Q1 Simplify exp (2 ln π΄ + 3 ln π΅ − 4 ln πΆ + 5 ln π·^{6}), where π΄, π΅, πΆ, π· are all positive quantities.
Q2 Revenue π (π, π) = ππ is a function of two variables: price π and quantity π. Sketch a contour map of π in the region 0 ≤ π ≤ 10, 0 ≤ π ≤ 20.
HINTS: (i) What mathematical shape is exemplified by the equation π₯π¦ = 12? What would a contour map of the function π(π₯, π¦) ≡ π₯π¦ look like?
Q3 
(a) 
By gathering berries for π‘ hours, Robinson Crusoe can obtain π(π‘) kg of 


berries, where π(π‘) = 5√π‘. How many kg of berries will he obtain in his fifth 


hour of labour? 

(b) 
Robinson works continuously, starting at 8:00 am. At noon, what is his 


marginal product of labourtime? 
Q4 
(a) 
National income π currently stands at $20 tn/yr (trillion dollars per year) and consumption πΆ = πΆ(π) at $16 tn/yr. If marginal consumption πΆ^{'}(π) = 0.7, 


use the Small Increment Formula to obtain an approximation to the level of 


consumption if π rises by $0.2 tn/yr. 

(b) 
The general price level π is rising over time according to the formula π(π‘) = 


π_{0}(exp(πΌπ‘) + ππ‘), where π_{0}, πΌ, and π are positive parameters. Calculate: 


(i) The rate of increase of prices, πΜ at π‘; 


(ii) The rate of growth of prices π^ at π‘. Then 


(iii) Evaluate both πΜ and π^ when π‘ = 0. 
Q5  Week 3: Unconstrained Optimization With A Single Choice Variable
Q5 Consider the optimization problem whose formal statement is
min π(π₯), where π(π₯) ≡ 3π₯^{4} − 4π₯^{3} − 12π₯^{2} + 10 [P]
x
Q6 (a) Mr Jones has utility function π(π΄, π΅) = π΄π΅. He loses π bananas, reducing his banana holding to π΅ − π. How many extra apples π will he need as compensation, to restore him to his former level of utility?
(b) From consuming π΄ apples and π΅ bananas, Ms Jones gets utility π(π΄, π΅) =
√π΄π΅ + π΄ + 2π΅. What is Ms Jones’ marginal utility with respect to apples, in terms of π΄ and π΅ ?
Q7 My utility function over apples (A) and bananas (B) is π(π΄, π΅) = π΄^{2}π΅. If apples cost
$6 per kg, and bananas cost $4 per kg, and I have $50 to spend. I wish to maximize utility, subject to the constraint that the value of my purchases equals by budget limit $50.
Q8 (a) Express the set {π₯: π₯ ∈ β_{(+)} ∧ (π₯^{2} < 16)} using interval notation.
HINT. Let π mean ‘The workforce is welleducated’. Let π‘ mean ‘The economy can thrive’.
Q9. (a) What is meant by [2,3)? Explain informally, then formally, using bound variable notation.
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