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. $$ f(x)=2 x^{2}+2 $$ on $$ [1,5] ; n=4 $$

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**a. Sketching the Graph**

The function is 
$$
f(x)=2x^2+2.
$$
This is a parabola opening upward with vertex at the smallest $$ x $$‐value on the interval. On the interval $$[1,5]$$, some key points are:

- At $$ x=1 $$: 
  $$
  f(1)=2(1)^2+2=2+2=4.
  $$
- At $$ x=3 $$:
  $$
  f(3)=2(3)^2+2=2(9)+2=18+2=20.
  $$
- At $$ x=5 $$:
  $$
  f(5)=2(5)^2+2=2(25)+2=50+2=52.
  $$

The graph will be a steeply rising upward-opening parabola starting at the point $$(1,4)$$, passing through $$(3,20)$$, and ending at $$(5,52)$$.

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

Since we are dividing the interval from $$x=1$$ to $$x=5$$ into $$n=4$$ subintervals, the width of each subinterval is
$$
\Delta x = \frac{5-1}{4} = 1.
$$
Thus, the grid points are:
$$
x_0 = 1,\quad x_1 = 2,\quad x_2 = 3,\quad x_3 = 4,\quad x_4 = 5.
$$

**c. Midpoint Riemann Sum Rectangles**

For the midpoint Riemann sum, we use the midpoint of each subinterval $$[x_{i-1}, x_i]$$ to determine the height of the rectangle. The midpoints are calculated as follows:
- Midpoint of $$[1,2]$$:
  $$
  m_1 = \frac{1+2}{2} = 1.5.
  $$
- Midpoint of $$[2,3]$$:
  $$
  m_2 = \frac{2+3}{2} = 2.5.
  $$
- Midpoint of $$[3,4]$$:
  $$
  m_3 = \frac{3+4}{2} = 3.5.
  $$
- Midpoint of $$[4,5]$$:
  $$
  m_4 = \frac{4+5}{2} = 4.5.
  $$

Each rectangle has:
- Base (width) $$ \Delta x = 1 $$
- Height $$ f(m_i) $$ where $$ m_i $$ is the midpoint of the $$i$$th subinterval.

**d. Calculating the Midpoint Riemann Sum**

We compute the value of $$ f(x) $$ at each midpoint:

1. For $$ m_1 = 1.5 $$:
   $$
   f(1.5)=2(1.5)^2+2=2(2.25)+2=4.5+2=6.5.
   $$
2. For $$ m_2 = 2.5 $$:
   $$
   f(2.5)=2(2.5)^2+2=2(6.25)+2=12.5+2=14.5.
   $$
3. For $$ m_3 = 3.5 $$:
   $$
   f(3.5)=2(3.5)^2+2=2(12.25)+2=24.5+2=26.5.
   $$
4. For $$ m_4 = 4.5 $$:
   $$
   f(4.5)=2(4.5)^2+2=2(20.25)+2=40.5+2=42.5.
   $$

The midpoint Riemann sum is then:
$$
\text{Midpoint Sum} = \Delta x \left[f(1.5)+f(2.5)+f(3.5)+f(4.5)\right].
$$
Substitute the values:
$$
\text{Midpoint Sum} = 1 \times \left[6.5+14.5+26.5+42.5\right].
$$

Adding the function values:
$$
6.5+14.5=21,\quad 26.5+42.5=69,\quad \text{and}\quad 21+69=90.
$$

Thus, the midpoint Riemann sum is:
$$
\boxed{90}.
$$
**a. Sketching the Graph** The function $$ f(x) = 2x^2 + 2 $$ is a parabola opening upward. On the interval $$[1,5]$$, it starts at $$(1,4)$$, passes through $$(3,20)$$, and ends at $$(5,52)$$. **b. Calculating $$ \Delta x $$ and Grid Points** Divide the interval into 4 equal parts: $$ \Delta x = 1 $$ Grid points: $$ x_0 = 1,\quad x_1 = 2,\quad x_2 = 3,\quad x_3 = 4,\quad x_4 = 5 $$ **c. Midpoint Riemann Sum Rectangles** Midpoints: $$ m_1 = 1.5,\quad m_2 = 2.5,\quad m_3 = 3.5,\quad m_4 = 4.5 $$ **d. Calculating the Midpoint Riemann Sum** Compute $$ f(m_i) $$ for each midpoint: $$ f(1.5) = 6.5,\quad f(2.5) = 14.5,\quad f(3.5) = 26.5,\quad f(4.5) = 42.5 $$ Midpoint Sum: $$ \text{Midpoint Sum} = 1 \times (6.5 + 14.5 + 26.5 + 42.5) = 90 $$ **Final Answer:** The midpoint Riemann sum is **90**.

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