The function is continuous on the interval [10, 20] with some of its values given in the table below. Estimate the average value of the function with a Left Hand Sum Approximation, using the intervals between those given points.
X: 10 12 15 19 20
f(x): -2 -5 -9 -12 -16
by this time I assume you know how to do a Riemann sum to approximate the area. The average value is just the area divided by the width of the interval.
This is just like figuring distance traveled by time * average speed.
To estimate the average value of the function using a Left Hand Sum Approximation, we need to calculate the sum of the areas of rectangles under the function between the given points.
First, let's label the given points and the intervals between them.
X: 10 12 15 19 20
f(x): -2 -5 -9 -12 -16
The intervals we have are: [10, 12], [12, 15], [15, 19], and [19, 20].
To calculate the sum of the areas, we will use the formula for the area of a rectangle, which is base times height. In this case, the base will be the width of each interval, and the height will be the function value at the left endpoint of each interval.
For the interval [10, 12]:
- Base: (12 - 10) = 2
- Height: -2
Area = Base * Height = 2 * (-2) = -4
For the interval [12, 15]:
- Base: (15 - 12) = 3
- Height: -5
Area = Base * Height = 3 * (-5) = -15
For the interval [15, 19]:
- Base: (19 - 15) = 4
- Height: -9
Area = Base * Height = 4 * (-9) = -36
For the interval [19, 20]:
- Base: (20 - 19) = 1
- Height: -12
Area = Base * Height = 1 * (-12) = -12
Now, sum up the areas of all the intervals:
Total area = (-4) + (-15) + (-36) + (-12) = -67
To estimate the average value of the function, divide the total area by the total width of the intervals:
Total width = (12 - 10) + (15 - 12) + (19 - 15) + (20 - 19) = 2 + 3 + 4 + 1 = 10
Average value = Total area / Total width = -67 / 10 = -6.7
Therefore, the average value of the function using a Left Hand Sum Approximation is approximately -6.7.