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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