Q. 7 - Calculus

Question

Suppose that the left-sum approximation with eight rectangles for the area between the graph of a function $f$ and the x-axis from $x=a$ to$x=b$ is equal to $8.2$ and that the corresponding right sum approximation is equal to $7.5$.

(a) What is the corresponding trapezoid sum approximation for this area? 

(b) Is the corresponding midpoint sum for this area necessarily between $7.5$ and $8.2$? If so, explain why. If not, sketch an example of a function f on an interval $[a,b]$ whose midpoint sum is not between the left sum and the right sum.

(c) What can you say about the corresponding upper sum for this area? The corresponding lower sum? 

(d) Is it necessarily true that $f$ is decreasing on the entire interval $[a,b]$? If so, explain why. If not, sketch an counterexample in which the left sum is greater than the right sum but $f$ is not decreasing on all of $[a,b].$

(e) Could the function $f$ be increasing on the entire interval $[a.b]$? If not, explain why not. If so, sketch a possible example in which the left sum is greater than the right sum and f is increasing on all of $[a,b]$.

Step-by-Step Solution

Answer

Part (a) The trapezoid sum is $7.85$

Part (b) No, the corresponding mid-point sum for the area is not necessary to be between $7.5$ and $8.2$.

Part (c) The upper sum is $\ge 8.2$ and its lower sum is $\le 7.5$

Part (d) No, it is not necessary that the function $f$ be decreasing on the entire interval $[a,b].$

Part (e) The answer is no.

Part (a) Step 1. Calculation

The objective is to find the approximated trapezoid sum for the above area.

The trapezoid sum is the average of the left-sum and right-sum.

So the trapezoid sum is,

$$\begin{gathered} \frac{8.2+7.5}{2} \\ =\frac{15.7}{2} \\ =7.85 \end{gathered}$$

Therefore, the trapezoid sum is $7.85$

Part (b) Step 1. Explanation

The objective is to determine if the corresponding mid-point sum for the area necessarily be between $7.5$ and $8.2.$

No, if the corresponding mid-point sum for the area need not necessarily be between $7.5$ and $8.2$

Therefore, the answer is no.

Part (c) Step 1. Explanation

The objective is say about the corresponding lower and the upper-sum.

The left-sum approximation with eight rectangles for the area between the graph of the function is $8.2$ and the right-sum is approximately equal to $7.5$

The upper sum is always greater than or equal to the actual signed area where as the lower sum is always less than or equal to the actual signed area.

So, the upper sum is $\ge 8.2$ and its lower sum is $\le 7.5$

Therefore, the upper sum is $\ge 8.2$ and its lower sum is $\le 7.5$

Part (d) Step 1. Explanation

No, it is not necessary that the function $f$ be decreasing on the entire interval $[a,b]$

Therefore, the answer is no.

Part (e) Step 1. Explanation

No, it is not necessary that the function $f$ be increasing on the entire interval $[a,b]$

Therefore, the answer is no