Function f(x) is positive, increasing 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 upper sum, lower sum, and trapezoidal rule approximations for the value of (b)∫(a) f(x)dx
Which one of the following statements is true?
- Lower sum < Trapezoidal rule Value < Upper sum
- Lower sum < Upper sum < Trapezoidal rule value
- Trapezoidal rule < Lower sum < Upper sum
- Cannot be determined without the x-values for the partitions
To determine the correct statement, we need to understand the concepts of upper sum, lower sum, and the trapezoidal rule in approximating the value of the definite integral.
1. Upper Sum: It is an approximation of the definite integral using the maximum value of the function within each subinterval of the partition. It gives an upper bound since we are considering the maximum value within each interval.
2. Lower Sum: It is an approximation of the definite integral using the minimum value of the function within each subinterval of the partition. It gives a lower bound since we are considering the minimum value within each interval.
3. Trapezoidal Rule: It approximates the definite integral by dividing the area under the curve into trapezoids. It uses the endpoints of each subinterval to calculate the area of each trapezoid and sums them up.
Given that the function is positive, increasing, and concave up on the closed interval [a, b], we can make the following observations:
- Since the function is positive, both the upper sum and the lower sum will be positive.
- As the function is increasing, the maximum value of the function will be at the right endpoint of each subinterval, resulting in a higher value for the upper sum.
- Similarly, the minimum value of the function will be at the left endpoint of each subinterval, resulting in a lower value for the lower sum.
- Since the function is concave up, the trapezoids used in the trapezoidal rule will underestimate the area under the curve, which means the trapezoidal rule approximation will be lower than the actual integral value.
Based on these observations, we can conclude that the correct statement is:
Lower sum < Upper sum < Trapezoidal rule value
Therefore, the second statement "Lower sum < Upper sum < Trapezoidal rule value" is true.
To determine which statement is true, let's analyze the properties of the lower sum, upper sum, and trapezoidal rule approximations.
1. Lower Sum:
The lower sum is calculated by taking the infimum (minimum) value of the function in each subinterval and summing them up. Since the function is positive and increasing on the interval, the lowest possible value for each subinterval will be the value at the left endpoint of the interval.
2. Upper Sum:
The upper sum is calculated by taking the supremum (maximum) value of the function in each subinterval and summing them up. Since the function is positive and increasing on the interval, the highest possible value for each subinterval will be the value at the right endpoint of the interval.
3. Trapezoidal Rule:
The trapezoidal rule approximates the definite integral by dividing the interval into subintervals and approximating the area under the curve in each subinterval as a trapezoid. The approximation is obtained by summing up the areas of all the trapezoids.
Given that the function is positive, increasing, and concave up on the closed interval [a, b], we can conclude the following:
- The lower sum will underestimate the value of the definite integral.
- The upper sum will overestimate the value of the definite integral.
- The trapezoidal rule will provide a better approximation than both the lower sum and the upper sum, as it takes into account both the left and right endpoints of each subinterval.
Therefore, the correct statement is:
- Lower sum < Trapezoidal rule Value < Upper sum
so, have you tried drawing such a curve and trying them out?
How about y=x^2 where 0 <= a <= b