Homework 1 — 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 is done on paper; Part 2 is a short typed text. Expect 2–3 hours in total. Both parts are formative: not graded, but the walkthrough is where you find out what you have really understood before the midterm and the final.

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

H1. A café has fixed costs of €9,000 a month. Each coffee sells for €2.50 and costs €0.70 to make. (a) Break-even in coffees per month. (b) The owner wants a profit of €3,000 a month. How many coffees? (c) The price is cut to €2.20 to fight a competitor. New break-even, and the sentence you would tell the owner.

H2. Demand: Q_d = 900 − 3p. Supply: Q_s = −100 + 2p. (a) Equilibrium price and quantity. (b) A subsidy of €25 per unit is paid to sellers (they respond to p + 25). New equilibrium. (c) Sketch both situations on one graph: axes labelled, both lines, both equilibria marked.

H3. A firm’s profit is π(q) = −q² + 80q − 1,200. (a) Quantities with zero profit. (b) The profitable range, written as an interval. (c) The quantity that maximises profit and the maximum profit. (d) For which fixed cost (replace the 1,200) would the firm have exactly one break-even point?

H4. Solve: (a) x² − 9 > 0; (b) −x² + 4x − 3 ≥ 0. Write each answer as an interval or union of intervals.

H5. Factor and simplify (x − 3)² / (3x − x²). Evaluate at x = 3.01 and x = 2.99. What is the sign on each side of 3?

H6. A subscription business has 4,000 subscribers growing 6% a month. (a) Subscribers after t months (formula) and after 12 months (number). (b) Months to reach 10,000. (c) Months to double.

H7. Solve for t: (a) 200 · 1.05ᵗ = 350; (b) e^{0.08t} = 3.

H8. (a) Solve |x − 12| ≤ 4 and |2x − 1| > 5. (b) Give the domain, as an interval, of f(x) = ln(2x − 6) and of g(x) = √(10 − x).

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

You are the plant manager of Arno Bikes. Using the numbers from class (fixed cost €60,000, variable cost €400, demand p = 1600 − 4q), write a memo to the Board answering one question: at what production levels does this business make money, and how robust is that window?

Rules: numbers and their meaning, no formulas in the text (put one small table or the equations in an appendix if you want). Say what happens to the window if fixed costs rise by €10,000. End with one recommendation.

This is a rehearsal of the register the midterm asks for (“prepare a very detailed report for the Board”). It will be read for clarity and for whether the numbers are right, not for length.

Self-check before submitting

Every equation is written in words first (“break-even means revenue equals cost”) and then in symbols.
Every quadratic: common factor divided out before the formula; discriminant computed and its sign commented.
Every inequality: roots found, sign line drawn, answer written as an interval.
Every log problem: logs taken of both sides, the exponent brought down with ln(aᵏ) = k ln a, calculator only at the end.
Intervals use [ for included endpoints and ( for excluded ones and for ±∞.

How this feeds the assessments

H1–H2 rehearse the midterm’s Q4 (a cost shock moves a schedule) and LO3 (supply and demand).
H3–H5 are the algebra of the midterm’s study of function (zeros, sign, vertex) and of the factoring step in Q5 of the final exam.
H6–H7 are the log manipulation that week 4 (compound interest, EAR) and the log integrals in Q5 rely on.
H8 is the interval notation the resit asks for when it says “the domain of f(x) is…”.