 0 for no

Assessment boundary

The learner’s own execution, failure analysis, interpretation and recommendation remain ungraded and need human review