1-The revenue from the sale of a product is, in dollars,
R = 1500x + 3000(4x + 3)−1 − 1000
where x is the number of units sold. Find the marginal revenue when 200 units are sold. (Round your answer to two decimal places.)
Interpret your result.
If the sales go from 200 units sold to
units sold, the revenue will increase by about $
2-The total physical output P of workers is a function of the number of workers, x. The function
P = f(x)
is called the physical productivity function. Suppose that the physical productivity of x construction workers is given by
P = 6(3x + 6)3 − 9.
Find the marginal physical productivity, dP/dx.
3-Suppose that the revenue (in dollars) from the sale of a product is given by
R = 30x + 0.9×2 − 0.002×3
where x is the number of units sold. How fast is the marginal revenue
x = 20?
4-The daily cost per unit of producing a product by the Ace Company is 30 + 0.5x dollars and the price on the competitive market is $70.
Find the revenue function, cost function, profit function, and marginal profit function (in dollars).
What is the maximum daily profit the Ace Company can expect on this product?
5-Analysis of daily output of a factory shows that, on average, the number of units per hour y produced after t hours of production is
y = 70t + 0.5t2 − t3, 0 ≤ t ≤ 8.
(a) Find the critical values of this function. (Assume
−∞ < t < ∞.
Enter your answers as a comma-separated list.)
Correct: Your answer is correct.
(b) Which critical values make sense in this particular problem? (Enter your answers as a comma-separated list.)
(c) For which values of t, for 0 ≤ t ≤ 8, is y increasing? (Enter your answer using interval notation.)
6-Suppose that a company’s daily sales volume attributed to an advertising campaign is given by the following equation.
t + 4
(t + 4)2
(a) Find how long it will be before sales volume is maximized.
(b) Find how long it will be before the rate of change of sales volume is minimized. That is, find the point of diminishing returns.
7-If the profit function for a product is
P(x) = 1500x + 20×2 − x3 − 12,000
dollars, selling how many items, x, will produce a maximum profit?
Find the maximum profit.
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