Find the value  
V
  of the Riemann sum  
V
=
n
∑
k
=
1
 
f
(
c
k
)
Δ
x
k
 
for the function  
f
(
x
)
=
x
2
−
2
  using the partition  
P
=
{
1
,
2
,
3
,
5
}
 , where  
c
k
  is the left endpoints of the partition.

To find the value of the Riemann sum for the function \( f(x) = x^2 - 2 \) using the partition \( P = \{1, 2, 3, 5\} \) with left endpoints, we need to compute the sum:

\[
V = \sum_{k=1}^n f(c_k) \Delta x_k
\]

where \( c_k \) represents the left endpoints of the subintervals, and \( \Delta x_k \) represents the width of each subinterval.

### Step-by-Step Solution

1. **Identify the subintervals and their widths \( \Delta x_k \):**

   The partition \( P = \{1, 2, 3, 5\} \) creates the following subintervals:
   - \( [1, 2] \)
   - \( [2, 3] \)
   - \( [3, 5] \)

   Therefore, the widths of the subintervals are:
   \[
   \Delta x_1 = 2 - 1 = 1, \quad \Delta x_2 = 3 - 2 = 1, \quad \Delta x_3 = 5 - 3 = 2
   \]

2. **Determine the left endpoints \( c_k \) for each subinterval:**

   Since we are using left endpoints:
   - For the interval \([1, 2]\), \( c_1 = 1 \)
   - For the interval \([2, 3]\), \( c_2 = 2 \)
   - For the interval \([3, 5]\), \( c_3 = 3 \)

3. **Evaluate \( f(c_k) \) for each left endpoint:**

   The function is \( f(x) = x^2 - 2 \). Let's compute \( f(c_k) \):
   \[
   f(c_1) = f(1) = 1^2 - 2 = -1
   \]
   \[
   f(c_2) = f(2) = 2^2 - 2 = 2
   \]
   \[
   f(c_3) = f(3) = 3^2 - 2 = 7
   \]

4. **Compute the Riemann sum \( V \):**

   Now, we calculate the Riemann sum using the formula:

   \[
   V = f(c_1) \Delta x_1 + f(c_2) \Delta x_2 + f(c_3) \Delta x_3
   \]

   Substitute the values we found:

   \[
   V = (-1)(1) + (2)(1) + (7)(2)
   \]

   Simplify the expression:

   \[
   V = -1 + 2 + 14 = 15
   \]

### Conclusion

The value of the Riemann sum \( V \) for the function \( f(x) = x^2 - 2 \) using the partition \( P = \{1, 2, 3, 5\} \) with left endpoints is **15**.

---

Would you like more details on any of these steps, or do you have any questions?

Here are 5 related questions to consider:

1. What would be the value of the Riemann sum if we used right endpoints instead of left endpoints?
2. How would the Riemann sum change if the function was \( f(x) = x^3 - 2x + 1 \)?
3. Can you compute the Riemann sum for a uniform partition with \( n = 4 \) on the interval \([1, 5]\)?
4. How does the choice of partition affect the accuracy of the Riemann sum approximation for the integral of a function?
5. What is the difference between a left Riemann sum, a right Riemann sum, and a midpoint Riemann sum?

**Tip:** When working with Riemann sums, always carefully consider the choice of endpoints and the width of each subinterval, as they directly impact the accuracy of the approximation for the integral.

Find the value of the Riemann sum V = ∑ f(ck)Δxk for the function f(x) = x^2 - 2 using the partition P = {1, 2, 3, 5}, where ck is the left endpoints of the partition.

Calculate the Riemann sum for the function f(x) = x^2 - 2 using the partition P = {1, 2, 3, 5} with left endpoints. This problem involves partitioning intervals and applying the Riemann sum formula, suitable for students in Grades 11-12 or early college.

Math Problem Statement

Find the value
V of the Riemann sum
V

n ∑ k

for the function
f ( x )

x 2 − 2 using the partition
P

Solution

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