MATH150-E01 Test #2 Summer 2016 Show all work. Name 1. Find an equation in slope-intercept form for the line through (4, 2) and (1, 3).

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1 1. Find an equation in slope-intercept form for the line through (4, 2) and (1, 3). 2. Let the supply and demand functions for sugar be given by p = S(q) = 1.4q 0.6 and p = D(q) = 2q where p is the price per pound and q is the quantity in thousands of pounds. a. Graph these on the same axes. b. Find the equilibrium quantity and equilibrium price.

2 3. In deciding whether to set up a new manufacturing plant, company analysts have decided that a linear function is a reasonable estimation for the total cost C(x) in dollars to produce x items. They estimate the cost to produce 10,000 items as $547,500, and the cost to produce 50,000 items as $737,500. a. Find a formula for C(x). b. Find the fixed cost. c. Find the total cost to produce 100,000 items. d. Find the marginal cost of the items to be produced in this plant. 4. Give the domain of the function f(x) = x2 3 x.

3 5. The cost to rent a midsized car is $54 per day or fraction of a day. If the car is picked up in Pittsburgh and dropped off in Cleveland, there is a fixed $44 drop-off charge. Let C(x) represent the cost of renting the car for x days, taking it from Pittsburgh to Cleveland. a. Find C( 3 4 ) b. Find the cost of renting the car for 2.4 days. c. Graph y = C(x). 6. Graph the following parabola and give its vertex, axis of symmetry, x-intercepts, and y-intercept. f(x) = 1 2 x2 + 6x + 24

4 7. The manager of a peach orchard is trying to decide when to arrange for picking the peaches. If they are picked now, the average yield per tree will be 100 lb., which can be sold for 80 per pound. Past experience shows that the yield per tree will increase about 5 lb. per week, while the price will decrease about 4 per pound per week. a. Let x represent the number of weeks that the manager should wait. Find the income per pound. b. Find the number of pounds per tree. c. Find the total revenue from a tree. d. When should the peaches be picked in order to produce maximum revenue? e. What is the maximum revenue? 8. Find any horizontal and vertical asymptotes and any holes that may exist in the graph of the following rational function. Draw the graph of the function, including ant x- and y-intercepts. y = 3 2x 4x+20

5 9. Suppose the average cost per unit C (x), in dollars, to produce x units of yogurt is given by C (x) = 600 x+20. a. Find C (10) and C (75). b. Give the equations of any asymptotes. Find any intercepts. c. Graph y = C (x). 5g(x) Let lim f(x) = 9 and lim g(x) = 27. Use the limit rules to find lim. x 4 x 4 x 4 1 f(x)

6 11. Use the properties of limits to help decide whether the following limit exists. If it does exist, find its (x+h)3 x3 value. lim h 0 h 12. Find all values of x = a where the function is discontinuous. For each value of x, give the limit of the function as x approaches a. Be sure to note when the limit doesn t exist. f(x) = 5+x x(x 2)

7 13. Find the value of the constant k that makes the function continuous. f(x) = { kx2 if x 2 x + k if x > Suppose that the total profit in hundreds of dollars from selling x items is given by P(x) = 2x 2 5x + 6. a. Find the average rate of change of profit for a change in x from 2 to 4. b. Find the instantaneous rate of change of profit with respect to the number of items produced when x = 2.

8 15. Suppose the position of an object moving in a straight line is given by s(t) = t 2 + 5t + 2. Find the instantaneous velocity at time t = 6.