Suppose a company's revenue function is given by R(q)=−q3+320q2R(q)=-q3+320q2 and its cost function is given by C(q)=290+20qC(q)=290+20q, where qq is hundreds of units sold/produced, while R(q)R(q) and C(q)C(q) are in total dollars of revenue and cost, respectively.

A) Find a simplified expression for the marginal profit function. (Be sure to use the proper variable in your answer.)

MP(q)=MP(q)=     

B) How many items (in hundreds) need to be sold to maximize profits? (Round your answer to two decimal places.)

2.  The profit function from manufacturing and selling xx BabCo Lounge Chairs is:
P(x)=30x−140−0.2x^2

a. Find the exact additional profit for manufacturing and selling 10 chairs instead of 9 chairs.

b. Find the marginal profit at x=9  

= per lounge chair.

3.  Acme Office Supplies manufactures file cabinets. The cost (in dollars) of producing x file cabinets is given by:

C(x)=1025+60x−x^2
a. Find the exact additional cost of producing 7 file cabinets instead of 6.

b. Find the marginal cost function. C'(x)=

c. Use the marginal cost function approximate the additional cost of producing 7 file cabinets instead of 6.

4. The cost function for the production of microwaves is given as

C(x)=50,000+40xC(x)=50,000+4

where x is the number of microwaves produced and C(x) is the total cost (in dollars) of producing x units.

Find the marginal cost as a function of x. C'(x)=

5. The total profit (in dollars) from the production and sales of xx espresso machines is

P(x)=40x−0.02x^2−260

a. How many espresso machines must be produced and sold to have a marginal profit of 32 dollars per unit:
machines

b. Find the marginal profit at a production/sales level of 350 machines:
dollars per espresso machine

c. Use the profit at 350 machines, which is $11290, and the marginal profit at 350 machines that you computed above to estimate the profit at an output/sales level of 351.

=$

6. The price-demand function for the sale of yo-yos is:

p=6−0.02x

where p is the price of a yo-yo in dollars, and x is the demand for yo-yos at a price of p dollars.
a. R'(290)=

b. What are the correct units for R'(290)?

7. The price-demand and cost functions for the production of microwaves are given as

p=250−x80

and C(x)=16000+30x

where x is the number of microwaves that can be sold at a price of p dollars per unit and C(x) is the total cost (in dollars) of producing x units

a. Find the profit function in terms of x.
P(x)=

b. Evaluate the marginal profit function at x=1500 microwaves rounded to the nearest cent.
P'(1500)=    per microwave

8.  AnselPix is an online company that makes and sells photographs of National Parks. The profit from selling xx prints of a scene in Arches National Park is P(x) dollars. AnselPix believes that the profit from making and selling 150photos will be 22,000 dollars. Assume that the marginal profit is P'(150)=−150. As AnselPix's financial advisor, would you recommend that they sell more photos or fewer photos? Why?

Fill in the first blank with either the word "more" or "fewer" and the second with the word "increase" or "decrease".

I would recommend that AnselPix sell (Select an answer, more/ fewer?) photos because the company will (Select an answer decrease/increase?) profit by approximately ($?) if they decide to make and sell the 151st photo instead of 150 photos.

9. The revenue (in dollars) from producing and selling xx navigation systems is

R(x)=x(2100−30x)

a. Find the marginal revenue function.
R'(x)=

Find the inverse of the matrix

[ 1 1 4 ]

[ 3 2 4 ]

[ 1 1 6 ]

It is a 3*3 matrix

5. The population of certain small organism grows according to the model: dy/dt = 5y where y=20 when t=0

t is measured in months. a) Construct the specific function for y. b) Find how many in the population of this organism after 8 months. c) Determine how many months until there are 228 of this organism. Round to the nearest 0.1

Find the local maximum and minimum values and saddle point(s) of the function. If you have three-dimensional graphing software, graph the function with a domain and viewpoint that reveal all the important aspects of the function. (Enter your answers as a comma-separated list. If an answer does not exist, enter DNE.) f(x, y) = x^3 + y^3 − 3x^2 − 3y^2 − 9x local maximum value(s) Correct: Your answer is correct. local minimum value(s) Incorrect: Your answer is incorrect. saddle point(s)

Use a triple integral to find the volume of the solid under the surfacez = x^2 y and above the triangle in the xy-plane with vertices (1.2) , (2,1) and (4, 0).

a) Sketch the 2D region of integration in the xy plane

b) find the limit of integration for x, y ,z

c) solve the integral

(sry abt this but, please read the question properly, i've already recieved 3 wrong answers because the one who answered didnt look the question properly, they jsut do the double integral somehow and leave it)

Check all of the following that are true for the series ∑n=1∞(n−8)cos(nπ)/(n^2).
Converge, diverge, integral test, comparison test, limit comparison, ratio test, and alternation test.

Same with ∑n=1∞8^n/((4^n)-1)

A car rental agency rents 420 cars per day at a rate of ​$40 per day. For each ​$1 increase in​ rate, 10 fewer cars are rented. At what rate should the cars be rented to produce the maximum​ income? What is the maximum​ income?

The rental agency will earn a maximum income of ​$ _____ when it charges ​$ _____ per day.

Find the mass, the center of mass, and the moment of inertia about the z-axis for the hemisphere x^2+y^2+z^2=1, z >(greater than or equal to) 0 if density is sqrt(x^2+y^2+z^2)

A spring with a mass of 2 kg has a damping constant 14 kg/s. A force of 3.6 N is required to keep the spring stretched 0.3 m beyond its natural length. The spring is stretched 0.7 m beyond its natural length and then released. Find the position of the mass at any time t. (Assume that movement to the right is the positive x-direction and the spring is attached to a wall at the left end.) What is x(t)?

Series Solutions of Ordinary Differential Equations For the following problems solve the given differential equation by means of a power series about the given point x0. Find the recurrence relation; also find the first four terms in each of two linearly independent solutions (unless the series terminates sooner). If possible, find the general term in each solution.

(4-x²)y"+2y=0, x₀

1. Determine if each of the infinite series below converges or diverge. State the criteria used to make the determination, and show the work.

1. (7 pts)      n=1∞sin(1/n)

2. (7 pts) n=1∞ln(n)/n

3. (7 pts)     n=1∞2n/n!

Find the third MacLaurin polynomial for the function f(x) = e2x sin(3x)

1. (8pt) Consider the function f(x) = (x + 3)(x + 5)^2 √ (7 − x) whose first and second derivatives are f'(x) = ((x + 5)(139 + 12x − 7x^2))/2 √ (7 − x) , f''(x) = (35x^3 − 225x^2 − 843x + 3481)/ 4(7 − x)^3/2 .

Note: 35x 3 − 225x 2 − 843x + 3481 has three roots: x1 ≈ 7.8835, x2 ≈ −4.3531 and x3 ≈ 2.8982.

(a) (1pt) What is the domain of f(x)? (b) (1pt) What are the x-intercepts and y-intercepts of f(x)? (c) (1pt) What are the critical numbers of f(x)? (d) (1pt) On what open interval(s) is f(x) increasing, and on what open interval(s) f(x) is decreasing? (e) (1pt) What are the x-values of potential points of inflection of f(x)? (f) (1pt) On what open interval(s) is f(x) concave-upward, and on what open interval(s) f(x) is concave-downward? (g) (2pt) Sketch y = f(x). Clearly label the intercepts, extrema, and points of inflection.