The function is continuous on the interval [10, 20] with some of its values given in the table above. Estimate the average value of the function with a Right Hand Sum Approximation, using the intervals between those given points.
x 10 12 15 19 20
f(x) –2 –5 –9 –12 –16
my bad - average = sum/(20-10)
To estimate the average value of the function with a Right Hand Sum Approximation, we need to use the intervals between the given points.
First, let's calculate the width of each interval:
Interval 1: [10, 12], width = 12 - 10 = 2
Interval 2: [12, 15], width = 15 - 12 = 3
Interval 3: [15, 19], width = 19 - 15 = 4
Interval 4: [19, 20], width = 20 - 19 = 1
Next, let's calculate the right-hand sum for each interval. The right-hand sum uses the right endpoint of each interval as the height:
Interval 1: [10, 12], right-hand sum = f(12) = -5
Interval 2: [12, 15], right-hand sum = f(15) = -9
Interval 3: [15, 19], right-hand sum = f(19) = -12
Interval 4: [19, 20], right-hand sum = f(20) = -16
Lastly, let's calculate the average value using the right-hand sum approximation:
Average value = (right-hand sum of interval 1 * width of interval 1 +
right-hand sum of interval 2 * width of interval 2 +
right-hand sum of interval 3 * width of interval 3 +
right-hand sum of interval 4 * width of interval 4) /
(total width of intervals)
Average value = ((-5 * 2) + (-9 * 3) + (-12 * 4) + (-16 * 1)) /
(2 + 3 + 4 + 1)
Average value = (-10 - 27 - 48 - 16) / 10
Average value = -101 / 10
Therefore, the average value of the function with a Right Hand Sum Approximation is approximately -10.1.
To estimate the average value of the function using a Right Hand Sum Approximation, you need to calculate the sum of the areas of the rectangles formed by each interval between the given points.
First, identify the width of each rectangle, which is the difference between consecutive x-values. In this case, the widths are 2, 3, 4, and 1 for the intervals [10,12], [12,15], [15,19], and [19,20], respectively.
Next, calculate the height of each rectangle, which is the value of the function at the right endpoint of each interval. The right endpoint values given in the table are -5, -9, -12, and -16 for the intervals [10,12], [12,15], [15,19], and [19,20], respectively.
Now, find the area of each rectangle by multiplying the width by the height. The areas are calculated as follows:
[10,12]: width = 2, height = -5, area = 2 * -5 = -10
[12,15]: width = 3, height = -9, area = 3 * -9 = -27
[15,19]: width = 4, height = -12, area = 4 * -12 = -48
[19,20]: width = 1, height = -16, area = 1 * -16 = -16
Finally, sum up the areas of all the rectangles:
Total area ≈ -10 + (-27) + (-48) + (-16) = -101
To estimate the average value of the function, divide the total area by the sum of the widths:
Average value ≈ -101 / (2 + 3 + 4 + 1) = -101 / 10 = -10.1
Therefore, the average value of the function using a Right Hand Sum Approximation is approximately -10.1.
using the right-hand sums, we get
(12-10)(-5) + (15-12)(-9) + (19-15)(-12) + (20-19)(-16) = -101
Now, the average is just the sum/4