91Ó°ÊÓ

In the function \( C = a t e^{-t} \), the parameter \( a \) plays a pivotal role in changing how the function appears on a graph. Changing \( a \) doesn't alter the fundamental behavior of the function; rather, it modifies the scale. This makes \( a \) a multiplicative factor that stretches or shrinks the graph vertically. For example, setting \( a = 2 \) will result in the graph of \( C \) becoming twice as tall throughout its domain, compared to when \( a = 1 \).

• When \( a \) increases, the peak of the graph becomes higher because the values of \( C \) multiply by a larger number.
• When \( a \) decreases, the peak becomes lower, compressing the graph vertically.

Importantly, the value of \( a \) has no effect on the position of the peak along the \( t \)-axis. The peak remains at \( t = 1 \), where the transition from growth due to the linear term to decay in the exponential term occurs.

The exponential decay component \( e^{-t} \) in the function \( C = a t e^{-t} \) means that as \( t \) increases, \( e^{-t} \) decreases exponentially. This decay impacts how the function behaves for larger values of \( t \). Initially, when \( t \) is small, the effect of the linear term \( t \) is dominant, so \( C \) increases. However, as \( t \) increases beyond 1, \( e^{-t} \) decreases rapidly, dominating the linear term and causing \( C \) to decrease.

• This creates the characteristic peak in the function's graph, as \( C \) grows first and then decays.
• The exponential decay ensures the function values never become negative and eventually approach zero.

This decay transforms the graph's shape into a quick rise followed by a gradual fall, regardless of the constant \( a \). However, higher values of \( a \) can magnify these changes in amplitude.

The linear term \( t \) in \( C = a t e^{-t} \) adds an essential element to the behavior of the function, especially at the beginning of \( t \'s \) range. This term represents a direct proportional relationship with time, meaning as \( t \) increases, so does \( t \), linearly impacting the initial growth of \( C \).

• The linear term helps the function initially rise as \( t \) moves from 0 to 1, fighting against the decay caused by \( e^{-t} \).
• After \( t = 1 \), this term is overwhelmed by the exponential decay, decreasing \( C \).

Therefore, while the linear term ensures there is an early increase in \( C \), its influence is limited to the initial interval of \( t \). Its linear growth pattern makes it an important part of why \( C \) peaks at \( t = 1 \), after which the exponential decline takes control.