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
I don't get it. Please help me!
To answer this question, we need to understand the properties of the left sum, right sum, and trapezoidal rule approximations.
1. Left sum approximation:
The left sum is calculated by taking the areas of rectangles with their heights determined by the function values at the left endpoints of each subinterval. This means the left sum tends to underestimate the value of the integral.
2. Right sum approximation:
The right sum is calculated by taking the areas of rectangles with their heights determined by the function values at the right endpoints of each subinterval. This means the right sum tends to overestimate the value of the integral.
3. Trapezoidal rule approximation:
The trapezoidal rule is calculated by taking the areas of trapezoids formed by connecting each pair of adjacent function values with a straight line segment. This method gives a better approximation than both the left sum and the right sum.
Now, let's consider the given information about the function f(x):
- It is positive: This means all function values are positive.
- It is decreasing: This means the function values decrease as x increases.
- It is concave up: This means the graph of the function is curved upwards.
Based on these properties, we can draw the following conclusions about the left sum, right sum, and trapezoidal rule approximations:
- Left sum: Since the function is decreasing, using left endpoints will yield greater function values. Since the function is also concave up, the left sum will underestimate the area under the curve even more.
- Right sum: Since the function is decreasing, using right endpoints will yield smaller function values. However, since the function is concave up, the right sum will still overestimate the area under the curve.
- Trapezoidal rule: Since the trapezoidal rule considers both left and right endpoints, it takes the concavity of the function into account. Thus, the trapezoidal rule will provide a better approximation than both the left sum and the right sum.
Therefore, the correct statement is:
a) Left sum < trapezoidal rule value < Right sum
just draw a graph, such as y = 1/x, which fits the criteria.
Make it big enough that you can draw the desired rectangles and trapezoids.
It should then be easy to see which ones over- and under-estimate, and then decide which order they belong in.