Let f(x) = x^3 − x on [−1, 1].

Find the true area of the shaded region using a limit of Riemann sums, taking the sample points to be the left endpoints. Hint: to ease the computations, you can use the fact that (a + b)^ 3 = a ^3 + 3a^2 b + 3ab^2 + b^3 (which follows by expanding the cube).

Verify your work by evaluating the corresponding definite integral using the FTC.

The quantity, q, of a certain skateboard sold depends on the selling price, p, in dollars, so we write q = f(p). You are given that f(100) = 15200 and f '(100) = −90.

(a) What does f(100) = 15200 tell you about the sales of skateboards?

When the price of the skateboard is $_________ , then ______ skateboards will be sold.

What does f '(100) = −90 tell you about the sales of skateboards?

If the price increases from $100 to $101, the number of skateboards sold would _______ (increase/decrease) by roughly________ skateboards

(b) The total revenue, R, earned by the sale of skateboards is given by R = pq. Find R '(p).(chose)

-R '(p) = p 'q

-R '(p) = pf '(p) + f(p)   

- R '(p) = p + q

-R '(p) = q

-R '(p) = f '(p) + f(p)

(c) If the skateboards are currently selling for $100, what happens to revenue if the price is increased to $101?
The revenue  ---Select--- (increases/decreases) by roughly $ _______.

.Suppose I am in a boat and I travel at the bearing N70E at 30 knots for 4 hours. Then, I turn 90 degrees clockwise and travel for 5 hours at the same speed. Find my bearing relative to the dock.

Determine over which intervals the following functions are increasing, decreasing, concave up, and concave down

y = x + sin(πx)

So we have a paraboloid x^2 + y^2 - 2 = z and the plane x + y +z = 1 how do we find the center of mass? For some reason we have to assume the uniform density is 8?

Seems complicated because I don't know where to start?

Water is leaking out of an inverted conical tank at a rate of 11000 cubic centimeters per min at the same time that water is being pumped into the tank at a constant rate. The tank has height 15 meters and the diameter at the top is 5.5 meters. If the water level is rising at a rate of 25 centimeters per minute when the height of the water is 1.5 meters, find the rate at which water is being pumped into the tank in cubic centimeters per minute. Round to the nearest whole number.

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)