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Chapter 5: Problem 3

Use a semilog graph to determine which of the following data sets are exponential. a.$$ \begin{array}{|c|c|} \hline \mathrm{t} & \mathrm{P}(\mathrm{t}) \\ \hline 0 & 5.00 \\ 1 & 3.53 \\ 2 & 2.50 \\ 3 & 1.77 \\ 4 & 1.25 \\ 5 & 0.88 \\ \hline \end{array} $$ b. $$ \begin{array}{|c|c|} \hline \mathrm{t} & \mathrm{P}(\mathrm{t}) \\ \hline 0 & 5.00 \\ 1 & 1.67 \\ 2 & 1.00 \\ 3 & 0.71 \\ 4 & 0.55 \\ 5 & 0.45 \\ \hline \end{array} $$ c. $$ \begin{array}{|c|c|} \hline \mathrm{t} & \mathrm{P}(\mathrm{t}) \\ \hline 0 & 5.00 \\ 1 & 3.63 \\ 2 & 2.50 \\ 3 & 1.63 \\ 4 & 1.00 \\ 5 & 0.63 \\ \hline \end{array} $$

Short Answer

Expert verified
Data sets a and c are exponential; b is not.

Step by step solution

01

Understand Exponential Relationships

For a data set to be exponential, it should show a constant ratio between consecutive terms. This means that each term is a constant multiple (the growth factor) of the previous term.
02

Plot Data Set on Semilog Graph

To determine if a data set is exponential, plot it on a semilogarithmic graph where the y-axis is logarithmic but the x-axis is linear. If the points form a straight line, the data is likely exponential.
03

Analyze Data Set a

The data set a has the values \( (0, 5.00), (1, 3.53), (2, 2.50), (3, 1.77), (4, 1.25), (5, 0.88) \). When plotted on a semilog graph, these points form a straight line, confirming that data set a is exponential.
04

Analyze Data Set b

The data set b has the values \( (0, 5.00), (1, 1.67), (2, 1.00), (3, 0.71), (4, 0.55), (5, 0.45) \). When plotted on a semilog graph, these points do not form a straight line, indicating that data set b is not exponential.
05

Analyze Data Set c

The data set c has the values \( (0, 5.00), (1, 3.63), (2, 2.50), (3, 1.63), (4, 1.00), (5, 0.63) \). When plotted on a semilog graph, these points do form a straight line, confirming that data set c is exponential.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Semilog Graphs
A semilog graph is a special type of graph used for plotting data that follows an exponential trend. It features a logarithmic scale on the y-axis while the x-axis remains linear. This unique scaling helps transform exponential relationships into straight lines, making it easier to identify such patterns.

When you plot data on a semilog graph, the log scale compresses larger values. This means that instead of jumping drastically, the changes between large numbers are reduced. On the other hand, the linear x-axis remains evenly spaced, providing a regular timeline or sequence view.

To check if a data set is exponential, plot the points on a semilog graph. If they align to form a straight line, it indicates an exponential relationship. The semilog graph essentially "unmasks" the exponential nature by aligning the curve into a straight line.

Using semilog graphs effectively requires familiarity with both reading logarithmic scales and plotting points accurately. Practicing this skill is key for clear data analysis and identifying exponential growth.
Data Analysis
Data analysis involves interpreting data sets to extract meaningful information, recognize patterns, and draw conclusions. In this exercise, we aim to determine if data sets are exponential by examining their behavior when plotted on a semilog graph. This requires both visual inspection and mathematical understanding.

To analyze a given set of data, one starts by gathering and organizing the data into a format that is easy to work with鈥攐ften as tabular data. Next, plotting is a critical step to visualize potential trends. In our case, using a semilog graph allows for simplifying the analysis of exponential data.

While examining semilog plots, look for a straight line trend. This is a clear indicator of an exponential behavior or relationship within the dataset. If the data points do not align into a straight line, this implies that the set does not exhibit exponential growth.

Analyzing data in the form of exponential growth patterns is particularly valuable in fields like natural sciences, economics, and engineering, where understanding the nature of change over time or conditions is crucial.
Exponential Growth
Exponential growth describes a situation where a quantity increases by a consistent percentage over equal time periods. This leads to the rapid increase of the quantity as each increment builds upon the previous amount.

