Equivalent growth functions The same exponential growth...

Equivalent growth functions The same exponential growth function can be written in the forms $y(t)=y_{0} e^{t f}, y(t)=y_{0}(1+r)^{t}$ and $y(t)=y_{0} 2^{1 / T_{2}}$. Write $k$ as a function of $r, r$ as a function of $T_{2}$ and $T_{2}$ as a function of $k .$

Key Concepts

Exponential Growth

Exponential growth is a fundamental concept where the rate of change of a quantity is proportional to its current value, which leads to growth processes that accelerate over time. This is seen in various forms, such as the continuous model y(t) = y?e^(kt) and the discrete model y(t) = y?(1+r)?, both of which describe the same underlying phenomenon with different parameters.

Conversion between Continuous and Discrete Growth Rates

This concept involves relating the continuous growth parameter k, which is used in the exponential expression, with the discrete growth rate r found in the compound interest style formulation. The relation k = ln(1+r) allows one to convert between the two representations, ensuring that the underlying growth process remains consistent regardless of the model used.

Doubling Time

Doubling time is the period required for a quantity undergoing exponential growth to double in size. It is connected to the continuous growth rate k through the formula T? = ln2/k. Additionally, when using a formulation like y(t) = y?2^(t/T?), one sees how the doubling time directly scales the growth process, providing a practical measure of how quickly the quantity increases.