To approximate \( \sqrt[3]{33} \) using the function \( f(x) = 3x \), we first need to establish that we are looking for the value of \( x \) such that \( f(x) = \sqrt[3]{33} \). In other words, we want to find \( x \) such that:
\[ 3x = \sqrt[3]{33} \]
To find this, we can rewrite it as:
\[ x = \frac{\sqrt[3]{33}}{3} \]
Next, we'll create a table where we can evaluate \( f(x) \) for various values of \( x \) that are close to \( \frac{\sqrt[3]{33}}{3} \).
First, let's calculate \( \sqrt[3]{33} \):
\[ \sqrt[3]{33} \approx 3.208 \quad (\text{to three decimal places}) \]
Now, we divide by 3:
\[ x \approx \frac{3.208}{3} \approx 1.0693 \]
Now we will evaluate \( f(x) \) for \( x \) values near \( 1.0693 \). We will use \( x \) values of \( 1.06, 1.07, 1.08, 1.069, 1.070, \) and \( 1.071 \).
Table of Values
| \( x \) | \( f(x) = 3x \) | |-----------|-------------------| | 1.060 | \( 3 \times 1.060 = 3.180 \) | | 1.067 | \( 3 \times 1.067 = 3.201 \) | | 1.069 | \( 3 \times 1.069 = 3.207 \) | | 1.070 | \( 3 \times 1.070 = 3.210 \) | | 1.071 | \( 3 \times 1.071 = 3.213 \) | | 1.080 | \( 3 \times 1.080 = 3.240 \) |
Summarized Results
Based on the table:
- For \( x = 1.060 \), \( f(x) = 3.180 \)
- For \( x = 1.067 \), \( f(x) = 3.201 \)
- For \( x = 1.069 \), \( f(x) = 3.207 \)
- For \( x = 1.070 \), \( f(x) = 3.210 \)
- For \( x = 1.071 \), \( f(x) = 3.213 \)
- For \( x = 1.080 \), \( f(x) = 3.240 \)
From the values calculated, we can see that \( f(1.069) = 3.207 \) is very close to \( \sqrt[3]{33} \approx 3.208 \).
Conclusion
Thus, based on the values obtained from the function \( f(x) \), and observing the output of the function as we get closer to \( x = 1.0693 \), we approximate:
\[ \sqrt[3]{33} \approx 3.208 \quad \text{(to three decimal places)} \]
The value of \( x \) that gives this output is roughly \( 1.0693 \).
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