Compare The Functions $f(x)=2x^3$ And $g(x)=3^x$ By Completing Parts (a) And (b).(a) Fill In The Table Below. Note That The Table Is Already Filled In For $x=3$.$\[ \begin{tabular}{|c|c|c|} \hline $x$ & $f(x)=2x^3$ &

Comparing the Functions $f(x)=2x^3$ and $g(x)=3^x$

In this article, we will compare the functions $f(x)=2x^3$ and $g(x)=3^x$ by completing parts (a) and (b). We will start by filling in the table below, which is already filled in for $x=3$.

Table

Part (a) - Filling in the Table

To fill in the table, we need to calculate the values of $f(x)=2x^3$ and $g(x)=3^x$ for each value of $x$.

Calculating $f(x)=2x^3$

To calculate $f(x)=2x^3$, we need to cube each value of $x$ and multiply the result by 2.

• For $x=1$, $f(1)=2(1)^3=2$
• For $x=2$, $f(2)=2(2)^3=16$
• For $x=3$, $f(3)=2(3)^3=54$ (already filled in)
• For $x=4$, $f(4)=2(4)^3=128$
• For $x=5$, $f(5)=2(5)^3=250$

Calculating $g(x)=3^x$

To calculate $g(x)=3^x$, we need to raise 3 to the power of each value of $x$.

• For $x=1$, $g(1)=3^1=3$
• For $x=2$, $g(2)=3^2=9$
• For $x=3$, $g(3)=3^3=27$ (already filled in)
• For $x=4$, $g(4)=3^4=81$
• For $x=5$, $g(5)=3^5=243$

Part (b) - Comparing the Functions

Now that we have filled in the table, we can compare the functions $f(x)=2x^3$ and $g(x)=3^x$.

Comparing the Growth Rates

From the table, we can see that both functions grow rapidly as $x$ increases. However, the growth rate of $f(x)=2x^3$ is faster than the growth rate of $g(x)=3^x$.

Comparing the Values

From the table, we can also see that the values of $f(x)=2x^3$ are always greater than the values of $g(x)=3^x$ for the same value of $x$.

Conclusion

In conclusion, the functions $f(x)=2x^3$ and $g(x)=3^x$ have different growth rates and values. While both functions grow rapidly as $x$ increases, the growth rate of $f(x)=2x^3$ is faster than the growth rate of $g(x)=3^x$. Additionally, the values of $f(x)=2x^3$ are always greater than the values of $g(x)=3^x$ for the same value of $x$.

The comparison of the functions $f(x)=2x^3$ and $g(x)=3^x$ has several implications.

Implications for Mathematics

The comparison of the functions $f(x)=2x^3$ and $g(x)=3^x$ has implications for mathematics. For example, it shows that the growth rate of a function can be affected by the power to which the variable is raised.

Implications for Real-World Applications

The comparison of the functions $f(x)=2x^3$ and $g(x)=3^x$ also has implications for real-world applications. For example, it can be used to model the growth of populations, the spread of diseases, and the growth of economies.

In our previous article, we compared the functions $f(x)=2x^3$ and $g(x)=3^x$ by filling in a table and analyzing the growth rates and values of the two functions. In this article, we will answer some frequently asked questions about the comparison of these two functions.

Q: What is the main difference between the functions $f(x)=2x^3$ and $g(x)=3^x$?

A: The main difference between the functions $f(x)=2x^3$ and $g(x)=3^x$ is the power to which the variable is raised. In $f(x)=2x^3$, the variable $x$ is raised to the power of 3, while in $g(x)=3^x$, the variable $x$ is raised to the power of 1.

Q: How do the growth rates of the two functions compare?

A: The growth rate of $f(x)=2x^3$ is faster than the growth rate of $g(x)=3^x$. This is because the power to which the variable is raised in $f(x)=2x^3$ is greater than the power to which the variable is raised in $g(x)=3^x$.

Q: What are some real-world applications of the comparison of these two functions?

A: The comparison of the functions $f(x)=2x^3$ and $g(x)=3^x$ has several real-world applications. For example, it can be used to model the growth of populations, the spread of diseases, and the growth of economies.

Q: Can you provide some examples of how the comparison of these two functions can be used in real-world applications?

A: Yes, here are a few examples:

• Modeling the growth of a population: If we assume that the population grows at a rate proportional to the cube of the current population, we can use the function $f(x)=2x^3$ to model the growth of the population.
• Modeling the spread of a disease: If we assume that the spread of a disease is proportional to the number of people infected, we can use the function $g(x)=3^x$ to model the spread of the disease.
• Modeling the growth of an economy: If we assume that the growth of an economy is proportional to the cube of the current GDP, we can use the function $f(x)=2x^3$ to model the growth of the economy.

Q: What are some limitations of the comparison of these two functions?

A: One limitation of the comparison of the functions $f(x)=2x^3$ and $g(x)=3^x$ is that it assumes that the growth rates of the two functions are constant over time. In reality, the growth rates of the two functions may change over time due to various factors such as changes in population growth rates, changes in disease transmission rates, and changes in economic conditions.

Q: Can you provide some suggestions for further research on the comparison of these two functions?

A: Yes, here are a few suggestions for further research:

• Investigate the effects of changes in population growth rates, disease transmission rates, and economic conditions on the growth rates of the two functions.
• Develop more complex models that take into account the interactions between the two functions.
• Investigate the applications of the comparison of these two functions in other fields such as physics, engineering, and computer science.

In conclusion, the comparison of the functions $f(x)=2x^3$ and $g(x)=3^x$ has several implications for mathematics and real-world applications. It shows that the growth rate of a function can be affected by the power to which the variable is raised, and it can be used to model the growth of populations, the spread of diseases, and the growth of economies. However, it also has some limitations, and further research is needed to fully understand the implications of the comparison of these two functions.