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From population growth and continuously compounded interest to radioactive decay and Newton’s law of cooling, exponential functions are ubiquitous in nature. In this section, we examine exponential growth and decay in the context of some of these applications.
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17 sie 2024 · Exponential growth and exponential decay are two of the most common applications of exponential functions. Systems that exhibit exponential growth follow a model of the form \(y=y_0e^{kt}\). In exponential growth, the rate of growth is proportional to the quantity present. In other words, \(y′=ky\).
Exponential growth and exponential decay are two of the most common applications of exponential functions. Systems that exhibit exponential growth follow a model of the form [latex]y= {y}_ {0} {e}^ {kt}. [/latex] In exponential growth, the rate of growth is proportional to the quantity present.
The relative continuous growth rate of $f(t)$ is defined as $$\frac{f'(t)}{f(t)}.$$ Your function is $f(t)=4 \cdot 2^{t/5}$, with $f'(t)=4\cdot (1/5)\ln(2)2^{t/5}.$ So its relative growth rate is $(1/5)\ln(2)$. Note how the initial value 4 "cancelled out" in finding the relative continuous growth rate.
Describe the relative growth rates of functions. Suppose the functions [latex]f [/latex] and [latex]g [/latex] both approach infinity as [latex]x\to \infty [/latex].
Learning Objectives. Use the exponential growth model in applications, including population growth and compound interest. Explain the concept of doubling time. Use the exponential decay model in applications, including radioactive decay and Newton’s law of cooling. Explain the concept of half-life.
The continuous growth formula is a mathematical expression used to model situations where a quantity grows at a constant rate continuously over time, rather than at discrete intervals. It is commonly represented as $$A = Pe^{rt}$$, where $$A$$ is the final amount, $$P$$ is the initial amount, $$r$$ is the growth rate, and $$t$$ is time.
