Function f(x) is positive, decreasing and concave up on the closed interval
[a, b]. The interval [a, b] is partitioned into 4 equal intervals and these are used to compute the left sum, right sum, and trapezoidal rule approximations for the value of integral from a to b f(x)dx. Which one of the following statements is true?
a) Left sum < trapezoidal rule value < Right sum
b) Left sum < Right sum < trapezoidal rule value
c) Right sum <trapezoidal rule value < left sum
d) Cannot be determined without the x-values for the partitions
My answer is C, is that correct for you Damon?
To determine which statement is true, let's first review what each type of approximation represents.
1. Left sum approximation: This approximates the integral by calculating the area of rectangles using the left endpoints of each subinterval. It tends to underestimate the actual area.
2. Right sum approximation: This approximates the integral by calculating the area of rectangles using the right endpoints of each subinterval. It tends to overestimate the actual area.
3. Trapezoidal rule approximation: This approximates the integral by calculating the area of trapezoids formed by connecting the endpoints of each subinterval. It provides a more accurate estimate than the left and right sum approximations.
Given that the function is positive, decreasing, and concave up, the left sum approximation will underestimate the integral since it uses rectangles with heights smaller than the function values. The right sum approximation will overestimate the integral since it uses rectangles with heights larger than the function values. The trapezoidal rule, being a combination of left and right sums, tends to provide a more accurate estimate.
Now, let's analyze the given answer choices:
a) Left sum < trapezoidal rule value < Right sum: This statement contradicts the properties of the left and right sums. The trapezoidal rule value should be between the left and right sums.
b) Left sum < Right sum < trapezoidal rule value: This statement is also incorrect because the trapezoidal rule value should be between the left and right sums.
c) Right sum < trapezoidal rule value < Left sum: This statement is also incorrect because the right sum should be larger than the trapezoidal rule value.
d) Cannot be determined without the x-values for the partitions: This statement is unnecessary because the information provided is sufficient to determine the correct answer.
Therefore, the correct statement is (a) Left sum < trapezoidal rule value < Right sum.
To determine which statement is true, we need to understand how the left sum, right sum, and trapezoidal rule work.
The left sum is an approximation of the area under the curve using left endpoints of each subinterval. This approximation will underestimate the actual value of the integral since it only considers the leftmost points.
The right sum, on the other hand, is an approximation of the area under the curve using right endpoints of each subinterval. This approximation will overestimate the actual value of the integral since it only considers the rightmost points.
The trapezoidal rule is an approximation that uses the average height of two consecutive points in each subinterval to calculate the area. It is more accurate than both the left and right sums because it takes into account more information.
Given that f(x) is positive, decreasing, and concave up on the closed interval [a, b], we can infer the following:
- Since f(x) is positive, the area under the curve will be positive as well.
- Since f(x) is decreasing, the left sum will yield a smaller approximation than the right sum. This is because the left sum only considers the lower values of the function, while the right sum considers the higher values.
- Since f(x) is concave up, the function will be curved in an upwards manner. This means that the trapezoidal rule, which considers both endpoints and the slope between them, will yield a higher approximation than both the left and right sums.
With these considerations, we can conclude that the correct statement is:
a) Left sum < trapezoidal rule value < Right sum