Math

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 $
.

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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.
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
MR
changing when
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).
R(x)
=

C(x)
=

P(x)
=

MP
=

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.)
t =

Correct: Your answer is correct.

(b) Which critical values make sense in this particular problem? (Enter your answers as a comma-separated list.)
t =

(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.
S(t) =
8
t + 4

40
(t + 4)2
+ 7
(a) Find how long it will be before sales volume is maximized.
t =
days

(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.
t =
days

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?
x =
items

Find the maximum profit.
$

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