Complete the following steps for the given function, interval, and value of $$ n $$. a. Sketch the graph of the function on the given interval. b. Calculate $$ \Delta x $$ and the grid points $$ x_{0}, x_{1}, \ldots, x_{n} $$. c. Illustrate the midpoint Riemann sum by sketching the appropriate rectangles. d. Calculate the midpoint Riemann sum. $$ f(x)=\frac{2}{x}+2 $$ on $$ [2,7] ; n=5 $$

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**a. Sketch the Graph**

We are given 
$$
f(x)=\frac{2}{x}+2.
$$
This function is defined for all $$ x\neq 0 $$ and is composed of a hyperbolic term $$ \frac{2}{x} $$ and a constant $$ 2 $$. On the interval $$[2,7]$$, the function is smooth and decreasing since as $$ x $$ increases, $$ \frac{2}{x} $$ decreases. (A sketch should show a curve that starts high at $$ x=2 $$ and decreases as $$ x $$ approaches 7, with the curve approaching the horizontal line $$ y=2 $$ for large $$ x $$.)

**b. Calculate $$\Delta x$$ and the Grid Points**

The interval is $$[a,b] = [2,7]$$ and we have $$ n=5 $$ subintervals. The width of each subinterval is
$$
\Delta x = \frac{b-a}{n} = \frac{7-2}{5} = 1.
$$
The grid points are then:
$$
\begin{aligned}
x_{0} &= 2, \
x_{1} &= 2+\Delta x = 3, \
x_{2} &= 3+\Delta x = 4, \
x_{3} &= 4+\Delta x = 5, \
x_{4} &= 5+\Delta x = 6, \
x_{5} &= 6+\Delta x = 7.
\end{aligned}
$$

**c. Illustrate the Midpoint Riemann Sum Rectangles**

For the midpoint Riemann sum, we use the midpoint of each subinterval as the height of the rectangle. The midpoints are computed as follows:
$$
\begin{aligned}
m_{1} &= \frac{x_0+x_1}{2} = \frac{2+3}{2} = 2.5, \
m_{2} &= \frac{x_1+x_2}{2} = \frac{3+4}{2} = 3.5, \
m_{3} &= \frac{x_2+x_3}{2} = \frac{4+5}{2} = 4.5, \
m_{4} &= \frac{x_3+x_4}{2} = \frac{5+6}{2} = 5.5, \
m_{5} &= \frac{x_4+x_5}{2} = \frac{6+7}{2} = 6.5.
\end{aligned}
$$
On a sketch, you would draw the $$x$$-axis from 2 to 7, mark the points $$2, 3, 4, 5, 6, 7$$, and then, above each interval $$[2,3]$$, $$[3,4]$$, etc., draw a rectangle whose height is the function value at the corresponding midpoint $$(2.5, 3.5, \ldots, 6.5)$$.

**d. Calculate the Midpoint Riemann Sum**

The midpoint Riemann sum is given by
$$
M_5=\Delta x\sum_{i=1}^{5} f(m_i),
$$
where, for each midpoint,
$$
f(m_i)=\frac{2}{m_i}+2.
$$
Calculate each term:

1. For $$ m_1 = 2.5 $$:
   $$
   f(2.5)=\frac{2}{2.5}+2=\frac{2}{\frac{5}{2}}+2=\frac{4}{5}+2=\frac{4+10}{5}=\frac{14}{5}.
   $$

2. For $$ m_2 = 3.5 $$:
   $$
   f(3.5)=\frac{2}{3.5}+2=\frac{2}{\frac{7}{2}}+2=\frac{4}{7}+2=\frac{4+14}{7}=\frac{18}{7}.
   $$

3. For $$ m_3 = 4.5 $$:
   $$
   f(4.5)=\frac{2}{4.5}+2=\frac{2}{\frac{9}{2}}+2=\frac{4}{9}+2=\frac{4+18}{9}=\frac{22}{9}.
   $$

4. For $$ m_4 = 5.5 $$:
   $$
   f(5.5)=\frac{2}{5.5}+2=\frac{2}{\frac{11}{2}}+2=\frac{4}{11}+2=\frac{4+22}{11}=\frac{26}{11}.
   $$

5. For $$ m_5 = 6.5 $$:
   $$
   f(6.5)=\frac{2}{6.5}+2=\frac{2}{\frac{13}{2}}+2=\frac{4}{13}+2=\frac{4+26}{13}=\frac{30}{13}.
   $$

Now, the midpoint Riemann sum is
$$
M_5=1\left(\frac{14}{5}+\frac{18}{7}+\frac{22}{9}+\frac{26}{11}+\frac{30}{13}\right).
$$
For an approximate numerical value, compute:
$$
\begin{aligned}
\frac{14}{5} &\approx 2.8,\[1mm]
\frac{18}{7} &\approx 2.5714,\[1mm]
\frac{22}{9} &\approx 2.4444,\[1mm]
\frac{26}{11} &\approx 2.3636,\[1mm]
\frac{30}{13} &\approx 2.3077.
\end{aligned}
$$
Thus,
$$
M_5\approx 2.8+2.5714+2.4444+2.3636+2.3077 \approx 12.4871.
$$

The final value of the midpoint Riemann sum is approximately $$\boxed{12.49}$$.
**a. Sketch the Graph** The function $$ f(x) = \frac{2}{x} + 2 $$ is a hyperbola shifted up by 2 units. On the interval $$[2,7]$$, it decreases from $$ f(2) = 3 $$ to $$ f(7) \approx 2.2857 $$. **b. Calculate $$\Delta x$$ and the Grid Points** $$ \Delta x = 1 $$ Grid points: $$ x_0 = 2, \quad x_1 = 3, \quad x_2 = 4, \quad x_3 = 5, \quad x_4 = 6, \quad x_5 = 7 $$ **c. Illustrate the Midpoint Riemann Sum Rectangles** Midpoints: $$ m_1 = 2.5, \quad m_2 = 3.5, \quad m_3 = 4.5, \quad m_4 = 5.5, \quad m_5 = 6.5 $$ Draw rectangles with heights $$ f(2.5), f(3.5), f(4.5), f(5.5), f(6.5) $$ over each subinterval. **d. Calculate the Midpoint Riemann Sum** $$ M_5 = 1 \times \left( \frac{14}{5} + \frac{18}{7} + \frac{22}{9} + \frac{26}{11} + \frac{30}{13} \right) \approx 12.49 $$ The midpoint Riemann sum is approximately **12.49**.

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