2) A certain company does research and believes that the quantity, Q, that will be demanded by consumers is related to the price by the function ?(?) = 25000 − 320?2.

a) Find a formula for revenue, R(p). Then use your formula to find the total revenue when selling the items at $1.50 each, and find the total revenue when selling the items at $8.50 each. Show your work.

b) Use the revenue function and calculus to solve for the price that creates the highest revenue.
Show all work/clearly explain all steps to use calculus to solve the problem. You can use Excel to solve the equation, but you must clearly show what equation is being solved and explain your process. Write a complete sentence to explain what your answer tells us about the revenue.

c) Find a formula for elasticity of demand E(p). Then use your formula to find the elasticity when
the price of each item is $1.50, and find the elasticity when the price of each item is $8.50. Show your work. Then write a complete sentence to interpret the meaning in context including an explanation of whether or not it is elastic or inelastic at each price as well as whether or not they should or should not raise the price.

d) Use the elasticity function to solve for when the price elasticity of demand is equal to −1. Show the equation you are solving. You can solve the equation using Excel or you can solve by hand. Write a complete sentence to explain what your answer tells us about the revenue.

e) What do you notice about your answers to part ii) and part iv)? Write a short paragraph describing two methods that can be used to solve for the maximum revenue if we have a revenue function.

1. Find the partial derivatives of f(x,y) = xe^(4xy)

f_x=? f_y=?

2. Find the partial derivative of f(x,y) = sqrt(x^3 + 4y^5)

f_x=? f_y=?

General question: how do you find constraints in an optimization problem?

For example, in this question:"Find two natural numbers whose sum is 16 and whose product is a maximum." the solution states that the constraint is that x would be between 1 and 15. Why not 0?

In another example, where the questions asks "The perimeter of a rectangle is 24 cm. Find the dimensions of the rectangle of maximum area. What is the maximum area?", the constraint is listed as length is between 0 and 12?

So why is 0 listed here and not in the first question?

Making buried food is an ancient technique used today. A restaurant makes pure buried food to please its diners. Food usually comes out at a temperature of 145 ° C. In the kitchen it is expected a quarter of an hour for the food temperature to be 100 ° C. If the room temperature is 20 ° C in the dining room

a) How long should you eat, if it is recommended that the food temperature be served at 50 ° C?

b) Assuming the same constant of proportionality and the same room temperature as in part a), at what temperature should the food be taken out so that it can be eaten after 25 minutes?

Give an example of a relation that fits the given requirements. If you think that there is no such relation, you must explain why. In each case, you may not use any of the relations that appears earlier in this problem set. Unless told otherwise, you may use any of the following ways of defining a relation: set-list notation, set-builder notation, or a directed graph.

(a) A relation on {a,b,c,d} that is not reflexive, not antireflexive, and not transitive. You must use set-list notation for your answer.

(b) A relation on {a,b,c,d} that is not symmetric and not anti-symmetric. You must use a directed graph to define this relation.

(c) A relation on the set of all text strings that is symmetric, but not transitive. You must use set-builder notation.

Please write each step neatly and nicely.

Let f(x, y, z) = 3x^(2)y + 4e^(2x+y) − 6 cos(xyz)

(a) Find fx
(b) Find fy
(c) Find fz
(d) Find fxy
(e) Find fyx
(f) Find fxzx
(g) Find fzzz

^{The number of new businesses established in the US since 1990 can be modeled by the function Nx=110.8x^3-5305.5x^2+76,701x+332,892 where x = 0 represents 1990 and the domain is [0, 25].}

6. Use the Second Derivative Test to discuss the concavity of this function on the given interval. What is the POI? Interpret the meaning of the POI in the context of this problem.           

A company manufactures and sells x cellphones per week. The weekly​ price-demand and cost equations are given below.

p=400-.05x and C(x)=20,000 + 135x

A) What price should the company charge for the​ phones, and how many phones should be produced to maximize the weekly​ revenue? What is the maximum weekly​ revenue? The company should produce ? phones each week at a price of ​$ ?. ​(Round to the nearest cent as​ needed.) The maximum weekly revenue is ​$ ? ​(Round to the nearest cent as​ needed.) ​

(B) What price should the company charge for the​ phones, and how many phones should be produced to maximize the weekly​ profit? What is the maximum weekly​ profit? The company should produce ? phones each week at a price of ​$ ?. ​(Round to the nearest cent as​ needed.) The maximum weekly profit is ​$ ?. ​(Round to the nearest cent as​ needed.)

Solve the following given differential equations by using the following method: method of variations of parameters.

1. ?′′′ −?′′ −?′ +? = ??

2. ?′′ − 2?′ + ? = ?? sin−1 ?

An open rectangular box is made from a 9 inch by 12 inch piece of cardboard by cutting squares of side length ? from the corners. Determine the length of the sides of the square which will maximize the volume. (Clearly identify the function in terms of one variable and state the domain, then solve.)

Retailers will buy 40 items from a wholesaler if the price is $175 and 60 items if the price is $150. The wholesaler is willing to supply 30 items if the price is $140 and 70 items if the price is $185. Assuming that the resulting supply and demand functions are linear, find the equilibrium point for the market. *please show all work*

Find the absolute maximum and absolute minimum values of f(x) = cos(2x)+2 sin(x) in the interval [0; pi]

f(x) = 7xex

(a) Find the intervals on which f is increasing or decreasing. (Enter your answers using interval notation.)

(b) Find the local maximum and minimum values of f. (If an answer does not exist, enter DNE.)

(c) Find the intervals of concavity and the inflection points. (Enter your answers using interval notation.)

prove that f(x)=x^2019 +x-1 has only one real root

For the series , go through the entire list of series tests, and say

• Why the test cannot be applied or
• Why the test can be applied

List of Tests

• Geometric Series
• Test for Divergence
• Integral Test
• p-series Test
• Comparison Test
• Limit Comparison Test
• Alternating Series Test
• Absolute Convergence Test
• Ratio Test
• Root Test