use four rectangles to find an estimate of each type for the area under the given graph of f from x = 1 to x…

use four rectangles to find an estimate of each type for the area under the given graph of f from x = 1 to x = 9.\n1. take the sample points from the left - endpoints.\nanswer: l4=\n2. is your estimate l4 an underestimate or overestimate of the true area?\n3. take the sample points from the right - endpoints.\nanswer: r4=\n4. is your estimate r4 an underestimate or overestimate of the true area?\n5. use the trapezoid rule with n = 4.\nanswer: t4=\n6. is your estimate t4 an underestimate or overestimate of the true area?

use four rectangles to find an estimate of each type for the area under the given graph of f from x = 1 to x = 9.\n1. take the sample points from the left - endpoints.\nanswer: l4=\n2. is your estimate l4 an underestimate or overestimate of the true area?\n3. take the sample points from the right - endpoints.\nanswer: r4=\n4. is your estimate r4 an underestimate or overestimate of the true area?\n5. use the trapezoid rule with n = 4.\nanswer: t4=\n6. is your estimate t4 an underestimate or overestimate of the true area?

Answer

Explanation:

Step1: Calculate width of sub - intervals

The interval is from $x = 1$ to $x=9$ and $n = 4$. The width of each sub - interval $\Delta x=\frac{9 - 1}{4}=2$. The sub - intervals are $[1,3]$, $[3,5]$, $[5,7]$, $[7,9]$.

Step2: Calculate $L_4$ (left - endpoint approximation)

We need to find the function values at the left - endpoints: $x_1 = 1$, $x_2=3$, $x_3 = 5$, $x_4=7$. Let's assume the function values are $f(1)=y_1$, $f(3)=y_2$, $f(5)=y_3$, $f(7)=y_4$. Then $L_4=\Delta x\left[f(1)+f(3)+f(5)+f(7)\right]=2(y_1 + y_2+y_3 + y_4)$. Without the actual function values from the graph, we leave it in this general form.

Step3: Calculate $R_4$ (right - endpoint approximation)

The right - endpoints are $x_1 = 3$, $x_2=5$, $x_3 = 7$, $x_4=9$. Let $f(3)=y_1$, $f(5)=y_2$, $f(7)=y_3$, $f(9)=y_4$. Then $R_4=\Delta x\left[f(3)+f(5)+f(7)+f(9)\right]=2(y_1 + y_2+y_3 + y_4)$.

Step4: Calculate $T_4$ (Trapezoid Rule)

The Trapezoid Rule formula is $T_4=\frac{\Delta x}{2}\left[f(1)+2f(3)+2f(5)+2f(7)+f(9)\right]=\frac{2}{2}\left[f(1)+2f(3)+2f(5)+2f(7)+f(9)\right]=f(1)+2f(3)+2f(5)+2f(7)+f(9)$.

Answer:

  1. $L_4 = 2\left[f(1)+f(3)+f(5)+f(7)\right]$
  2. $R_4 = 2\left[f(3)+f(5)+f(7)+f(9)\right]$
  3. $T_4=f(1)+2f(3)+2f(5)+2f(7)+f(9)$