Math 1314 Test 3 Review Material covered is from Lessons The total weekly cost of manufacturing x cameras is given by the cost function: 3 2

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1 Math 1314 Test 3 Review Material covered is from Lessons The total weekly cost of manufacturing x cameras is given by the cost function: 3 2 C( x) = x + 0.4x + 800x + 3, 000. A. Find the marginal cost function. B. Use the marginal cost function to approximate the cost of producing the 157 th camera. 2. A company has the given demand function: p = 0.02x A. Find the revenue function. B. Find the marginal revenue function. C. Use the marginal revenue function to approximate the revenue realized from the sale of the 234 th unit. 3. A music company produces a variety of electric guitars. The total cost of producing x 1 2 guitars is given by the function C( x) = x x where C(x) is given in dollars. 5 Find the average cost of producing 130 guitars. C( x) Recall: C( x) = x Math 1314 Test 3 Review 1

2 4. The tables below give the costs associated with the production of x items and pricing information when x items are demanded. Assume that the cost data is exponential and the demand data is linear. x ,000 Cost in dollars of producing x items 112, , , , ,300 x ,000 Price when x items are demanded Create two lists in GGB then: A. Find the exponential cost function. B. Find the linear demand function. C. Find the revenue function. D. Find the profit function. E. Use GGB and the marginal profit function to approximate the actual profit realized on the sale of the 651 st item. Recall: Demand is said to be elastic if E( p ) > 1. Demand is said to be unitary if E( p ) = 1. Demand is said to be inelastic if E( p ) < Suppose E(p) = 1 4 when the price of the item is p. Then the demand is a. Elastic b. Unitary c. Inelastic Math 1314 Test 3 Review 2

3 6. Suppose the demand equation of a product is given by p = -0.04x where the function gives the unit price in dollars when x units are demanded. Compute E(p) when p = 535 and interpret the results. p f ( p) Recall: E( p) = f ( p) 7. The sales from company ABC for the years are given below. Year Profits in millions of dollars Rescale the data so that x = 0 corresponds to A. Find an exponential regression model. B. Find the rate at which the company's sales were changing in Math 1314 Test 3 Review 3

4 8. The number of deer present in a nature preserve can be expressed using the model 125 N( t) = 0.6t 1 31e, where N(t) gives the number of deer and t gives the number of months + since the initial count of deer was taken. Enter the function in GGB. A. How many deer will be present after 6 months? B. At what rate is the population changing after 6 months? 9. At the beginning of an experiment, a researcher has 511 grams of a substance. If the half-life of the substance is 16 days: A. Identify two points given in the problem. B. Find an exponential regression model using the two points in part a. C. How many grams of the substance are left after 25 days? D. What is the rate of change after 10 days? 10. The graph given below is the first derivative of a function, f. A. Find any critical numbers of f. B. Find any intervals where the f is increasing/decreasing. Math 1314 Test 3 Review 4

5 11. Let f ( x) = 0.2x x x 5. Enter the function in GGB. A. Find any critical numbers of f. B. Interval(s) on which f is increasing; interval(s) on which f is decreasing. C. Coordinates of any relative extrema. D. Interval(s) on which f is concave upward; interval(s) on which f is concave downward. Math 1314 Test 3 Review 5

6 E. Coordinates of any inflection points. F. Find any vertical asymptotes and any horizontal asymptotes. 12. Let 3 2 x 4x 7 f ( x) =. Enter the function in GGB. x A. Interval(s) on which f is increasing; interval(s) on which f is decreasing. B. Coordinates of any relative extrema. Math 1314 Test 3 Review 6

7 C. Interval(s) on which f is concave upward; interval(s) on which f is concave downward. D. Coordinates of any inflection points. E. Find any vertical asymptotes and any horizontal asymptotes. 13. Find the absolute maximum and absolute minimum of this function. Math 1314 Test 3 Review 7

8 14. Find the value of x that gives the absolute maximum value of the interval [-1, 2]. Enter the function in GGB. 4 5 f ( x) = 2x 5x 2 on 15. The mosquito population is a function of rainfall, and can be approximated by the 2 3 formula N( x) = x x, where x is the number of inches of rainfall. Note that x is non-negative. Math 1314 Test 3 Review 8

9 16. An open top box is constructed from a rectangular sheet of material by cutting equal squares from each corner and folding up the flaps. The dimensions of the sheet are 18 inches by 9 inches. A. Write a function that will give the volume of the box. B. Find the critical numbers for the function. C. Find the dimensions of the box with maximum volume. D. Find the maximum volume. Math 1314 Test 3 Review 9

10 17. A farmer has 420 feet of fencing to enclose 2 adjacent rectangular pig pens sharing a common side. The two adjacent pens have the same dimensions. A. Write a function that models the enclosed area. B. Find the critical numbers for the function. C. What dimensions should be used for each pig pen so that the enclosed area will be a maximum? D. What is the maximum area? Math 1314 Test 3 Review 10