The five rectangles in the graph below illustrate a + (select a description) midpoint Riemann sum for f(x) = 2* on the interval [0, 3]. The value of this Riemann sum is 18 17 16 15 14 13 12 11 10 9 8 Riemann sum for y = 2* on [0, 3]
Unitary Method
The word “unitary” comes from the word “unit”, which means a single and complete entity. In this method, we find the value of a unit product from the given number of products, and then we solve for the other number of products.
Speed, Time, and Distance
Imagine you and 3 of your friends are planning to go to the playground at 6 in the evening. Your house is one mile away from the playground and one of your friends named Jim must start at 5 pm to reach the playground by walk. The other two friends are 3 miles away.
Profit and Loss
The amount earned or lost on the sale of one or more items is referred to as the profit or loss on that item.
Units and Measurements
Measurements and comparisons are the foundation of science and engineering. We, therefore, need rules that tell us how things are measured and compared. For these measurements and comparisons, we perform certain experiments, and we will need the experiments to set up the devices.
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![### Understanding Riemann Sums with Midpoint Rectangles
The five rectangles in the graph below illustrate a **midpoint** Riemann sum for \( f(x) = 2^x \) on the interval \([0, 3]\).
**Interactive Element:**
- Dropdown to select the description: midpoint, left endpoint, and right endpoint.
- Input box for entering the calculated Riemann sum value.
**Graph Explanation:**
The graph depicts the function \( y = 2^x \) plotted on the interval \([0, 3]\). The blue curve represents the continuous exponential function \( f(x) = 2^x \). To approximate the area under the curve from \( x = 0 \) to \( x = 3 \), the interval is divided into 5 subintervals, each of equal width.
**Midpoint Riemann Sum:**
- Each rectangle's height is determined by the function's value at the midpoint of the respective subinterval.
- The base of each of these rectangles spans the width of the subinterval, \(\Delta x\). In this case, since the interval \([0, 3]\) is divided into five rectangles, \(\Delta x = \frac{3}{5} = 0.6\).
**Graph Features:**
- **X-axis:** The horizontal axis, labeled from 0 to 4, represents the x-values.
- **Y-axis:** The vertical axis, labeled from 0 to 19, represents the function's values.
- **Blue Curve:** Represents \( y = 2^x \), showing exponential growth.
- **Rectangles:**
- There are five transparent blue rectangles.
- The height of each rectangle corresponds to the value of the function at the midpoint of the subinterval \( [x_i - \frac{\Delta x}{2}, x_i + \frac{\Delta x}{2}] \), providing an approximation of the area under the curve.
**Calculation of the Midpoint Riemann Sum:**
1. Determine the midpoint of each subinterval.
2. Evaluate the function \( f(x) \) at these midpoints.
3. Multiply each function value by the width of the subinterval \( \Delta x \) to find the area of each rectangle.
4. Sum the areas of the rectangles to approximate the total area under the curve.
The graph and the input](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F686df84f-2572-4ea8-99e4-a6aa8d03e207%2F72cad662-2e8e-4386-b217-646153c26f48%2Fu23akbb_processed.png&w=3840&q=75)
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