13–14. 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 + 10.5t, g(t) = 100e^(t/10)

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Identify the given functions: the linear function \(f(t) = 100 + 10.5t\) and the exponential function \(g(t) = 100e^{\frac{t}{10}}\).

Calculate the growth rate of the linear function \(f(t)\) by finding its derivative with respect to \(t\): \(f'(t) = \frac{d}{dt}(100 + 10.5t)\).

Observe that the derivative \(f'(t)\) is a constant value, which shows that the absolute growth rate of the linear function is constant.

Calculate the relative growth rate of the exponential function \(g(t)\) by first finding its derivative: \(g'(t) = \frac{d}{dt}\left(100e^{\frac{t}{10}}\right)\).

Express the relative growth rate of \(g(t)\) as \(\frac{g'(t)}{g(t)}\) and simplify to show that this ratio is a constant, confirming the relative growth rate of the exponential function is constant.

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주요 개념

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Absolute Growth Rate

The absolute growth rate of a function measures how much the function's value changes per unit increase in the input variable. For a linear function like f(t) = 100 + 10.5t, this rate is constant and equals the slope, which is 10.5. It represents a steady, fixed increase over time.

Relative Growth Rate

The relative growth rate is the rate of change of a function relative to its current value, often expressed as (f'(t)/f(t)). For exponential functions like g(t) = 100e^(t/10), this rate is constant, reflecting proportional growth. It indicates the percentage increase per unit time.

Differentiation of Functions

Differentiation is the process of finding the derivative, which gives the instantaneous rate of change of a function. Calculating derivatives of f(t) and g(t) allows us to determine their growth rates. Understanding how to differentiate linear and exponential functions is essential to analyze their growth behavior.

13–14. 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 + 10.5t, g(t) = 100e^(t/10)