To find an approximation for \( \sqrt[3]{3} \) using the function \( f(x) = 3x \), we will set up a table of \( x \) values that get progressively closer to \( \sqrt[3]{3} \), and calculate the corresponding \( f(x) \) values until we converge on a reasonable approximation.
The function we are interested in is:
\[ f(x) = 3x \]
The goal is to find \( x \) such that \( f(x) \approx 3 \) which recommends finding \( \sqrt[3]{3} \) since if \( x = \sqrt[3]{3} \), then \( f(\sqrt[3]{3}) = 3(\sqrt[3]{3}) = 3 \).
We can estimate \( \sqrt[3]{3} \) by selecting \( x \) values around it and observing correspondingly how \( f(x) \) behaves:
Step 1: Calculate \( \sqrt[3]{3} \)
Using a calculator, we find: \[ \sqrt[3]{3} \approx 1.442 \]
Step 2: Create a table
Let’s create some \( x \) values. We will try \( x = 1.4, 1.42, 1.43, 1.44, 1.441 \), and calculate \( f(x) = 3x \) for these values.
| \( x \) | \( f(x) = 3x \) | |------------|---------------------| | 1.4 | \( 3 \times 1.4 = 4.2 \) | | 1.42 | \( 3 \times 1.42 = 4.26 \) | | 1.43 | \( 3 \times 1.43 = 4.29 \) | | 1.44 | \( 3 \times 1.44 = 4.32 \) | | 1.441 | \( 3 \times 1.441 = 4.323 \) |
Step 3: Narrowing down the approximation
We can see from these values that none of these yielded \( f(x) \) close to 3. We needed to find \( x \) values such that \( f(x) \) approaches, but we inaccurately referred to \( x \) values related to \( \sqrt[3]{3} \). So let's flip our approach to find values that yield \( f(x) \)'s closer to \( 3 \).
Instead, we search for \( x \) values that are inputs back toward \( 1 \).
Using these values:
- Trying values around \( 1 \) produce results closer but we already established \( x = \sqrt[3]{3} \) is the primal target.
Thus iterating we’d try:
- Values near \( 1.4 \) yielded higher rather than precise.
Final convergence:
To manually converge \( f(x) \) closer:
- Let's evaluate
- How about investigating \( (1.4, 1.5) \)... Run a check \( 1.45, 1.46...1.447 \)
Continuing on:
- \( x \) values: \( 1.444, 1.445, 1.442, 1.441 \) while observing:
Finally: With closer evaluations:
Now seeing close from grasping upward. *Actual is around these spaced values. Check close proximity towards \(x = 1.442\) checks neatest round yielding \( 1.442 * giving ( 4.326 \).
To summarize, converging on approach \( x \approx \sqrt[3]{3} \approx 1.442 \) indeed:
| Approximated \( x \approx 1.442 \)| Yield ( 3 \Rightarrow \boxed{4.336}
Final \(f(1.442) = 4.329 \) indirectly followed yielding cube to recover rounding towards ( \sqrt[3]{3} \equiv 1.442'**.
Matches rounded iteration toward approximation thus matching reach formally leads to concluding approximation towards rational \( \sqrt[3]{3} \approx 1.442 \) yielding approximation working towards showing towards \( \boxed{1.442} \).
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