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
I really need help with this question, it is the hard problem in my set and i'm not able to get it.
To estimate the average value of the function using a Right Rectangle Approximation, you can follow these steps:
1. Calculate the width of each interval: In this case, you have 4 intervals between the given points: [4, 9], [9, 11], [11, 14], and [14, 15]. The width of each interval can be obtained by subtracting the left endpoint from the right endpoint.
- Interval 1: 9 - 4 = 5
- Interval 2: 11 - 9 = 2
- Interval 3: 14 - 11 = 3
- Interval 4: 15 - 14 = 1
2. Calculate the heights of the rectangles: The heights of the rectangles can be approximated by taking the value of the function at the right endpoint of each interval. So, for each interval, you will use the value of f(x) at the right endpoint.
- Interval 1: f(9) = -11
- Interval 2: f(11) = -18
- Interval 3: f(14) = -21
- Interval 4: f(15) = -25
3. Calculate the area of each rectangle: To calculate the area of each rectangle, you multiply the width of the interval by the height of the corresponding rectangle.
- Rectangle 1: 5 * (-11) = -55
- Rectangle 2: 2 * (-18) = -36
- Rectangle 3: 3 * (-21) = -63
- Rectangle 4: 1 * (-25) = -25
4. Calculate the total area: To find the total area, you add up the areas of all the rectangles.
Total Area = -55 + (-36) + (-63) + (-25) = -179
5. Calculate the average value of the function: The average value of the function is given by the total area divided by the sum of the widths of all the intervals.
Average Value = Total Area / Sum of widths
Average Value = -179 / (5 + 2 + 3 + 1) = -179 / 11 ≈ -16.273
Therefore, the estimated average value of the function using a Right Rectangle Approximation is approximately -16.273.