Section 68 problems 3 5 12 2 section 69 problem 10 3

1. Section 6.8 Problems 3, 5, 12
2. Section 6.9 Problem 10
3. Section 6.10 Problem 5
4. Section 6.11 Problems 4, 5, 8
5. Section 7.1 Problem 6
6. Section 7.3 Problem 1
7. Section 7.4 Problems 6, 9
8. Section 7.7 Problems 1, 5
9. Section 8.2 Problems 5, 6, 7
10. Section 8.4 Problem 2
11. Section 8.5 Problems 3, 5
12. Section 8.6 Problems 2, 6
13. Section 8.7 Problems 3, 5
The remaining problems mostly relate chapter 8 to some earlier economics concepts.
14. Find the quantity which maximizes profit if the total revenue and total cost (in dollars) are given by
R ( q ) = 5 q - 0 . 003 q
2
C ( q ) = 300 + 1 . 1 q
where q is the quantity and 0 ≤ q ≤ 1000 units. What production level gives the minimum profit?
15. The total revenue and total cost curves for a product are given in Figure below.
(a) Sketch the marginal revenue and marginal cost, MR and MC, on the same axes. Mark the two quantities
where marginal revenue equals marginal cost. What is the significance of these two quantities? At which
quantity is profit maximized?
(b) Sketch the profit function π ( q ).
q
3
q
4
q
(quantity)
$
R ( q )
C ( q )

16. Show that if marginal cost is greater than average cost, then the derivative of average cost with respect to quantity
is positive.
17. If a ( t ) and b ( t ) are positive valued differentiable functions, and if A, α, β are constants find expressions for
y
0
y
.
(a) y = ( a ( t ))
2
b ( t )
(b) y =
a ( t )
5
b ( t )
(c) y = A ( a ( t ))
α
( b ( t ))
β
, where A is a constant.
18. If F ( x ) = f ( x
n
g ( x )), find a formula for F
0
( x ).
19. For the total cost function
C ( y ) = y
2
- 3 y + 2 , y > 0
show that and (illustrate on a graph)
(a) Marginal cost ( MC ) is less than average cost ( AC ) when average cost is decreasing
(b) MC = AC when the derivative of AC is zero.
(c) MC exceeds AC when AC is increasing.
20. A store has been selling 200 DVD burners a week at $350 each. A market survey indicates that for each $10
rebate offered to buyers, the number of units sold will increase by 20 a week. Find the demand function and the
revenue function. How large a rebate should the store offer to maximize its revenue?
21. (a) Show that the critical points of an average cost function occur when the marginal cost equals the average
cost.
(b) If C ( x ) = 16000 + 200 x + 4 x 3 / 2 , in dollars, find (i) the cost, average cost, and marginal cost at a production
level of 1000 units (ii) the production level that minimizes the average cost (iii) the minimum average cost.
22. (a) Show that if the profit is at a maximum, then the marginal revenue equals the marginal cost.
(b) If C ( x ) = 16000 + 500 x - 1 . 6 x 2 + 0 . 004 x 3 is the cost function and p ( x ) = 1700 - 7 x is the demand function,
find the production level that maximizes the profit.