Mathematically, exponential growth can be expressed with the formula: \[ P(t) = P_0 \times (1 + r)^t \]where:
  • \( P(t) \) is the amount at time \( t \),
  • \( P_0 \) is the initial amount,
  • \( r \) is the rate of growth,
  • \( t \) is the time period.
Exponential growth is characterized by doubling patterns over regular intervals, commonly seen in populations, investments, and certain types of reactions in chemistry.

Recognizing exponential growth through visual methods, like semilog graphs, enables quick identification of such trends. This understanding is essential in making informed predictions and decisions based on historical data. Notably, exponential growth models are highly sensitive to changes in the growth rate \( r \), making precision in initial assumptions critical for accurate forecasts. Such growth is often unsustainable in the long term because resources can become limited, leading to eventual decline or stabilization.

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Most popular questions from this chapter

Use one rule for each step and identify the rule to differentiate a. \(P(t)=3 \ln t+e^{3 t}\) b. \(P(t)=t^{2}+\ln 2 t\) c. \(P(t)=\ln 5\) d. \(P(t)=\ln \left(e^{2 t}\right)\) e. \(P(t)=\ln \left(t^{2}+t\right)\) f. \(P(t)=e^{t^{2}-t}\) g. \(P(t)=e^{1 / x}\) h. \(P(t)=e^{\sqrt{x}}\) i. \(P(t)=\ln \left((t+1)^{2}\right)\) j. \(P(t)=e^{-t^{2} / 2}\)

Suppose your number system is that of Early Greek mathematicians and includes only rational numbers. Does it satisfy the Axiom of Completion?

Consider the kinetics of penicillin that is taken as a pill in the stomach. The diagram in Figure Ex. \(5.4 .9(\) a \()\) may help visualize the kinetics. We will find in Chapter 17 that a model of plasma concentration of antibiotic \(t\) hours after ingestion of an antibiotic pill yields an equation similar to $$C(t)=5 e^{-2 t}-5 e^{-3 t} \quad \mu \mathrm{g} / \mathrm{ml}$$ A graph of \(C\) is shown in Figure Ex. \(5.4 .9 .\) At what time will the concentration reach a maximum level, and what is the maximum concentration achieved? As we saw in Section 3.5 .2 and may be apparent from the graph in Figure Ex. 5.4 .9 , the highest concentration is associated with the point of the graph of \(C\) at which \(C^{\prime}=0 ;\) the tangent at the high point is horizontal. The question, then, is at what time \(t\) is \(C^{\prime}(t)=0\) and what is \(C(t)\) at that time?

Plasma penicillin concentration is $$P(t)=5 e^{-0.3 t}-5 e^{-0.4 t}$$ \(t\) hours after ingestion of a penicillin pill into the stomach. A small amount of the drug diffuses into tissue and the tissue concentration, \(C(t)\), is $$ C(t)=-e^{-0.3 t}+0.5 e^{-0.4 t}+0.5 e^{-0.2 t} \quad \mu \mathrm{g} / \mathrm{ml} $$ a. Use your technology (calculator or computer) to find the time at which the concentration of the drug in tissue is maximum and the value of \(C\) at that time. b. Compute \(C^{\prime}(t)\) and solve for \(t\) in \(C^{\prime}(t)=0\). This is really bad, for you must solve for \(t\) in $$ 0.3 e^{-0.3 t}-0.2 e^{-0.4 t}-0.1 e^{-0.2 t}=0 $$ Try this: $$ \text { Let } \quad Z=e^{-0.1 t} \quad \text { then solve } \quad 0.3 Z^{3}-0.2 Z^{4}-0.1 Z^{2}=0 . $$ c. Solve for the possible values of \(Z\). Remember that \(Z=e^{-0.1 t}\) and solve for \(t\) if possible using the possible values of \(Z\). d. Which value of \(t\) solves our problem?

In Section 1.3 we found from the discrete model of light extinction, \(I_{d+1}=I_{d}-0.18 I_{d}\) that the solution \(\quad I_{d+1}=0.82 I_{d}\) Light decrease in water is continuous, however. Find a value of \(k\) for which the solution to the continuous model, \(I^{\prime}(x)=-k I(x),\) matches the data.
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