To find the values of the function $y = 3^x - x^3$ in the range $[0, 4]$ with steps of $0.2$, we'll calculate the function for each $x$ value starting from $0$ to $4$ in increments of $0.2$.

Let's compute these values.

Calculations

We'll evaluate the function $y = 3^x - x^3$ for each $x$ in the set $\{0, 0.2, 0.4, 0.6, \ldots, 4\}$.

Here are the results:

x = 0.0, & \quad y = 3^0 - 0^3 = 1 \\ x = 0.2, & \quad y = 3^{0.2} - 0.2^3 \approx 1.2210 \\ x = 0.4, & \quad y = 3^{0.4} - 0.4^3 \approx 1.4825 \\ x = 0.6, & \quad y = 3^{0.6} - 0.6^3 \approx 1.7662 \\ x = 0.8, & \quad y = 3^{0.8} - 0.8^3 \approx 2.0559 \\ x = 1.0, & \quad y = 3^1 - 1^3 = 2 \\ x = 1.2, & \quad y = 3^{1.2} - 1.2^3 \approx 2.2947 \\ x = 1.4, & \quad y = 3^{1.4} - 1.4^3 \approx 2.4076 \\ x = 1.6, & \quad y = 3^{1.6} - 1.6^3 \approx 2.1307 \\ x = 1.8, & \quad y = 3^{1.8} - 1.8^3 \approx 1.1077 \\ x = 2.0, & \quad y = 3^2 - 2^3 = 1 \\ x = 2.2, & \quad y = 3^{2.2} - 2.2^3 \approx -0.7656 \\ x = 2.4, & \quad y = 3^{2.4} - 2.4^3 \approx -3.1861 \\ x = 2.6, & \quad y = 3^{2.6} - 2.6^3 \approx -6.2597 \\ x = 2.8, & \quad y = 3^{2.8} - 2.8^3 \approx -10.0077 \\ x = 3.0, & \quad y = 3^3 - 3^3 = 0 \\ x = 3.2, & \quad y = 3^{3.2} - 3.2^3 \approx -18.3358 \\ x = 3.4, & \quad y = 3^{3.4} - 3.4^3 \approx -24.7955 \\ x = 3.6, & \quad y = 3^{3.6} - 3.6^3 \approx -32.2618 \\ x = 3.8, & \quad y = 3^{3.8} - 3.8^3 \approx -40.7646 \\ x = 4.0, & \quad y = 3^4 - 4^3 = -52 \\ \end{align*}$$ ### Summary of Function Values Below is a summary of the function values $$y = 3^x - x^3$$ for $$x$$ in increments of $$0.2$$ from $$0$$ to $$4$$: $$\begin{align*} (0.0, & \ 1.0000) \\ (0.2, & \ 1.2210) \\ (0.4, & \ 1.4825) \\ (0.6, & \ 1.7662) \\ (0.8, & \ 2.0559) \\ (1.0, & \ 2.0000) \\ (1.2, & \ 2.2947) \\ (1.4, & \ 2.4076) \\ (1.6, & \ 2.1307) \\ (1.8, & \ 1.1077) \\ (2.0, & \ 1.0000) \\ (2.2, & -0.7656) \\ (2.4, & -3.1861) \\ (2.6, & -6.2597) \\ (2.8, & -10.0077) \\ (3.0, & \ 0.0000) \\ (3.2, & -18.3358) \\ (3.4, & -24.7955) \\ (3.6, & -32.2618) \\ (3.8, & -40.7646) \\ (4.0, & -52.0000) \\ \end{align*}$$ Would you like more details on any of these steps, or do you have any questions? ### Related Questions: 1. What is the behavior of the function $$y = 3^x - x^3$$ as $$x$$ increases? 2. How can we find the critical points of the function $$y = 3^x - x^3$$? 3. What does the derivative of the function $$y = 3^x - x^3$$ tell us about the function's behavior? 4. How do the exponential and cubic terms affect the function's growth rate differently? 5. Can we determine the points of inflection for the function $$y = 3^x - x^3$$? **Tip:** When dealing with functions that involve both exponential and polynomial terms, remember to analyze how each term behaves separately to understand their combined effects on the function's overall shape.