Question

Explain the statement “No matter what the growth rate of a linear function is,
that of an exponential function will always exceed it.” Explain how the linear
function S = 50w and thr exponential function A = P(1 + r)^t demonstrates that this statement is true.

Explain the answers simply, as short and simple as possible. Dont make it very long, please.

Answer 1

The statement means that as time goes on, an exponential function will grow faster than a linear function, regardless of their initial growth rates.

1. **Linear Function (S = 50w)**: This function increases by a constant amount (50) for each unit increase in \(w\). It adds the same 50 each time, so it grows steadily.

2. **Exponential Function (A = P(1 + r)^t)**: This function increases by a percentage based on its current value. As \(t\) (time) increases, each increase is larger than the last because it's a percentage of a growing number.

For example, if \(r = 0.05\):
- At \(t = 1\), \(A\) increases by 5% of \(P\).
- At \(t = 2\), \(A\) increases by 5% of the new total (which is higher).

Over time, the gap between the values of the linear and exponential functions widens, illustrating that the exponential growth rate eventually exceeds the linear growth rate. Thus, no matter how fast the linear function grows, the exponential function will eventually outpace it.