(a) Teddy J is a manufacturer of dish washing liquid. If his monthly demand function for 750ml size is q = 4000 − 250p and his total cost function is C(q) = 500 + 0.2q. Derive an expression, R(q) for Teddy J′s total revenue curve. Derive an expression, Π(q) for Teddy J ′ s profit function. Determine whether Teddy J′s profit is increasing or decreasing when he produces 5 hundred, 750ml

Problem 1

(a) Teddy J is a manufacturer of dish washing liquid. If his monthly demand function for 750ml size is q = 4000 − 250p and his total cost function is C(q) = 500 + 0.2q.

• Derive an expression, R(q) for Teddy J′s total revenue curve.
• Derive an expression, Π(q) for Teddy J ′ s profit function.
• Determine whether Teddy J′s profit is increasing or decreasing when he produces 5 hundred, 750ml bottles of dish washing liquid.

(iv) How many 750ml bottles of dish washing liquid should Teddy J produce per month if he wishes to maximize his profits.

(b) A firm has an average cost function A(q) = 125 q + q 2 16 − 4. Where q is the firm′ s output.

(i) Determine the level of output for average costs are minimum.

(ii) Hence determine the range of values for which average costs are decreasing.

(iii) What part of the decreasing range is practically feasible?

• Write an equation for the total cost function. (v) Hence calculate the level of output for which total costs are minimum.

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(a) The sales of a book publication are expected to grow according to the function S = 300000(1 − e −0.06t), where t is the time, given in days.

(i) Show using differentiation that the sales never attains an exact maximum value.

(ii) What is the limiting value approached by the sales function?

(b) A poll commissioned by a politician estimates that t days after he makes a statement denegrating women,the percentage of his constituency (those who support him at the time he made the statement) that still supports him is given by S(t) = 75(t 2 − 3t + 25) t 2 + 3t + 25 The election is 10 days after he made the statement.

(i) If the derivative S’(t) may be thought of as an approval rate, derivate the a function for his approval rate.

(ii) When was his support at its lowest level?

(iii) What was his minimum support level? (iv) Was the approval rate positive or negative on the date of the election?

 (c) Lara offers 100 autograph bats. If each is priced at p dollars, it is that the demand curve for the bast will be p = 250 − q 4 . If price elasticily is E(p) = dq q ÷ dp p . When |E(p)| < 1, demand is inelastic and when |E(p)| > 1, demand is elastic. (i) Find the price elasticity of demand for Lara′ s bats. [5 mks] (ii) Is demand inelastic or elastic? [1 mk]

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(a) A town has a population of 5000 persons, but is expected to grow by 2% every year.

(i) What would be the population size in 7 years?

(ii) Find the sum of the first eight terms of the sequence 1/8, − 1/4, 1/2,.....

(b) A landscape contractor is hired to cultivate ornamental plants in three new residential developments. The contractor charges the developer for each tree cultivated an hourly rate to cultivate the ornamental plants, and a fixed delivery charge. In one development it took 211 labour hours to cultivate 244 ornamental plants for a cost of $9394. In a second development it took 128 labour hours to cultivate 283 ornamental plants for a cost of $8270. In the final development it took 165 labour hours to cultivate 386 ornamental plants for a cost of $10938. In total 504 labour hours were taken and 913 ornamental plants were cultivated. Using Cramer’s Rule of the Inverse Method, determine the cost for each tree,the hourly labour charge, and the delivery charge.

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(i) Limx→∞ 5x −3 − 4 2x −2 + 9

(ii) Limx→∞ (x − 3) 2 x 2 2 − 2x − 3

(b) During a nationwide program to immunize the population against a new strain of the flu, public health officials determined that the cost of inoculating x% of the susceptible population would be approximately C(x) = 1.85x 100 − x million dollars.

(i) What would it cost to providing immunization to the first 20% of the susceptible population?

(ii) What would it cost to providing immunization to the next 30% of the susceptible population?

(iii) Suppose 17 million dollars are available for providing immunization. What percentage of the susceptible population will not receive immunization?

(iv) If money was not a problem will they be able to providing immunization to the entire susceptible population?

 (c) Determine the values of x for which the function f(x) = { 2x 2 − 4 x < 2 x + 2 2 < x > 5 7 x ≥ 5 is discontinuous.