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.
$$f(t)=100+10.5 t, g(t)=100 e^{t / 10}$$

Key Concepts

Absolute Growth Rate

Absolute growth rate refers to the actual rate at which a function changes with respect to its independent variable. It is typically determined by taking the derivative of the function, which gives a measure of the instantaneous increase or decrease per unit change in the independent variable. In the case of linear functions, this rate remains constant throughout the domain.

Relative Growth Rate

Relative growth rate is the ratio of the derivative of a function to the function itself, often expressed as f'(t)/f(t). This measure indicates the rate of change in relation to the current size or value of the function, effectively giving the percentage change per unit time. For exponential functions, this ratio is constant, which is a hallmark of their multiplicative growth behavior.

Linear Functions

Linear functions form expressions of the type f(t) = mt + b, where m is the slope and b is the y-intercept. Their defining characteristic is a constant rate of change, meaning that for every unit increase in the independent variable, the function increases by a fixed amount, as indicated by the slope m.

Exponential Functions

Exponential functions are expressed in the form g(t) = a * e^(kt), where k is a constant that determines the rate of growth or decay, and a is the initial value. A key feature is that their relative growth rate, given by the derivative divided by the function itself, is constant. This property explains why the percentage increase remains consistent over time, making exponential functions suitable for modeling phenomena with continuous proportional change.

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