Homework 2 — 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 a short typed text. Expect 2–3 hours. Formative: not graded, but the walkthrough is where you find out what you have really understood.

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

H1. State the domain of each, as an interval, and say in one clause what the restriction means for a firm. (a) f(q) = √(2q − 30) (b) g(q) = ln(40 − q) (c) h(q) = (q + 5)/(q² − 9)

H2. A workshop has C(q) = 0.4q² + 8q + 200 and sells at a fixed price of €60. (a) Write R(q) and π(q). (b) Evaluate π(30), π(65), π(100). (c) From those three numbers alone, between which two quantities must the best output lie? Say how you know, without computing it.

H3. Evaluate, showing the factoring step: (a) lim (q→4) (q² − 16)/(q − 4) (b) lim (q→2) (q² − q − 2)/(q − 2) (c) lim (q→0⁺) (q² + 3q)/q

H4 (exam form). Evaluate lim (q→2⁺) (q − 2)²/(2q − q²). Write out the factoring and comment on the sign of each factor as q approaches 2 from above.

H5. Evaluate and give the business reading of each: (a) lim (q→∞) (0.4q² + 8q + 200)/q — average cost per unit as the run gets long. Why does this one not settle down, and what does that say about the workshop? (b) lim (q→∞) (200 + 15q)/q.

H6. Differentiate: (a) π(q) = −3q² + 900q − 40,000 (b) C(q) = 0.4q² + 8q + 200 (c) R(q) = 45 ln(q + 1) (d) f(q) = 2q⁴ − 6q² + q − 9

H7. For π(q) = −3q² + 900q − 40,000: (a) Compute π'(50) and write the one sentence you would say to the owner. (b) Compute π'(200) and do the same. (c) Solve π'(q) = 0 and state what that quantity is.

H8 (exam form). Find f(q) given f'(q) = 9q² − 4q + 2 and f(1) = 10.

Part 2 — Memo to the Board (150–250 words, typed)

You are the plant manager of Arno Bikes. Last month you produced 80 bikes. The Board has asked for one number: should the next bike be built?

Answer it with the derivative. Give the marginal profit at 80, say what it means in euros, and state the quantity at which the answer changes from yes to no. Then add one sentence on what you would need to check before trusting the number — the model says demand is p = 1600 − 4q and cost is 60,000 + 400q; say which of those two you would verify first, and why.

Numbers and their meaning. No formulas in the text; put the working in an appendix if you want it seen.

Self-check before submitting

Every domain is written as an interval, with [ or ( chosen deliberately.
Every limit that starts as 0/0 shows the factoring before the substitution.
Every derivative is done term by term, and the constant term has disappeared.
In H8 the constant of integration is found from the given condition, not left as c.
The memo says what one more unit does, in euros, and where the sign changes.

How this feeds the assessments

H1, H5 are the domain and asymptote questions the resit asks in exactly this form.
H3, H4 are the limit item of the final exam’s Q5: a 0/0 that factors, and a one-sided limit.
H6, H8 are the other two Q5 items: differentiate term by term, and recover a function from its derivative with an initial condition.
H2, H7 and the memo are the midterm: study a profit function and write the result for a Board.