.values the function y = 3^x− x^3 in the range [0, 4] in steps of 0.2. Find the places where the

graph crosses X-axis.

To find the places where the graph of the function \( y = 3^x - x^3 \) crosses the X-axis within the range \([0, 4]\) in steps of 0.2, we can proceed with the following steps:

### 1. **Evaluate the function at each point in the range:**
   We calculate the value of \( y = 3^x - x^3 \) for \( x \) in the range \([0, 4]\) with steps of 0.2.

### 2. **Identify where \( y = 0 \):**
   The graph crosses the X-axis where \( y = 0 \). We need to find the corresponding \( x \) values where this occurs.

Let's calculate:

\[
\begin{array}{|c|c|}
\hline
x & y = 3^x - x^3 \\
\hline
0.0 & 3^0 - 0^3 = 1 - 0 = 1 \\
0.2 & 3^{0.2} - 0.2^3 \approx 1.2457 - 0.008 \approx 1.2377 \\
0.4 & 3^{0.4} - 0.4^3 \approx 1.5518 - 0.064 \approx 1.4878 \\
0.6 & 3^{0.6} - 0.6^3 \approx 1.9332 - 0.216 \approx 1.7172 \\
0.8 & 3^{0.8} - 0.8^3 \approx 2.4082 - 0.512 \approx 1.8962 \\
1.0 & 3^{1.0} - 1.0^3 = 3 - 1 = 2 \\
1.2 & 3^{1.2} - 1.2^3 \approx 3.5742 - 1.728 \approx 1.8462 \\
1.4 & 3^{1.4} - 1.4^3 \approx 4.2768 - 2.744 \approx 1.5328 \\
1.6 & 3^{1.6} - 1.6^3 \approx 5.1397 - 4.096 \approx 1.0437 \\
1.8 & 3^{1.8} - 1.8^3 \approx 6.2031 - 5.832 \approx 0.3711 \\
2.0 & 3^{2.0} - 2.0^3 = 9 - 8 = 1 \\
2.2 & 3^{2.2} - 2.2^3 \approx 10.384 - 10.648 \approx -0.264 \\
2.4 & 3^{2.4} - 2.4^3 \approx 12.245 - 13.824 \approx -1.579 \\
2.6 & 3^{2.6} - 2.6^3 \approx 14.609 - 17.576 \approx -2.967 \\
2.8 & 3^{2.8} - 2.8^3 \approx 17.515 - 21.952 \approx -4.437 \\
3.0 & 3^{3.0} - 3.0^3 = 27 - 27 = 0 \\
3.2 & 3^{3.2} - 3.2^3 \approx 33.696 - 32.768 \approx 0.928 \\
3.4 & 3^{3.4} - 3.4^3 \approx 41.905 - 39.304 \approx 2.601 \\
3.6 & 3^{3.6} - 3.6^3 \approx 52.212 - 46.656 \approx 5.556 \\
3.8 & 3^{3.8} - 3.8^3 \approx 65.794 - 54.872 \approx 10.922 \\
4.0 & 3^{4.0} - 4.0^3 = 81 - 64 = 17 \\
\hline
\end{array}
\]

### 3. **Determine where the graph crosses the X-axis:**
   The graph crosses the X-axis where \( y = 0 \). From the table:

- Between \( x = 2.0 \) and \( x = 2.2 \), \( y \) changes from positive to negative.
- At \( x = 3.0 \), \( y = 0 \).

### Conclusion:
The graph of the function \( y = 3^x - x^3 \) crosses the X-axis approximately between \( x = 2.0 \) and \( x = 2.2 \), and exactly at \( x = 3.0 \).

---

Would you like further details or have any questions?

Here are some related questions for further exploration:

1. How can we more precisely find the exact value where the function \( y = 3^x - x^3 \) crosses the X-axis between 2.0 and 2.2?
2. What happens to the function \( y = 3^x - x^3 \) as \( x \) increases beyond 4?
3. How does the behavior of the function change when the base of the exponential function is different, say \( y = 2^x - x^3 \)?
4. What are the roots of the equation \( 3^x = x^3 \)?
5. How would you graph this function using a graphing calculator or software?

**Tip:** When analyzing functions like \( y = 3^x - x^3 \), checking values at small intervals helps to capture the behavior more accurately, especially near critical points like where the function crosses the X-axis.

Explore how to find the x-intercepts of the function y = 3^x - x^3 within the range [0, 4]. This problem involves evaluating exponential and polynomial functions, identifying roots of equations, and analyzing graphical intersections. Suitable for advanced high school students and anyone interested in algebraic functions.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation