What are the dimensions of a rectangle whose perimeter is 5600 units and whose area is as large as possible?

Consumers' Surplus The demand function for a certain model of Blu-ray player is given by p = 600 0.5x + 1 where p is the unit price in dollars and x (in units of a thousand) is the quantity demanded per week. What is the consumers' surplus if the selling price is set at $300/unit? (Round your answer to the nearest dollar.)

A) Find the equation of the tangent line to the curve y = 5e^-8x at the point (0, 5).

B) Solve for t.

e^0.09t = 9

C) Rancher Johann wants to build a three-sided rectangular fence near a river, using 280 yards of fencing. Assume that the river runs straight and that Johann need not fence in the side next to the river.

Johann wants to build a fence so that the enclosed area is maximized.

• What should be the length of each side running perpendicular to the river?
  yards
  
• What should be the length of the side running parallel to the river?
  yards
  
• What is the largest total area that can be enclosed?
  square yards

D) Find the absolute maximum and minimum values on the closed interval [-3,3] for the function below. If a maximum or minimum value does not exist, enter NONE.

f(x) = (4x)/(x² + 1)

E) When a baseball park owner charges $5.00 for admission, there is an average attendance of 100 people. For every $0.25 increase in the admission price, there is a loss of 2 customers from the average number.

1. What admission price should be charged in order to maximize revenue
   
2. What is the maximum revenue?

F) Find the derivative.

f(x) = x⁶ · e^2x

By cutting away identical squares from each corner of a rectangular piece of cardboard and folding up the resulting flaps, an open box may be made. If the cardboard is 15 in. long and 7 in. wide,

(a) Find the dimensions (in inches) of the box that will yield the maximum volume. (Round your answers to two decimal places if necessary.)

(b) Which theorem did you use to find the answer?

Is there a fourth degree polynomial that takes these values?

Let C be the curve of intersection of the parabolic cylinder x2 = 2y, and the surface 3z = xy. Find the exact length of C from the origin to the point

(5, 25/2,125/6)

An application of modeling with differential equations from a particular field, such as Physics,Biology, Business, Chemistry, etc.

1) Find the critical numbers of the function. 

f(θ) = 16 cos θ + 8 sin^2 θ

θ=?

2) Find the absolute maximum and absolute minimum values of f on the given interval.

f(x) = x/(x^2 − x + 9), [0, 9]

3) f(x) = 3x³ + 4x² + 7x + 5,    a = 5

(f ⁻¹)'(a) = ?

Suppose f is defined by f(x)=3x/(4+x^2), −1≤x<3.

What is the domain of f?

Find the intervals where f is positive and where f is negative.

Does f have any horizontal or vertical asymptotes. If so, find them, and show your supporting calculations. If not, briefly explain why not.

Compute f′ and use it to determine the intervals where f is increasing and the intervals where f is decreasing.

Find the coordinates of the local extrema of f

Make a rough sketch of the graph of f using only the information from the previous steps.

Solve the differential equation by variation of parameters.

2y'' + y' = 6x

Find the area of the following region. The region inside the inner loop of the​ limaçon r = 7 - 14 cos theta

Consider a single source procurement situation: You have a demand rate of 151 units per period, replenishment rate of 2,550 units per period, lead time of 8 periods, item cost of 21.18, cost per procurement of 7.77, holding cost per unit of 7.63, and per unit shortage cost of 10.43. Find the minimum cost procurement quantity..

Solve : y''+2y'+y=e^-x + sinx by Undetermined Coefficients method and Variation of Parameters

Consider the equation below. (If an answer does not exist, enter DNE.) f(x) = x3 − 9x2 − 21x + 9

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

(b) Find the local minimum and maximum values of f. local minimum value local maximum value

(c) Find the inflection point. (x, y) = Find the interval on which f is concave up. (Enter your answer using interval notation.) Find the interval on which f is concave down. (Enter your answer using interval notation.)

Consider the function f(x, y) = 4xy − 2x ⁴ − y ² .

(a) Find the critical points of f.

(b) Use the second partials test to classify the critical points.

(c) Show that f does not have a global minimum.