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Chapter 1: Problem 26

In the form \(P=P_{0} a^{t}\). Which represent exponential growth and which represent exponential decay? $$P=7 e^{-\pi t}$$

Short Answer

Expert verified
Exponential decay

Step by step solution

01

Identify the components of the given function

The function given is \(P = 7 e^{-\pi t}\). Here, the initial value \(P_0 = 7\) and the base of the exponential function is \(e^{-\pi}\). In the form \(P = P_0 a^t\), \(a\) corresponds to the base of the exponential term, which in this case is \(e^{-\pi}\).
02

Determine the value of the exponential base

The base \(a\) here is \(e^{-\pi}\). Since \(e\) is a mathematical constant approximately equal to 2.71828, \(e^{-\pi}\) means \(e\) raised to the power of \(-\pi\), which is a negative exponent.
03

Analyze the sign of the exponent

Because the exponent of \(e\) is negative (\(-\pi\)), the exponential form \(e^{-\pi}\) will result in a value that is a positive fraction less than one. This is because any positive number with a negative exponent results in a fraction.
04

Determine the type of exponential function

In exponential functions, if the base \(a\) is less than 1, the function represents exponential decay. Since \(e^{-\pi}\) is a fraction less than one, the function \(P = 7 e^{-\pi t}\) represents exponential decay.

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Key Concepts

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

Exponential Functions
Exponential functions are mathematical expressions that grow or decay at a constant percentage rate. They are unique in that a constant, usually greater than one, is raised to the power which varies over time. These functions can model a vast range of real-world phenomena such as population growth, radioactive decay, and even the cooling of a hot object. The general form of an exponential function is given by:
  • \( P = P_0 a^t \)
Where:
  • \( P \) is the final amount or value.
  • \( P_0 \) is the initial amount or value.
  • \( a \) is the base of the exponential function, determining the nature of growth or decay.
  • \( t \) is the time variable.
When the base \( a \) is greater than one, the function models exponential growth, meaning it increases over time. Conversely, if the base \( a \) is between zero and one, the function will represent exponential decay, illustrating decrease over a period.
Negative Exponent
In mathematics, a negative exponent signifies that the base should be moved to the denominator of a fraction. It shows how division works in exponential terms and provides a direct connection to how values decrease in exponential decay functions.For example, \( e^{- ext{value}} \) is equivalent to \( \frac{1}{e^{ ext{value}}} \). Here's what a negative exponent does:
  • Makes large numbers small: A negative power of a large constant like \( e \) results in a very small number or fraction.
  • Indicates inversion: It tells you to take the reciprocal of the base raised to a positive exponent.
  • In exponential decay: It ensures that the exponential function decreases over time.
So when the function is \( e^{- ext{something}} \), like \( e^{- ext{蟺}} \) in our example, it creates a fraction, causing the overall exponential function to decay as time \( t \) increases. This aligns with the premise that for decay, the exponential term must be constantly less than one.
Mathematical Constant e
The mathematical constant \( e \), known as Euler's number, plays a critical role in exponential functions, especially those representing natural phenomena. Approximately equal to 2.71828, \( e \) is the base of natural logarithms and is pivotal in describing continuously compounding growth or decay processes.Key traits of \( e \):
  • It is an irrational number, meaning it cannot be expressed as a simple fraction.
  • \( e \) originates from the concept of continuously compounded interest, which is why it naturally appears in growth and decay contexts.
  • The function \( e^x \) is its own derivative, making calculus operations simpler in many exponential scenarios.
In our example \( P = 7 e^{- ext{蟺} t} \), \( e \) forms the base of the exponential part, capturing the natural decrease dictated by a negative coefficient of \( t \). This not only helps illustrate decay but also simplifies calculations related to changes over continuous intervals.

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Most popular questions from this chapter

Pregnant women metabolize some drugs at a slower rate than the rest of the population. The half-life of caffeine is about 4 hours for most people. In pregnant women, it is 10 hours. \({ }^{61}\) (This is important because caffeine, like all psychoactive drugs, crosses the placenta to the fetus.) If a pregnant woman and her husband each have a cup of coffee containing \(100 \mathrm{mg}\) of caffeine at \(8 \mathrm{am}\), how much caffeine does each have left in the body at \(10 \mathrm{pm}\) ?

In 2000 , there were about 213 million vehicles (cars and trucks) and about 281 million people in the US. The number of vehicles has been growing at \(4 \%\) a year, while the population has been growing at \(1 \%\) a year. If the growth rates remain constant, when is there, on average, one vehicle per person?

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From October 2002 to October 2006 the number \(N(t)\) of Wikipedia articles was approximated by \(N(t)=\) \(N_{0} e^{t / 500}\), where \(t\) is the number of days after October 1,2002 . Find the doubling time for the number of Wikipedia articles during this period.

Sketch graphs of the functions. What are their amplitudes and periods? $$y=3 \sin 2 x$$
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