Question
Term number 1 2 3 4 5 6
Sequence for 20 40 60 80 100 120
g(x)=20x
Sequence for 2 4 8 16 32 64
f(x)=2^x
Soya is comparing the growth rates of g(x)=20x and f(x)=2^x using their sequences as shown in the table. They notice that the terms of f(x)=2^x are always less than the terms of g(x)=20x and so conclude that f(x) =2^x grows slower than g(x)=20x. Where is their mistake in comparing the growth rates
Sequence for 20 40 60 80 100 120
g(x)=20x
Sequence for 2 4 8 16 32 64
f(x)=2^x
Soya is comparing the growth rates of g(x)=20x and f(x)=2^x using their sequences as shown in the table. They notice that the terms of f(x)=2^x are always less than the terms of g(x)=20x and so conclude that f(x) =2^x grows slower than g(x)=20x. Where is their mistake in comparing the growth rates
Answers
Answered by
GPT-4o mini
Soya's conclusion that \(f(x) = 2^x\) grows slower than \(g(x) = 20x\) based on the finite sequence provided is misleading due to several reasons:
1. **Finite vs. Infinite Behavior**: The sequences provided only depict the terms for \(x = 1\) to \(x = 6\), which is a limited range. While it is true that for these values \(2^x\) results in smaller values compared to \(20x\), growth rates must be analyzed in the context of their behavior as \(x\) approaches infinity.
2. **Relative Growth Rates**: The essential point to consider is the rate at which each function grows as \(x\) increases.
- The function \(g(x) = 20x\) is a linear function, which grows at a constant rate.
- The function \(f(x) = 2^x\) is an exponential function, which grows increasingly rapidly as \(x\) increases.
3. **Exponential Growth**: Even though \(2^x\) may show smaller values at lower \(x\), its rate of growth surpasses that of linear functions as \(x\) gets sufficiently large. For instance, at \(x = 10\), we have:
- \(g(10) = 20 \times 10 = 200\)
- \(f(10) = 2^{10} = 1024\)
At \(x = 10\), \(2^{10}\) clearly exceeds \(20 \times 10\).
4. **Conclusion**: The mistake is using only a small subset of values to judge their growth rates instead of looking at their long-term behavior. In mathematics, exponential growth outpaces linear growth, and this phenomenon becomes prominent as \(x\) increases. Thus, Soya's conclusion is incorrect.
In summary, the correct interpretation reveals that while \(f(x) = 2^x\) yields smaller values than \(g(x) = 20x\) for small \(x\), it eventually grows much faster as \(x\) becomes larger.
1. **Finite vs. Infinite Behavior**: The sequences provided only depict the terms for \(x = 1\) to \(x = 6\), which is a limited range. While it is true that for these values \(2^x\) results in smaller values compared to \(20x\), growth rates must be analyzed in the context of their behavior as \(x\) approaches infinity.
2. **Relative Growth Rates**: The essential point to consider is the rate at which each function grows as \(x\) increases.
- The function \(g(x) = 20x\) is a linear function, which grows at a constant rate.
- The function \(f(x) = 2^x\) is an exponential function, which grows increasingly rapidly as \(x\) increases.
3. **Exponential Growth**: Even though \(2^x\) may show smaller values at lower \(x\), its rate of growth surpasses that of linear functions as \(x\) gets sufficiently large. For instance, at \(x = 10\), we have:
- \(g(10) = 20 \times 10 = 200\)
- \(f(10) = 2^{10} = 1024\)
At \(x = 10\), \(2^{10}\) clearly exceeds \(20 \times 10\).
4. **Conclusion**: The mistake is using only a small subset of values to judge their growth rates instead of looking at their long-term behavior. In mathematics, exponential growth outpaces linear growth, and this phenomenon becomes prominent as \(x\) increases. Thus, Soya's conclusion is incorrect.
In summary, the correct interpretation reveals that while \(f(x) = 2^x\) yields smaller values than \(g(x) = 20x\) for small \(x\), it eventually grows much faster as \(x\) becomes larger.