Question

How does linear growth differ from exponential growth?

Answer 1

Hint: In the given question, we have been asked how do two types of same concepts differ from each other. To solve any question of such kind, we must first know the meaning of the given concept. We must know their formulae. Then after that, we differentiate the two formulae, and then we can infer their meaning and their differences from there.

Complete step-by-step answer:
A linear function, \[f\left( x \right) = ax + c\] has a derivative of \[f'\left( x \right) = a\].
This means that the function has no growth after attaining the growth of “\[a\]” (a constant). No matter how big the other part of the function gets \[\left( c \right)\], the growth is going to be the same.
An exponential function, \[f\left( x \right) = b{e^x} + c\] has a derivative of \[f'\left( x \right) = b{e^x}\].
This means that the function keeps on increasing as the argument increases. As the value of “\[x\]” increases, its derivative increases and hence, the growth increases to no limit.
Hence, a linear growth does not increase after a point whereas an exponential growth never stops.

Note: In the given question, we were asked the difference between linear growth and exponential growth. To do that, we must know the meaning of the two given growths, their formulae. Then, we used differentiation to find the change in its growth. Linear growth has a constant growth rate of one for every argument but exponential growth has faster growth as the argument increases; gets steeper and steeper.