Question

Use the function f(x)=3x
to find the approximation 33√
, correct to three decimal places, by creating a table with x-values closer and closer to 3–√
and the corresponding f(x)
values.

Answer 1

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 \( \boxed{1.442} \).