Find the upper sum for  
f
(
x
)
=
1
−
x
5
  on the interval  
[
−
3
,
1
]
  using  
2
  rectangles.

Submit your answer using an exact value. For instance, if your answer is  
10
3
 , then enter this fraction as your answer in the response box.

To find the upper sum for the function \( f(x) = 1 - x^5 \) on the interval \([-3, 1]\) using 2 rectangles, we need to divide the interval into 2 subintervals, find the maximum value of the function on each subinterval, and then multiply each maximum value by the width of the corresponding subinterval.

### Step 1: Determine the Interval and Subintervals

The interval is \([-3, 1]\). To divide this into 2 equal subintervals, we calculate the width of each subinterval:

\[
\text{Width of each subinterval} = \frac{1 - (-3)}{2} = \frac{4}{2} = 2
\]

Thus, the two subintervals are:
1. \([-3, -1]\)
2. \([-1, 1]\)

### Step 2: Find the Maximum Value of \(f(x)\) on Each Subinterval

For the function \( f(x) = 1 - x^5 \), we find the maximum value of \(f(x)\) on each subinterval.

#### Subinterval 1: \([-3, -1]\)

We need to determine the maximum value of \(f(x)\) on \([-3, -1]\). Since \( f(x) = 1 - x^5 \) is decreasing on this interval (because \(x^5\) is increasing for \(x < 0\)), the maximum occurs at \(x = -3\).

\[
f(-3) = 1 - (-3)^5 = 1 - (-243) = 1 + 243 = 244
\]

#### Subinterval 2: \([-1, 1]\)

For the interval \([-1, 1]\), the function \(f(x) = 1 - x^5\) is decreasing from \(x = -1\) to \(x = 1\). The maximum value occurs at \(x = -1\).

\[
f(-1) = 1 - (-1)^5 = 1 - (-1) = 1 + 1 = 2
\]

### Step 3: Calculate the Upper Sum

The upper sum is the sum of the products of the width of each subinterval and the maximum value of the function on that subinterval:

\[
\text{Upper sum} = \text{Width of first subinterval} \times \text{Maximum on first subinterval} + \text{Width of second subinterval} \times \text{Maximum on second subinterval}
\]

\[
\text{Upper sum} = 2 \times 244 + 2 \times 2 = 488 + 4 = 492
\]

### Final Answer

The upper sum for \(f(x) = 1 - x^5\) on the interval \([-3, 1]\) using 2 rectangles is:

\[
\boxed{492}
\]

Would you like more details or have any questions? Here are 5 related questions for further exploration:

1. How would the upper sum change if we used 4 rectangles instead of 2?
2. What is the lower sum for \(f(x) = 1 - x^5\) on the same interval using 2 rectangles?
3. How can we find the exact area under the curve \(f(x) = 1 - x^5\) on the interval \([-3, 1]\) using integration?
4. What are the general steps to find an upper sum for any function over a given interval?
5. How does the choice of subintervals affect the upper sum approximation?

**Tip:** When dealing with piecewise integration or summation, always check the behavior of the function over each subinterval to identify maximum and minimum values accurately.

Learn how to calculate the upper sum for the function f(x) = 1 - x^5 on the interval [-3, 1] using 2 rectangles. Understand the process of dividing the interval, finding maximum values, and computing the upper sum. Exact solution provided.

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Find the upper sum for
f ( x )

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