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Chapter 3: Problem 65

Describe a difference between exponential growth and logistic growth.

Short Answer

Expert verified
Exponential growth refers to growth that becomes increasingly rapid in relation to the growing total quantity or size, and it doesn't consider potential limiting factors, hence theoretically it can continue indefinitely. Logistic growth, on the other hand, is initially fast and then slows down over time until it reaches a maximum value, considering limiting environmental factors such as resources. It forms an S-shaped curve upon graphing.

Step by step solution

01

Exponential Growth Explanation

Exponential growth can be defined as growth whose rate becomes ever more rapid in proportion to the growing total number or size. It can be illustrated mathematically with the function \( Y=a(1+r)^x \), where Y represents the final amount of whatever is growing, 'a' stands for the initial amount, 'r' is the rate of growth (as a decimal), and 'x' signifies the time period.
02

Logistic Growth Explanation

Logistic growth is growth that increases rapidly at first, then slows down over time until it reaches a maximum value known as the carrying capacity (K). It can be represented mathematically through the function \( Y = K/(1+e^{-r(x-x_0)}) \), where Y is the quantity of what's growing, K equals the carrying capacity, 'r' is the maximum growth rate, 'x' represents time, and \(x_0\) is the inflection point.
03

Comparison of Exponential and Logistic Growth

The primary difference between exponential and logistic growth is that exponential growth is consistent and continuous, while logistic growth is initially rapid and then tapers off, creating an S-shaped curve. In exponential growth, there are no limiting factors, implying that growth can theoretically continue indefinitely. On the contrary, logistic growth takes into consideration limiting factors (like limited resources), hence the growth rate decreases as the population approaches the carrying capacity.

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Most popular questions from this chapter

The exponential growth models describe the population of the indicated country, \(A\), in millions, \(t\) years after 2006 $$\begin{array{l}\mathrm{Camada}\quadA=33.1e^{0.009\mathrm{t}}\\\\\mathrm{U}_{\mathrm{ganda}}\quad A=28.2 e^{0.034 t}\end{array}$$ In Exercises \(81-84,\) use this information to determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. The models indicate that in \(2013,\) Uganda's population will exceed Canada's.

Hurricanes are one of nature's most destructive forces. These low-pressure areas often have diameters of over 500 miles. The function \(f(x)=0.48 \ln (x+1)+27\) models the barometric air pressure, \(f(x),\) in inches of mercury, at a distance of \(x\) miles from the eye of a hurricane. Use this function to solve Exercises \(133-134\) The function \(W(t)=2600\left(1-0.51 e^{-0.075 t}\right)^{3}\) models the weight, \(W(t),\) in kilograms, of a female African elephant at age \(t\) years. (1 kilogram \(\approx\) 2.2 pounds) Use a graphing utility to graph the function. Then \([\text { TRACE }]\) along the curve to estimate the age of an adult female elephant weighing 1800 kilograms.

From 1970 through \(2010 .\) The data are shown again in the table. Use all five data points to solve Exercises \(70-74\). $$\begin{array}{cc}\hline \begin{array}{c}x, \text { Number of Years } \\\\\text { after } 1969 \end{array} & \begin{array}{c}y, \text { U.S. Population } \\\\\text { (millions) }\end{array} \\ \hline 1(1970) & 203.3 \\\11(1980) & 226.5 \\\21(1990) & 248.7 \\\31(2000) & 281.4 \\\41(2010) & 308.7 \end{array}$$ a. Use your graphing utility's exponential regression option to obtain a model of the form \(y=a b^{x}\) that fits the data. How well does the correlation coefficient, \(r,\) indicate that the model fits the data? b. Rewrite the model in terms of base \(e\). By what percentage is the population of the United States increasing each year?

In Exercises \(125-132,\) use your graphing utility to graph each side of the equation in the same viewing rectangle. Then use the \(x\) -coordinate of the intersection point to find the equation's solution set. Verify this value by direct substitution into the equation.. $$3^{x}=2 x+3$$

a. Graph \(y=e^{x}\) and \(y=1+x+\frac{x^{2}}{2}\) in the same viewing rectangle. b. Graph \(y=e^{x}\) and \(y=1+x+\frac{x^{2}}{2}+\frac{x^{3}}{6}\) in the same viewing rectangle. c. Graph \(y=e^{x}\) and \(y=1+x+\frac{x^{2}}{2}+\frac{x^{3}}{6}+\frac{x^{4}}{24}\) in the same viewing rectangle. d. Describe what you observe in parts \((a)-(c) .\) Try generalizing this observation.
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