SECTION A (25 MARKS)
Question one (25 Marks)
a. Given a function ?(?)=10?5−5?4+2?2+? find:
i. ?(2)(2 Marks)
ii. ?′(−3)(2 Marks)
b. The fixed costs of producing a good are 10 and the variable costs are
4+6? per unit
i. Find expressions for total cost, TC and average cost, AC(4 Marks)
ii. Evaluate TC and AC when ?=14(4 Marks)
c. Suppose that the total cost to an electronics company of producing Q flat screens televisions is ??=780?+10000 obtain an expression for the average cost function and find the average cost of production When Q is very large (4 Marks)
d. Differentiate the following functions:
i. √?3(2 Marks)
ii. ?=1?8(2 Marks)
e. Differentiate the function (?)=?6+2? . Hence calculate the slope of the graph of ?=?6+2? at the point ?=4(3 Marks)
f. Find expression for ???? for the profit function ?=2?3+30?2−20?−10
(2 Marks)
SECTION B (45 MARKS)
Question two (15 Marks)
a. If the average cost function of a good is ??=2?+6+13?
i. Find an expression for marginal cost, MC (3 Marks)
ii. If the current output is 15, estimate the effect on TC of a 3-unit decrease in Q(4 Marks)
b. If the demand function is ?=120−3?
i. Find an expression for TR interms of Q(3 Marks)
ii. Find the value of MR at ?=10 using differentiation and a 1 unit increase approach(5 Marks)
Page 3 of 3
Question three (15 Marks)
a. Differentiate ?=3−2?3+2?(4 Marks)
b. Find ?′ and ?′′ given ?2−??+?2=3(6 Marks)
c. If the total revenue function, TR of a good is given by 100?−?2
i. Write down an expression for the marginal revenue function, MR
(2 Marks)
ii. If the current demand is 70 estimate the change in the value of TR due to a 3-unit increase in Q(3 Marks)
Question four (15 Marks)
a. Determine the elasticity of demand when the price falls from 136 to 119 given the demand function ?=200−?2(5 Marks)
b. Given the demand function ?=50−2? find the elasticity when the price is 30. Is the demand inelastic, unit inelastic or elastic at this price?(5 Marks)
c. Given the demand function ?=−?2−4?+96 find the price elasticity of demand when ?=51. If this price rises by 2%, calculate the corresponding percentage change in demand(5 Marks)
Question five (15 Marks)
a. Differentiate the following functions
i. ?=(?2+4)2(2?3−1)3(4 Marks)
ii. ?=(?2−3)4(3 Marks)
b. Find ?′,?′′and?′′′ at:
i. the point (2,1) on ?2−?2−?=1(4 Marks)
ii. the point (1,1) on ?3+3?2?−6??2+2?3=0(4 Marks)
