Homework 3 — bring it to the next session

Bring it to class, done on paper. Nothing to hand in or send: the first 15–20 minutes of next session are spent going through it together at the board — you walk us through your working, we correct it there and then. Mistakes are the useful part, so bring what you tried even where you got stuck.

Two parts. Part 1 on paper. Part 2 typed. Expect 2–3 hours. This is the closest rehearsal of the midterm you will get before writing it, so treat Part 2 as a dry run.

Part 1 — Drill set (paper, show all work)

H1 (the full study). Study π(q) = −2q² + 640q − 24,000 completely and sketch it. Give, in this order: domain · zeros · sign line · π' and the stationary point · π'' and its nature · behaviour as q → ∞ · a labelled sketch with all four key points marked.

H2. C(q) = 0.2q² + 4q + 80 (€ thousand), q ≥ 0. (a) Fixed cost. (b) Average cost AC(q), its domain, and its limits as q → 0⁺ and q → ∞. (c) The quantity that minimises AC, and the value of AC and of C' there. Comment in one line on what you find.

H3 (optimisation). A firm has R(q) = 40 ln(q + 1) and C(q) = 0.2q² + 4q + 80, both € thousand. (a) Write π(q) and π'(q). (b) Solve π'(q) = 0, showing the quadratic it reduces to. (c) Confirm the nature of the point with π''. (d) Is the firm profitable there? Give the number — and if the answer surprises you, say in one line what it means for the decision. (Maximising profit and making a profit are not the same thing.)

H4. f(q) = q³ − 9q² + 24q on [0, 6]. (a) Stationary points and their nature. (b) Inflection point. (c) Sketch.

H5 (integration, exam form). Evaluate: (a) ∫ (q⁴ − 2q² + 5)/q² dq (b) ∫ (6q² − 4q + 3) dq (c) ∫ (5/q − 2√q) dq

H6 (exam form). Find f(q) given f'(q) = 15q² − 8q + 3 and f(1) = 12.

H7. Marginal cost is MC(q) = 1.2q + 5; fixed cost is 40. (a) Recover C(q). (b) Compute ∫₁₀²⁰ MC(q) dq and say, in one clause, what that number is.

Part 2 — Board report (400–500 words, typed)

This is a rehearsal of the midterm. A firm has a cost function C(q) = 0.2q² + 4q + 80 and a revenue function R(q) = 40 ln(q + 1), both in € thousand per month.

Write a short report for the Board that:

  1. States the profit function and studies it: domain, sign, the quantity that maximises it, and proof that it is a maximum.
  2. Includes a sketch of the profit function, hand-drawn, with axes labelled and the maximum marked.
  3. Answers this question in plain English: at the optimum, how much room for error does the firm have? Support it with the profit at the optimum and at one quantity either side.
  4. Ends with a recommendation of one sentence.

Structure it with headings. Numbers in the text, working in an appendix. The midterm asks for a longer version of exactly this, with an added qualitative section and Harvard referencing — which we brief next.

Self-check before submitting

  • The study follows the seven steps in order, and each one is labelled.
  • π'' is computed and its sign is stated, not just the formula: that is what proves maximum or minimum.
  • The sketch has: both axes labelled, the zeros, the maximum, and the vertical intercept.
  • Every integral is split term by term before integrating, and every indefinite integral ends in + c.
  • In H6 the constant is found from f(1) = 12.

How this feeds the assessments

  • H1 and Part 2 are the midterm: study a function, sketch it, tell a Board what it means.
  • H2 and H3 are the resit’s “full function study” question, which uses an average-cost or a log-revenue firm.
  • H5 and H6 are two of the five items of the final exam’s Q5, in the exact form they appear; three of its five items are integrals.