Linear programming (internal)
Internal 3 credits Constraints Feasible region Objective function Vertices OptimisationAS 91574, 3 credits. Turn a real situation into inequalities, graph the feasible region, and optimise an objective function over it. Short standard, very predictable structure — and the marks concentrate in the interpretation, not the graphing.
1 What the grades require
Same ladder for all three Calculus internals.| Grade | What it takes | The half people drop |
|---|---|---|
| Achieved | Select and use methods; know the concepts and terms; communicate with appropriate representations | Showing no working — the method IS the evidence |
| Merit | A logical sequence of steps, connecting concepts, or forming and using a model | The “and also”: relate findings back to the context |
| Excellence | Devise a strategy, build a chain of reasoning or a proof, or form a generalisation | The “and also”: correct mathematical statements, or a stated insight |
Both Merit and Excellence are “one or more of … AND ALSO …”. A perfect answer that never interprets its result in context cannot pass the Achieved ceiling.
2 Setting up the model
Define, then constrain — before any graphing.The order that never loses marks
- Define each variable with its unit, in one sentence each.
- One inequality per real limit, plus x ≥ 0 and y ≥ 0.
3 Graphing the feasible region
Shade each constraint, then find the overlap.Graphing checklist
- Draw every boundary line — solid for ≤/≥, dashed for </>.
- Test a convenient point (usually the origin) to decide which side to shade.
- Label the feasible region clearly once every constraint is drawn.
- Check whether the region is bounded — an unbounded region may have no finite maximum (or minimum).
4 Vertices and the objective function
The optimum sits at a corner — or an entire edge.Finding and testing vertices
- List the vertices as coordinates, solving simultaneously where boundary lines meet.
- Evaluate the objective function at each vertex in a small table.
- If two vertices tie, the whole edge between them is optimal — say so.
5 Integer constraints
When variables must be whole, the vertex is only a starting point. ✕The unrounded answer
An optimum of 6.4 machines is not an answer. Check both 6 and 7 against every constraint — 7 may be infeasible — and quote the best feasible whole-number point, saying which constraint binds.6 Reaching Excellence
Devise the strategy yourself, and generalise how the optimum responds to change. Internal-style Sensitivity: how far can profit per unit of B rise before B becomes optimal? The current optimum makes only product A; find the threshold where B overtakes it. Write the objective as P = 12A + kB, where k is B's (currently unknown, to be varied) profit per unit. At the current optimal vertex, A is made exclusively because 12A > kB there for the given k. Find the value of k that makes the objective EQUAL at the current vertex and the vertex where B is maximised — that is the threshold. State the generalisation: for k below the threshold, keep making A; above it, switch to B. ✓ This is "forming a generalisation" plus a stated chain of reasoning — the exact Excellence criteria, applied to a real business question about changing conditions.Practice
Recall it first, then use it — testing yourself builds the memory, where rereading the notes above only feels easier.
Recall and application shuffled together — you have to work out what each item wants before you can answer it, which is the part the exam actually tests. Both kinds feed the same spacing schedule, so anything you miss comes back sooner.
Say the answer out loud before flipping — that attempt is where the learning happens, even when you get it wrong.
Exam-style only. Instant feedback, shuffled each attempt, best score saved on this device.