Question

Marshall is comparing the growth rate of f(x)=6*1.5^x and g(x)=5x+10 bases on the graph he concludes that the growth rate of g(x)=5x+10 is always greater than the growth rate of f(x) =6*1.5^x. Where is his mistake

Answer 1

Marshall's mistake arises from comparing the growth rates of the two functions without fully understanding their behaviors, especially as \( x \) approaches larger values.

To assess which function grows faster, we need to examine the growth rates analytically:

1. **Exponential Function:** \( f(x) = 6 \times 1.5^x \)
- This function is exponential because it has a base of 1.5 raised to the power of \( x \). Exponential functions generally grow faster than polynomial functions as \( x \) becomes large.

2. **Linear Function:** \( g(x) = 5x + 10 \)
- This is a linear function, which grows at a constant rate. As \( x \) increases, \( g(x) \) increases linearly.

To compare these functions more rigorously, we should evaluate their limits as \( x \) approaches infinity:

- For \( f(x) \):
\[
\lim_{x \to \infty} f(x) \text{ grows without bound since it is exponential.}
\]

- For \( g(x) \):
\[
\lim_{x \to \infty} g(x) \text{ also grows without bound but at a linear rate.}
\]

To see the growth rates, we can look at the derivatives of each function:

- The derivative of \( f(x) = 6 \times 1.5^x \) is:
\[
f'(x) = 6 \times 1.5^x \ln(1.5)
\]
This shows that \( f'(x) > 0 \) and is increasing for all \( x \).

- The derivative of \( g(x) = 5x + 10 \) is:
\[
g'(x) = 5
\]
This is constant.

For sufficiently large \( x \), \( f'(x) \) (which represents the rate of growth of \( f(x) \)) will exceed \( g'(x) \). So, as \( x \) becomes large enough, \( f(x) \) will grow faster than \( g(x) \).

**The crucial point** in Marshall's analysis is the misunderstanding that at all \( x \), \( g(x) \) is greater than \( f(x) \). While it may be true for small values of \( x \) (where the linear term could dominate due to its initial values), eventually, as \( x \) increases, the exponential function \( f(x) = 6 \times 1.5^x \) will outpace the linear function \( g(x) = 5x + 10 \).

Therefore, Marshall's conclusion overlooks the asymptotic behavior of the functions, where the exponential function eventually grows faster than the linear one for large \( x \).