Absolute and relative growth rates Two functions f and g are...

Absolute and relative growth rates Two functions $f$ and $g$ are given. Show that the growth rate of the linear function is constant and the relative growth rate of the exponential function is constant.
$$f(t)=2200+400 t, g(t)=400 \cdot 2^{t / 20}$$

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Key Concepts

Absolute Growth Rate

Absolute growth rate refers to the derivative of a function with respect to time. It provides the actual increase (or decrease) in the function's value per unit time. For a linear function, the derivative is constant, meaning the function grows by the same fixed amount over equal time intervals.

Relative Growth Rate

Relative growth rate is the ratio of the absolute growth rate to the current value of the function. It expresses how fast the function is growing in percentage terms rather than absolute increments. In exponential functions, this rate remains constant, indicating that the percentage change per unit time does not vary.

Linear Function Growth

A linear function is one in which the change in the output is proportional to the change in the input, represented by a constant slope. This constant rate of change results in a constant absolute growth rate, making linear functions particularly simple to analyze over time.

Exponential Function Growth

An exponential function is characterized by a constant relative growth rate, where the rate at which the function increases is proportional to its current value. This leads to rapid growth over time, as each increase is compounded with the function's already large base.

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