Determine the level of production and the corresponding

Question

Question 1:

Some years ago it was estimated that the demand for steel approximately satisfied the equation p = 240 - 30x, and the total cost of producing x units of steel was C(x) = 168 + 60x. (The quantity x was measured in millions of tons and the price and total cost were measured in milions of dollars.)

Determine the level of production and the corresponding price that maximize the profits.

The maximum profit occurs at x= ___(million tons).

The corresponding price is ___ dollars per ton.

Question 2:

Use the given intonation to make a good sketch of the function f(x) near x = 4

f(4) = - 5, f'(4) = 2, f''(4) = 1

Choose the correct graph below:

Question 3:

A pharmacist wants to establish an optimal inventory policy for a now antibiotic that requires refegeration in storage. The pharmacist expects to sell 4500 packages of this antibiotic at a steady rate during the next year. She plans to place several orders of the same size spaced equally throughout the year. The ordering cost for each delivery is 515. and carrying costs, based on the average number of packages in inventory, amount to 56 per year for one package. Use this information to answer the following questions.

(a) Let x be the order quantity and r the number of orders placed dunng the year Find the inventory cost (ordering cost plus carrying cost) in terms of x and r.

(b) Find the constraint function

(c) Determine the economic order quantity that minimizes the inventory cost and then find the minimum inventory cost.

Question 4:

The graph of the following function has one relative maximum point and one relative minimum point. Find these points using the first derivative test.

f(x)= x³ - 3x² - 9x + 1

The relative maximum is __.
The relative minimum __.

Question 5:

The number of farms in the United States t years after 1925 is f(t) million, where f(t), f'(t), and f''(t) are graphed below. Use the scale on the right for f'(t) and f''(t). Use the scale on the left for f(t).

(a) Approximately how many farms were there in 1990?
(Type an integer or decimal rounded to the nearest tenth as needed.)

(b) At what rate was the number of farms declining in 1990?
(Type a whole number.)

(c) In what year were there about 4.3 million farms?

(d) When was the number of farms declining at the rate of 90,000 farms per year?
(Use a comma to separate answers as needed.)

(e) When was the number of farms declining fastest?

Question 6:

Sketch the graph of the following function.

f(x) = 3 - 11x + 3x² - x³

Choose the correct graph below.

Question 7:

The graph of the function has one relative extreme point. Plot this point and check the concavity thew. Using only this information, sketch the graph.

f(x)= -x² -6x - 5

The relative extreme point on the graph is __. (Type an ordered pair.)

Since the value of f'' at the relative extreme point is __, the graph is ___ at this point.

Which of the following is the graph of f(x)?

Question 8:

Draw the graph of the function y = f(x) that satisfies the following properties.

Both the function and the slope decrease as x increases.

Select all graphs that satisfy both properties.

Question 9:

Sketch the following curve, indicating all relative extreme points and inflection points

y = 3 - 3x² - x³

The relative extreme points are __.
(Type an ordered pair. Use a comma to separate answers as needed.)

The inflection points are __.
(Type an ordered pair. Use a comma to separate answers as needed )

Use the graphing tool to graph the curve.

Question 10:

The first and second derivatives of the function f(x) have the values given in the table below. Find the x-coordinates of all relative extreme points and find the x-coordinates of all inflection points.

x	f'(x)	f''(x)
2 ≤ x< 4	Positive	Negative
4	0	Negative
4 < x < 5	Negative	Negative
5	Negative	0
5 < x < 6	Negative	Positive
6	0	0
6 < x ≤ 8	Negative	Negative

The x-coordinates for the relative extreme points are ___ (Use a comma to separate answers as needed.)

The x-coordinates for the inflection points are ___ (Use a comma to separate answers as needed )

Question 11:

A cable runs along the wall from C to Pat a cost of $3 per meter, and straight from P to M at a cost of $5 per meter. Let x be the distance from C to P. If M is 8 meters from the nearest point A on the wall where P lies, and A is 18 motors from C, find x such that the cost of installing the cable is minimized and find this cost.

The cost is minimized when x is ___ meters.
The minimum installation cost is $___.

Question 12:

Find the minimum value of f(t) = 2t³ - 6t² + 30, t ≥ 0, and give the value of I where this minimum occurs.

The minimum value occurs when t = __ with a value of __.

Question 13:

Describe the following graph. Your description should include each of the six categories.

On which interval(s) is the function increasing? Select the correct choice below and, if necessary, fill in the answer box to complete your choice.
A. __
(Type an Inequality. Use a comma to separate answers as needed )

B. There is no interval on which the function is increasing.
On which interval(s) is the function decreasing? Select the correct choice below and, if necessary, fill in the answer box to complete your choice-

A.
(Type an inequality. Use a comma to separate answers as needed.)

B. There is no interval on which the function is decreasing
Where is/are the relative maximum point(s)? Select the correct choice below and, if necessary, fill in the answer box to complete your choice.

A.  x = __ (Use a comma to separate-answers as needed )
B. There are no relative maximum points
Where is/are the relative minimum point(s)? Select the correct choice below and, if necessary, fill in the answer box to complete your choice.

A. x = (Use a comma to separate answers as needed.)
B. There are no relative minimum points
What is the absolute maximum value? Select the correct choice below and. if necessary, fill in the answer box to complete your choice