Download Business Calculus

Business Calculus
Optimization Worksheet
On this sheet R(x) will refer to a revenue function, C (x) will refer to a cost function, P(x) will refer to a profit
function and lowercase p(x) will refer to a price (or demand) function. Lowercase x will represent the number of
items sold of a particular product and therefore will not ever be a negative amount.
FACTS:
1. Revenue = number of items sold multiplied by the price of the item = xp(x)
2. Profit = Revenue minus Cost =
R( x)  C ( x)
3. Average cost = Cost divided by the number of items sold =
C ( x)
x
4. A company will “break even” when the revenue is equal to cost or when the profit is zero.
5. Anytime you see “marginal of” that means take the first derivative of that particular function.
***************************************************************************************
If p( x)  5  0.001x is a price (demand) function and C( x)  35  1.5x is the cost function for a product,
find the following:
1. The revenue function.
2. The profit function.
If p( x)  70  0.002x is a price (demand) function and C ( x)  8000  50 x  0.03x 2 is the cost function
for a product, find the following:
3. The revenue function.
4. The profit function.
Find the value(s) of x that give a BREAK EVEN POINT given the following information:
5. P( x)  1500  20 x  x 2
6. R( x)  500  115x  x 2 and C ( x)  20x
7. p( x)  40  x and C( x)  10x  200
8. P ( x)  14000  680x  x 2
9. R( x)  400  95x  x 2 and C ( x)  20x
Find the requested marginal for the given function:
10. marginal profit if P ( x)  x 3  50 x  2000.
11. marginal revenue if R( x)  50 x  0.5 x 2 .
12. marginal revenue if R( x)  100x  0.2 x 2 .
13. marginal revenue if R( x)  800x  x 2 .
14. marginal cost if C ( x)  900  300 x .
15. marginal average cost if C( x)  15x  3000.
16.
17.
18.
19.
marginal profit if R( x)  500  115x  x 2 and C( x)  20x.
marginal profit if p( x)  40  x and C( x)  10x  200.
marginal profit if p( x)  400  0.3x and C( x)  200x  1100.
marginal average cost if C( x)  3500x  14000.
Use marginals to find the value of x that produces the following:
20. maximum revenue if R ( x)  800x  0.2 x 2 .
21. minimum average cost if C ( x)  1.25x 2  25x  800.
22. maximum revenue if R ( x)  48x 2  0.02 x 3 .
23. minimum average cost if C ( x)  x 3  300x  400.
24. maximum revenue if R( x)  500x  x 2 .
25. minimum average cost if C ( x)  x 3  3500x  16000.
26. maximum profit if P( x)  300x  0.01x 2  5000.
27. maximum profit if R( x)  150x  0.03x 2 and C( x)  90x  4000.
28. maximum profit if p( x)  500  0.001x and C( x)  100x  15000.
29. maximum profit if p( x)  90  x and C( x)  100  30x.
30. maximum profit if p( x)  6000  0.4 x 2 and C( x)  2400x  5200.
31. price that gives the maximum profit in #29.
32. price that gives the maximum profit in #30.
Answers:
1. 5x  0.001x 2
2. 3.5x  0.001x 2  35
3. 70x  0.002x 2
4. 20x  0.032x 2  8000
5. 50
6. 100
7. 10 and 20
8. 700
9. 80
10. 3x 2  50
11. 50  x
12. 100 0.4x
13.
400  x
800x  x 2
150
14.
x
 3000
15.
x2
16. 95  2 x
17. 30  2 x
18. 200 0.6x
19.
20.
21.
22.
23.
24.
25.
26.
27.
28.
29.
30.
31.
32.
 14000
x2
2,000
640  25
1600
3
200  5.84
250
20
15,000
1,000
200,000
30
3000  55
$60
$4,790