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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.
- 4.2 Linear Approximations and Differentials
Analysis. Using a calculator, the value of 9.1 9.1 to four...
- 5.6 Integrals Involving Exponential and Logarithmic Functions
As mentioned at the beginning of this section, exponential...
- 3.9 Derivatives of Exponential and Logarithmic Functions
Assume the population is increasing at a rate of 5% per...
- 6.2 Determining Volumes by Slicing
When the solid of revolution has a cavity in the middle, the...
- 6.4 Arc Length of a Curve and Surface Area
Arc Length of the Curve x = g(y). We have just seen how to...
- Introduction
Figure 1.1 A portion of the San Andreas Fault in California....
- Key Terms
This free textbook is an OpenStax resource written to...
- Key Equations
6.8 Exponential Growth and Decay; 6.9 Calculus of the...
- 4.2 Linear Approximations and Differentials
17 sie 2024 · In exponential growth, the rate of growth is proportional to the quantity present. In other words, \(y′=ky\). Systems that exhibit exponential growth have a constant doubling time, which is given by \((\ln 2)/k\). Systems that exhibit exponential decay follow a model of the form \(y=y_0e^{−kt}.\)
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.
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.
In this section we look at some applications of the derivative by focusing on the interpretation of the derivative as the rate of change of a function. These applications include acceleration and velocity in physics, population growth rates in biology, and marginal functions in economics.
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.
