let f: [0,2] to all real numbers given when f(x) 1 is x=0 and 1/x from 0 is less than x which is less and equal to 2; set up a riemann sum for x so that your riemann sum will be greater than 100 you can choose your ci and xi

To set up a Riemann sum for the function \( f(x) \) defined on the interval \([0, 2]\), where \( f(x) \) is given by:

\[
f(x) = 
\begin{cases} 
1 & \text{if } x = 0, \\
\frac{1}{x} & \text{if } 0 < x \leq 2,
\end{cases}
\]

We aim to ensure that the Riemann sum exceeds 100. We can choose the partition points \( x_i \) and sample points \( c_i \) within each subinterval to achieve this.

### Step 1: Define the Partition
Let's partition the interval \([0, 2]\) into \( n \) subintervals:

\[
0 = x_0 < x_1 < x_2 < \dots < x_n = 2
\]

The width of each subinterval is:

\[
\Delta x_i = x_i - x_{i-1}
\]

### Step 2: Choose Sample Points \( c_i \)
For each subinterval \([x_{i-1}, x_i]\), choose a sample point \( c_i \) where \( f(c_i) \) will be evaluated. To make the sum large, choose \( c_i \) close to 0 where \( f(x) = \frac{1}{x} \) becomes large.

### Step 3: Set Up the Riemann Sum
The Riemann sum \( S \) is given by:

\[
S = \sum_{i=1}^{n} f(c_i) \Delta x_i
\]

We need \( S > 100 \).

### Step 4: Choose an Example Partition and Sample Points
Let's consider a simple partition with \( n = 2 \) subintervals:

- \( x_0 = 0 \), \( x_1 = 0.01 \), \( x_2 = 2 \)
- Choose \( c_1 = 0.005 \) in \([0, 0.01]\) and \( c_2 = 1 \) in \([0.01, 2]\)

Now compute the Riemann sum:

\[
S = f(c_1) \cdot \Delta x_1 + f(c_2) \cdot \Delta x_2
\]

\[
S = \frac{1}{0.005} \cdot (0.01 - 0) + \frac{1}{1} \cdot (2 - 0.01)
\]

\[
S = 200 \cdot 0.01 + 1 \cdot 1.99 = 2 + 1.99 = 3.99
\]

This sum is not greater than 100, so we need more subintervals or different sample points to achieve a larger sum.

### Adjust the Partition to Increase the Sum
To exceed 100, consider refining the partition further near 0, where \( f(x) = \frac{1}{x} \) becomes large:

- \( x_0 = 0 \), \( x_1 = 0.0001 \), \( x_2 = 0.01 \), \( x_3 = 0.1 \), \( x_4 = 2 \)
- Choose \( c_1 = 0.00005 \), \( c_2 = 0.005 \), \( c_3 = 0.05 \), \( c_4 = 1 \)

Now, compute the new

Let f: [0,2] to all real numbers given when f(x) = 1 if x = 0 and f(x) = 1/x if 0 < x ≤ 2; set up a Riemann sum for x so that your Riemann sum will be greater than 100. You can choose your ci and xi.

Explore how to set up a Riemann sum for a piecewise function defined on the interval [0, 2], ensuring the sum exceeds 100. Understand the use of sample points and partitions in calculus to achieve the desired result. Suitable for college-level Calculus I students.

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