To approximate \( \sqrt[3]{33} \) using the function \( f(x) = 3x \), we first recognize that we want to find the value of \( x \) such that \( f(x) = 33 \).
Given that \( f(x) = 3x \), this leads us to solve:
\[ 3x = 33 \quad \Rightarrow \quad x = \frac{33}{3} = 11 \]
However, since we are interested in finding an approximation for \( \sqrt[3]{33} \), we need to approach the function differently. We'll look for values \( x \) near \( \sqrt[3]{33} \) to find \( f(x) \).
To start, let's calculate \( \sqrt[3]{33} \) mathematically. We know:
\[ \sqrt[3]{33} \approx 3.208 \]
Now, we'll create a table of values for \( x \) around 3 and calculate \( f(x) = 3x \):
| \( x \) | \( f(x) = 3x \) | |--------------|-------------------| | 3 | 9 | | 3.1 | 9.3 | | 3.2 | 9.6 | | 3.3 | 9.9 | | 3.5 | 10.5 | | 3.6 | 10.8 | | 3.7 | 11.1 | | 3.9 | 11.7 | | 3.1 | 9.3 | | 3.2 | 9.6 | | 3.3 | 9.9 | | 3.4 | 10.2 | | 3.5 | 10.5 | | 3.6 | 10.8 |
Now, since we know \( f(x) = 3x \) more about what we want, notice that this table doesn’t give good convergence towards \( \sqrt[3]{33} \).
Instead, let's narrow down our values closer to \( \sqrt[3]{33} \approx 3.2 \) to get the output closer to \( 33 \) or 3.208
Here's a better table:
| \( x \) | \( f(x) = 3x \) | |--------------|------------------| | 3.0 | 9 | | 3.1 | 9.3 | | 3.2 | 9.6 | | 3.3 | 9.9 | | 3.4 | 10.2 | | 3.5 | 10.5 | | 3.6 | 10.8 | | 3.7 | 11.1 | | 3.8 | 11.4 | | 3.9 | 11.7 |
We can review f(x) = 3x passing 3.2 where we could expect 33 we check a few.
Based on Table where we know \(3 \cdot 3.3 = 9.9 \), we don't reach it with these inputs.
To improve the accuracy we can work:
\[ 0 < x \to f(x) < 3.3 \]
We find:
If \( x= 3.2 \), f(3.2) = 9.6 < 33 If \( x = 3.3 \), f(3.3)= 9.9 < 33
We see we need to solve \( 3x \) as we know \(10<x \to 11\)
| \( x \) | \( f(x) = 3x \) | |--------------|------------------| | 3.0 | 9 | | 3.2 | 9.6 | | 3.3 | 10.2 | | 3.24 | 10.8 | | 3.28 | 10.8 | | 3.3 | 11.1 |
From the above approximations, we can see enough; as x gets closer to 3.2 the more shift we may compact it.
Thus, with additional iterations, we can further narrow down this value of \( x \) to 3.208.