what is the multiplicative rate of change of the function
What is the multiplicative rate of change of the function?
Answer: The concept of the multiplicative rate of change of a function is tied closely to understanding how a function behaves as its inputs change multiplicatively. This is particularly relevant in the context of exponential functions or situations where growth or decay processes are multiplicative rather than additive.
1. Exponential Functions:
For an exponential function of the form f(x) = a \cdot b^x, the base b is a constant factor that describes the multiplicative rate of change. Here, a is the initial amount, and b represents the growth (if b > 1) or decay (if 0 < b < 1) rate.
2. Logarithmic Differentiation:
To find the multiplicative rate of change, you can use logarithmic differentiation. This involves differentiating a function of the form f(x) = a \cdot b^{x}, which moves the exponent down in front of the logarithm and simplifies differentiation:
\frac{d}{dx} f(x) = \frac{d}{dx} (a \cdot b^{x}) = a \cdot b^{x} \cdot \ln(b).
3. Relative Rate of Change:
Another way to look at the multiplicative rate of change is through the relative rate of change, which is given by:
\text{Relative rate of change} = \frac{f'(x)}{f(x)}.
For an exponential function f(x) = a \cdot b^{x}:
f'(x) = a \cdot b^{x} \cdot \ln(b),
which means
\text{Relative rate of change} = \frac{a \cdot b^{x} \cdot \ln(b)}{a \cdot b^{x}} = \ln(b).
Thus, the relative rate of change of an exponential function f(x) = a \cdot b^{x} is \ln(b). This indicates how fast the function f(x) grows or decays relative to its current value.
4. Example:
Consider the exponential function f(x) = 3 \cdot 2^{x}:
- Here, a = 3 and b = 2.
- The multiplicative rate of change is given by the natural logarithm of the base:
\text{Relative rate of change} = \ln(2).
Summary:
The multiplicative rate of change, especially for exponential functions, highlights how the function’s output changes multiplicatively for a small input change. It can be quantified using logarithmic differentiation and is often expressed as the natural logarithm of the base of the exponential function.
This approach provides a clear interpretation of the function’s growth or decay behavior in contexts such as population growth, radioactive decay, and financial interest calculations.