Suppose that we have five websites: A, B, C, D, and E. Let's also suppose that the links between the sites are depicted in the graph below:

Here, the arrow pointing from C to D means that there is a hyperlink on site C that takes you to site D. For small sets of objects, graphs like this one are a convenient way to depict connections.

1. Create a linking matrix L containing the information of which site links to which, just as we did in the popularity example. Remember to normalize, and be sure that your input is exact. (For example, make sure you enter 1/3 instead of 0.3333—this is important for the next part of this exercise, since our columns must sum to 1.) Include all input and output from MATLAB.

2. Use the rref command to find all solutions x to the matrix equation (L - I)x = 0. Include all input and output from MATLAB. If you get an error message, be sure to double-check your answer for the first part of this exercise.

3. Which website has the highest PageRank? Explain your answer, especially in light of any negative numbers that may have appeared in your solutions. List the remaining websites in decreasing PageRank order.

Let V be the vector space of 2 × 2 real matrices and let P2 be the vector space of polynomials of degree less than or equal to 2. Write down a linear transformation T : V ? P2 with rank 2. You do not need to prove that the function you write down is a linear transformation, but you may want to check this yourself. You do, however, need to prove that your transformation has rank 2.

SOLVE the IVP: (D^2+1)y = e^t, y(0) = -1 and y'(0) = 1. Thank you.

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) = x³ − 27xy + 27y³

An investment firm recommends that a client invest in bonds rated​ AAA, A, and B. The average yield on AAA bonds is 4​%, on A bonds 5​%, and on B bonds 8​%. The client wants to invest twice as much in AAA bonds as in B bonds. How much should be invested in each type of bond under the following​ conditions?

A The total investment is ​$9,000​, and the investor wants an annual return of ​$470 on the three investments.

B The values in part A are changed to ​$24,000 and ​$1,250​, respectively.

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1 point) A company has found that the relationship between the price p and the demand x for a particular product is given approximately by p=1281−0.17x2. The company also knows that the cost of producing the product is given by C(x)=870+398x. Find P(x), the profit function. P(x) = Now use the profit function to do the following: (A) Find the average of the x values of all local maxima of P. Note: If there are no local maxima, enter -1000. Average of x values = (B) Find the average of the x values of all local minima of P. Note: If there are no local minima, enter -1000. Average of x values = (C) Use interval notation to indicate where P(x) is concave up. Note: Enter 'I' for ∞, '-I' for −∞, and 'U' for the union symbol. If you have extra boxes, fill each in with an 'x'. Concave up: (D) Use interval notation to indicate where P(x) is concave down. Concave down:

A retailer has been selling 1200 tablet computers a week at $350 each. The marketing department estimates that an additional 80 tablets will sell each week for every $10 that the price is lowered.
what is p(x) =
what price should be set for max revenue
If the retailer's weekly cost function is C(x) = 35,000 + 130x what price should it choose in order to maximize its profit?

A system of differential equations solved by the Laplace transform has led to the following system:

(s-3) X(s) +6Y(s) = 3/s

X(s) + (s-8)Y(s) = 0

Obtain the subsidiary equations and then apply the inverse transform to determine x (1)

1. Solve linear system using Gaussian elimination

a)

x₁ + 2x₂ + x₃ = 2

-x₁ − 3x₂ + 2x₃ = -3

  x₁ − 6x₂ + 3x₃ = -6

b)

-2b + 2c = 10

3a + 12b -3c = -6

6a + 18b + 0c = 19

c)

4x - 1y + 4z + 3t = 5

1x - 4z + 6t = 7

5x - 5y + 1z + 2t = -5

4x + 1y + 3z + 3t = 6

1). A group of students plans a tour. The charge per student is $66 if 25 students go on a trip. If more that 25 students participated, the charge per student is reduced by $2 times the number of students over 25. Find the number of students that will furnish the maximum revenue. The number of students that will furnish the maximum revenue is ______? The maximum revenue is $_______?

2) A motorboat is capable of traveling at a speed of 15 miles per hour in still water. On a particular​ day, it took 30 minutes longer to travel a distance of 18 miles upstream than it took to travel the same distance downstream. What was the rate of current in the stream on that​ day?

Consider the region illustrated below, bounded above by ? = ?(?) = 10 cos ( ?? 9 ) and below by ? = ?(?) = ? −?+3 . The curves intersect at the points (?, ?(?)) and (?, ?(?)), where 0 < ? < ? < 5. Do not try to find or estimate the values of ? or ?.

a. The total area of the region.

b. The volume of the solid that has this region as its base, where cross-sections perpendicular to the ?-axis are semicircles with their diameters on the xy-plane.

c. The volume of the solid that results by revolving this region about the x-axis.

d. The volume of the solid that results by revolving this region about the line ? = 5.

For the following find: (Show detailed working)

1) general solution of the associated homogns DE

2)  form of the particular solution associated with the undetermined coefficients method. Do not evaluate coefficients

a) (d² y / dx²) - dy/dx = x - 3

b) (d² y / dx²) + 2 dy/dx + y = (1+x)e^-x + x²

^c) (d² y / dx²) + y = sin(2x)

1. Explain how the first derivative test of a function determines where the function is increasing and decreasing.
2. Explain how to apply the second derivative test.
3. What is an inflection point?
4. Why are special methods such as L'Hopital's Rules, needed to evaluate indeterminate forms?