1. Sketch the graph of the cubic function f(x) = 6+12x + 3x² − 2x³ for −2 ≤ x ≤3.
2. Sketch the graph of the cubic function f(x) = 8x³ +30x² +13x−15 for −4 ≤ x ≤ 2.
1. The fixed costs of producing a good are 12 and the variable costs are 7 per unit. Find expressions for TC and AC. Evaluate TC and AC when Q = 4 and Q = 12. Sketch the graph of AC as a function of Q.
2. The fixed costs of producing a good are 9 and the variable costs are 4+3Q per unit. Find expressions for TC and AC. Evaluate TC and AC when Q = 5 and Q = 10. Sketch the graph of AC as a function of Q.
3. Suppose that the total cost to a furniture company of producing Q desks is TC = 50Q + 40 000. Obtain an expression for the average cost function, AC. What value does AC approach when Q is very large?

1. Find the inverses of the following functions:
a) f(x) = −3x + 2,
b) f(x) = 5x + 3,
c) f(x) = (x − 3)² + 2, 3 ≤ x.
2. Sketch the functions y = 2x and y = 3x on the same graph for −2 ≤ x ≤ 2.
3. Evaluate (a) log3 9, (b) log4 2, (c) log7(1/7).
4. An economy is forecast to grow continuously at an annual rate of 3% so that the gross national product (GNP), measured in billions of euros, after t years is given by: GNP = 60e⁰·⁰³ᵗ
a) Calculate the current value of GNP and its future value in four years’time.
b) After how many years is GNP forecast to be 90 billion euros?
5. The percentage, y, of households possessing dishwashers t years after they have been introduced in a country is modelled by y = 30 − 25ᵉˉ⁰·²ᵗ

a) Find the percentage of households that have dishwashers
i. at their launch;
ii. after 1 year;
iii. after 10 years;
iv. after 20 years.
b) After how many years will the percentage of households possessing dishwashers first reach 15%?

6. A firm’s turnover, y, measured in millions of pounds, after t years is given by: y = 8e⁰·⁰⁹ᵗ
a) What is its turnover in its initial year of trading and after two years oftrading?
b) After how many years will its turnover have doubled since it started trading?
Extra exercises
Solve the equation ln(𝑥 + 4)² = 3 𝑓 or 𝑥
Express log𝑎 3 + log𝑎 4 − log𝑎6 as a single logarithm

1. Sketch the graph of the cubic function f(x) = 6+12x + 3x² − 2x³...