Let C(x) represent the cost of producing x items and p(x) be the sale price per item if x items are sold. The prot P(x) of selling x items is P(x) = xp(x) - C(x) (revenue P(x) dP minus costs). The average prot per item when x items are sold is T and the marginal prot is a. The marginal prot approximates the prot obtained by selling one more item given that x items have already been sold. Consider the following cost functions C and price functions p. Complete parts a through d below. C(x) = 0.04x2 + 80x + 300, p(x) = 200 - 0.1x, a = 700 E) 3. Find the prot function P. The prot function is P(x) = |:|. 90 p2+3 A store manager estimates that the demand for an energy drink decreases with increasing price according to the function d(p) = , which means that at price p (in dollars). d(p) units can be sold. The revenue generated at price p is R(p) = p - d(p) (price multiplied by number of units). a. Find and graph the revenue function. h. Find and graph the marginal revenue R'(p). c. From the graphs of R and R', estimate the price that should be charged to maximize the revenue. a. R(p) = D A spring hangs from the ceiling at equilibrium with a mass attached to its end. Suppose you pull downward on the mass and release it 10 in. below its equilibrium position. The distance x (in inches) of the mass from its equilibrium position after t seconds is given by the function x(t) = 10 sin t - 10 cos t, where x is positive when the mass is above the equilibrium position. 3. Graph and interpret this function. dx b. Find a and interpret the meaning of this derivative. x>0 c. At what times is the velocity of the mass zero? ,1. , TO d. The function given here for x is a model for the motion of a spring. In what ways is this model unrealistic? Eggkffum ' ' ' r ' ' ' xi x 0 > o > o > )- 2 2' )- D En E: El -1 1 -1 -1 'l Awoman attached to a bungee cord jumps from a bridge that is 34 rn above a river. Her height in meters above the river t seconds after the jump is y(t)=17(1+ e "t cost) , fort20. a. Determine her velocity at t= 1 and t= 4. b. Use a graphing utility to determine when she is moving downward and when she is moving upward during the rst 10 s. c. Use a graphing utility to estimate the maximum upward velocity. E) a. Her velocity at time t is given by the function v(t) = D