You may be given raw data concerning costs and revenues. In that case, you ll need to start by finding functions to represent cost and revenue.

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1 Example 2: Suppose a company can model its costs according to the function 3 2 Cx ( ) x 0.04x 200x 70, 000 where Cxis ( ) given in dollars and demand can be modeled by p 0.02x 300. a. Find the revenue function. b. Find the break even point. c. Find the smallest positive quantity for which all costs are covered. You may be given raw data concerning costs and revenues. In that case, you ll need to start by finding functions to represent cost and revenue. 3

2 Example 3: Suppose you are given the cost data and demand data shown in the tables below. quantity produced total cost quantity demanded price in dollars a. Find a cubic regression equation that models costs, and a quadratic regression equation that models demand. Begin by entering the data in GGB. Then create lists. Cubic Cost Model: Quadratic Demand Model: b. State the revenue function. c. Find the break-even point. d. Find the smallest positive quantity for which all costs are covered. 4

3 Market Equilibrium The price of goods or services usually settles at a price that is dictated by the condition that the demand for an item will be equal to the supply of the item. If the price is too high, consumers will tend to refrain from buying the item. If the price is too low, manufacturers have no incentive to produce the item, as their profits will be very low. Market equilibrium occurs when the quantity produced equals the quantity demanded. The quantity produced at market equilibrium is called the equilibrium quantity and the corresponding price is called the equilibrium price. Mathematically speaking, market equilibrium occurs at the point where the graph of the supply function and the graph of the demand function intersect. We can solve problems of this type either algebraically or graphically. Example 4: Suppose that a company has determined that the demand equation for its product is 5x 3p 30 0 where p is the price of the product in dollars when x of the product are demanded (x is given in thousands). The supply equation is given by 52x 30 p 45 0, where x is the number of units that the company will make available in the marketplace at p dollars per unit. Find the equilibrium quantity and price. 5

Example 5: The quantity demanded of a certain electronic device is 8000 units when the price is $260. At a unit price of $200, demand increases to 10,000 units.

Assume that both the supply equation and the demand equation are linear. Find the supply equation, the demand equation and the equilibrium quantity and price.

4 Example 5: The quantity demanded of a certain electronic device is 8000 units when the price is $260. At a unit price of $200, demand increases to 10,000 units. The manufacturer will not market any of the device at a price of $100 or less. However for each $50 increase in price above $100, the manufacturer will market an additional 1000 units. Assume that both the supply equation and the demand equation are linear. Find the supply equation, the demand equation and the equilibrium quantity and price. Question 4: A manufacturer has a monthly fixed cost of $80, and a production cost of $22 for each unit produced. The product sells for $33 per unit. Find the break-even point. a. 0, 3200 b , c , d , e. None of the above 6

5 Math 1314 Test 2 Review Lessons Given f ( x) = 2x x 2. A. Find any zeros of f. B. Find any local (relative) extrema of f. C. Find f '( 0.25) and f ''( 0.25) 2. Given f ( x) = 2 x 2 2e + 3x 2 x 1. A. Find any zeros of f. B. Find any extremum of f Test 2 Review 1

6 3. The following table of values gives a company s annual profits in millions of dollars. Rescale the data so that the year 2003 corresponds to x = 0. Create a list of points. Year Profits (in millions of dollars) A. Find the cubic regression model for the data. B. Find the R 2 value for the cubic regression model. C. Use the cubic regression model to predict the company's profits in D. Find the exponential regression model for this data. 4. The graph of f ( x ) is shown below. A. x 4 B. + x 4 C. x Test 2 Review 2

7 5. Suppose 2x 5, x 1 f x x x 2 ( ) = 8, 1 < > x 1, x 2 Determine, if they exist, A. B. C. + x 1 x 1 x x lim x 1 x 5 7. x + 3 lim x 4 2 x 8 2 x + 5x lim x 2 x x x lim x x lim x + 5x 7x 1 x x x 1314 Test 2 Review 3

8 3 x lim x 2 x 7x Enter the function into GGB. Look at the graph to determine your answer. 12. The graph of f ( x ) is shown below. Which of the following statements is true? I. x 2 exists and is equal to 3. II. x 5 exists and is equal to 3. III. IV. x 6 x 2 does not exist. does not exist; there is a hole where x = 2. V. does not exists; there is unbounded behavior as x approaches 4. x Test 2 Review 4

9 13. The graph of f ( x ) is shown below. Which of the following statements is true? I. The function is continuous at x = 3. II. The function is discontinuous at x = 3 because x 3 does not exist. III. The function is discontinuous at x = 3 because f(3) does not exist. IV. The function is discontinuous at x = 3 because even though f(3) exists and exists, the two quantities are not equal. x Find the first and second derivative: f x = x x + x + x ( ) x 5 x 15. Let f ( x) = 2x ln( x 1) + e A. Find the slope of the tangent line at x = 3. B. Write the equation of the tangent line at the given point Test 2 Review 5

10 2 16. Find the average rate of change of f ( x) = 0.28x 0.11x on the interval f ( x + h) f ( x) [ 1.5, 4 ]. Recall: = average rate of change/difference quotient h 17. The model gives the number of bacteria in a culture t hours after an experiment begins. What will be the bacteria population 6 hours after the experiment begins? 18. A country s gross domestic product (GDP) in billions of dollars, t years from now, is projected to be for 0 t 5. What will be the rate of change of the country s GDP 2 years from now? 19. A ball is thrown upwards from the roof of a building at time t = 0. The height of the ball in feet is given by, where t is measured in seconds. Find the velocity of the ball after 3 seconds. 20. Suppose a manufacturer has monthly fixed costs of $250,000 and production costs of $24 for each item produced. The item sells for $40. Assume all functions are linear. State the: A. cost function. B. revenue function. C( x) = cx + F R( x) = sx c = cost/unit; F = fixed costs s = selling price C. profit function. P( x) = R( x) C( x) 1314 Test 2 Review 6

11 D. Find the break-even point. Recall: R( x) = C( x) 21. Cost data and demand data for a company's best-selling product are given in the tables below. Create two lists. Quantity produced 1,000 2,000 3,000 4,000 Total cost $13,400 $14,200 $14,900 $15,400 Quantity demanded 1,000 2,000 3,000 4,000 Price in dollars $10.75 $10.15 $9.85 $9.70 A. Find linear regression model for cost. B. Find the linear regression model for demand. Then find the revenue function. Linear Demand Equation: Linear Revenue Equation: Recall: R( x) = px D. Use the linear cost and revenue function to find the number of items that must be sold to break even on that product. Round your answer to the nearest unit Test 2 Review 7

12 22. Suppose that a company has determined that the demand equation for its product is 5x + 3p 30 = 0 where p is the price of the product in dollars when x of the product are demanded (x is given in thousands). The supply equation is given by 52x 30 p + 45 = 0, where x is the number of units that the company will make available in the marketplace at p dollars per unit. Find the equilibrium quantity and price. The following formulas will be provided with Test 2. It will be a link. f ( x + h) f ( x) f ( b) f ( a) = h b a C( x) = cx + F R( x) = sx or R( x) = xp P( x) = R( x) C( x) 1314 Test 2 Review 8

Example 11: A country s gross domestic product (in millions of dollars) is modeled by the function

Example 11: A country s gross domestic product (in millions of dollars) is modeled by the function Math 1314 Lesson 7 With this group of word problems, the first step will be to determine what kind of problem we have for each problem. Does it ask for a function value (FV), a rate of change (ROC) or