A table of values of an increasing function f is shown. Use the table to find lower and upper estimates for 30f(x) dx.10lower estimate     upper estimate

To find the lower and upper estimates for the integral $\int_{10}^{30} f(x) \, dx$ using a table of values, we will follow these steps:

1. Break Down the Problem

We need to estimate the area under the curve represented by $f(x)$ between $x = 10$ and $x = 30$ using the provided values.

2. Relevant Concepts

• The lower estimate can be calculated using the left endpoint values of the function.
• The upper estimate can be calculated using the right endpoint values of the function.

If $f(x)$ is given at evenly spaced intervals, we can use:

• Lower estimate $L = \sum (f(x_i) \cdot \Delta x)$ using left endpoints.
• Upper estimate $U = \sum (f(x_{i+1}) \cdot \Delta x)$ using right endpoints.

Where $\Delta x$ is the width of each interval.

3. Analysis and Detail

• Assuming we have a table of values for $f(x)$ at equal intervals from $x = 10$ to $x = 30$.
• Let’s denote the values as $f(10), f(11), \ldots, f(30)$, and let $n = 30 - 10 + 1 = 21$.
• If the interval is uniform, e.g., $\Delta x = 1$, the lower estimate uses the first $n-1$ values and the upper estimate uses the last $n-1$ values.

4. Verify and Summarize

• Ensure all values from the table are correctly substituted into the calculations for both estimates.

Assuming we have:

• Lower estimate values: $f(10), f(11), \ldots, f(29)$
• Upper estimate values: $f(11), f(12), \ldots, f(30)$

Final Answer

The final answers for lower and upper estimates will be given as:

• Lower Estimate: $L = \sum_{i=10}^{29} f(i)$
• Upper Estimate: $U = \sum_{i=11}^{30} f(i)$

Please substitute the respective function values from the table for a specific numerical answer.