The function f is continuous on the interval [4, 15], with some of its values given in the table above. Estimate the average value of the function with a Right Rectangle Approximation, using the 4 intervals between those given points.

x 4 9 11 14 15
f(x) -6 -11 -18 -21 -25

-11.545
-14
-16.273
-18.75

Please help me. I really don't know what to do.

You have to find the average value of the function.

average value = [1/(b-a)] [the integral of f(x) dx]

The integral will be solved using the right approximation.
a and b are the domain, so a = 4 and b = 15

average value = [1/(15 - 4)] [(-11)(5) + (-18)(2) + (-21)(3) + (-25)(1)]
average value = 1/11 [-179]
average value = -179/11 = -16.273

Oh dear, it seems like you've stumbled into the land of math! Don't worry, I'll do my best to guide you through this problem with a touch of humor.

We're given some values of the function f(x) at specific points, and we want to estimate the average value using a Right Rectangle Approximation. To do this, we're going to create rectangles that "approximate" the area under the curve (which in this case is the function) between those given points.

First, let's label the intervals between the given points as I₁, I₂, I₃, and I₄. These intervals can be thought of as the bases of our rectangles.

Now, to approximate the area under the curve, we'll take the length of each interval (b) and multiply it by the height of the rectangle (h). The height of each rectangle can be found by taking the value of f(x) at the right endpoint of the interval.

So, for I₁, the length (b) is 5 (9 - 4) and the height (h) is -11.

For I₂, the length (b) is 2 (11 - 9) and the height (h) is -18.

For I₃, the length (b) is 3 (14 - 11) and the height (h) is -21.

Finally, for I₄, the length (b) is 1 (15 - 14) and the height (h) is -25.

Now, let's calculate the area of each rectangle by multiplying the base and height:

Area of I₁ = 5 * (-11) = -55
Area of I₂ = 2 * (-18) = -36
Area of I₃ = 3 * (-21) = -63
Area of I₄ = 1 * (-25) = -25

To estimate the average value of the function, we need to take the sum of these areas and divide it by the total length of the interval, which is 11 (15 - 4).

Sum of areas = -55 + (-36) + (-63) + (-25) = -179

Average value = Sum of areas / Length of interval = -179 / 11 ≈ -16.273

So, my dear friend, the estimated average value of the function using the Right Rectangle Approximation is approximately -16.273. Just remember, math may not always be a laughing matter, but with a little humor, we can make it more enjoyable! Keep up the good work!

To estimate the average value of a function using a Right Rectangle Approximation, you need to calculate the area of each rectangle and then divide the total area by the number of rectangles.

In this case, you have 4 intervals between the given points, which means you can create 4 rectangles.

First, calculate the width of each rectangle by subtracting the x-values of consecutive points. In this case, the width of the first rectangle is (9 - 4) = 5, the width of the second rectangle is (11 - 9) = 2, the width of the third rectangle is (14 - 11) = 3, and the width of the fourth rectangle is (15 - 14) = 1.

Next, calculate the height of each rectangle. To approximate the height of each rectangle, you can use the right endpoint (since it's a right rectangle approximation). The right endpoint of each interval is the f(x) value given in the table. So, for the first rectangle, the height is -11, for the second rectangle the height is -18, for the third rectangle the height is -21, and for the fourth rectangle the height is -25.

Now, calculate the area of each rectangle by multiplying the width by the height. For the first rectangle, the area is 5 * (-11) = -55. For the second rectangle, the area is 2 * (-18) = -36. For the third rectangle, the area is 3 * (-21) = -63. For the fourth rectangle, the area is 1 * (-25) = -25.

Add up the areas of all the rectangles: -55 + (-36) + (-63) + (-25) = -179.

Finally, to estimate the average value of the function, divide the total area by the total width: -179 / (5 + 2 + 3 + 1) = -179 / 11 ≈ -16.273.

Therefore, the estimated average value of the function using a Right Rectangle Approximation with the 4 intervals between the given points is approximately -16.273.

So, the correct answer is -16.273.

Use the mid-point rule with n = 4 to approximate the area of the region bounded by y = x3 and y = x

The x values given are the right-left edges of the rectangles. You want right-side values, so figure f(x) at

x = 9,11,14,15

Multiply each height by its interval's width, and the approximation is just

5*11+2*18+2*21+1*25 = 158

So, the average height is the area divided by the width: 158/11 = 14.36 (minus, of course) or -14

As for y=x^3, note that the graphs intersect at x=0,1.

So, you want 4 intervals of width 0.25, and all you do is find

f(x) at x = 1/8, 3/8, 5/8, 7/8

multiply each value by 0.25 and add 'em up.

It's just a bunch of rectangles, not something scary.