import numpy as np import pandas as pd

Define the function f(x) = x^3

def f(x): return x**3

Given values

initial_x = -3 final_x = 3 increment_x = 0.5

Create an array of x values

x_values = np.arange(initial_x, final_x + increment_x, increment_x)

Compute the corresponding y values

y_values = f(x_values)

Create a DataFrame for the results

df_results = pd.DataFrame({ 'x': x_values, 'f(x) = x^3': y_values })

import ace_tools as tools; tools.display_dataframe_to_user(name="Function Results: f(x) = x^3", dataframe=df_results) x f(x) = x^3 0 -3.0 -27.000 1 -2.5 -15.625 2 -2.0 -8.000 3 -1.5 -3.375 4 -1.0 -1.000 x f(x) = x^3 0 -3.0 -27.000 1 -2.5 -15.625 2 -2.0 -8.000 3 -1.5 -3.375 4 -1.0 -1.000successI've provided a table showing the values of $f(x) = x^3$ for $x$ ranging from -3 to 3 with increments of 0.5. Please let me know if you would like further details or have any questions.

Here are five related questions to explore further:

1. What is the derivative of $f(x) = x^3$ and how does it change with respect to $x$?
2. How does the graph of $f(x) = x^3$ look when plotted within this range of $x$?
3. What happens to the function's output when the increment is changed to 0.1 or 1.0?
4. Can you find the inflection point of the function $f(x) = x^3$ and explain its significance?
5. What other polynomial functions can you compare to $f(x) = x^3$ and analyze similarly?

Tip: Exploring the derivative of a function gives insights into its rate of change, which can be crucial for understanding the behavior of functions like $f(x) = x^3$.