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Chapter 2: Problem 1

In Problems \(1-4\), produce a table for \(t=0,1,2, \ldots, 5\) and graph the function \(N_{t}\). $$ N_{t}=3^{t} $$

Short Answer

Expert verified
The function \( N_{t} = 3^{t} \) grows exponentially, as shown by the increasing values in the table and the graph.

Step by step solution

01

Identify the Function

The given function is expressed as \( N_{t} = 3^{t} \). We will evaluate this function for values of \( t \) from 0 to 5.
02

Calculate \( N_{t} \) for each \( t \)

Compute the value of \( N_{t} = 3^{t} \) for each \( t \) in the range \( t=0,1,2,3,4,5 \).\( N_{0} = 3^{0} = 1 \)\( N_{1} = 3^{1} = 3 \)\( N_{2} = 3^{2} = 9 \)\( N_{3} = 3^{3} = 27 \)\( N_{4} = 3^{4} = 81 \)\( N_{5} = 3^{5} = 243 \)
03

Create a Table of Values

Using the computed values for \( N_{t} \), construct a table. \[\begin{array}{|c|c|}\hline t & N_{t} \ \hline 0 & 1 \ \hline 1 & 3 \ \hline 2 & 9 \ \hline 3 & 27 \ \hline 4 & 81 \ \hline 5 & 243 \ \hline\end{array} \]
04

Graph the Function

Plot the function \( N_{t} = 3^{t} \) on a graph using the table of values. Place \( t \) on the x-axis and \( N_{t} \) on the y-axis, and plot the points: - \((0, 1)\)- \((1, 3)\)- \((2, 9)\)- \((3, 27)\)- \((4, 81)\)- \((5, 243)\)Connect these points to display the exponential growth of the function.

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Key Concepts

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

Understanding Table of Values
A table of values is a fundamental tool in understanding functions, particularly when dealing with exponential functions such as \( N_{t} = 3^{t} \). This table helps us organize computed outputs for specified inputs, making patterns in the function easier to see. Let's break down how to construct this useful tool for our function.
  • **Define your range:** Choose the values of \( t \) that you want to examine. For the given exercise, we are using \( t = 0, 1, 2, 3, 4, 5 \).
  • **Calculate each value:** Plug each of these \( t \) values into the function \( N_{t} = 3^{t} \) to get the corresponding \( N_{t} \). This results in the values: 1, 3, 9, 27, 81, and 243 for each respective \( t \).
Once you have calculated these values, you can efficiently display them in a structured table format. This makes it much easier to analyze and graph the function's behavior. Each row of the table represents a unique pair of \( t \) and \( N_{t} \), clearly showing how the function evolves as \( t \) changes.
Graphing Exponential Functions
Graphing exponential functions like \( N_{t} = 3^{t} \) provides a visual representation of how the function behaves over a range of values. Creating a graph starts with taking your table of values and plotting them on a coordinate plane.

Here is how you can graph it effectively:
  • **Axes Setup:** Begin by setting the x-axis to represent the variable \( t \), the independent variable, while the y-axis will represent \( N_{t} \), the dependent variable.
  • **Plotting Points:** Use the pairs from your table to plot points like \((0, 1)\), \((1, 3)\), \((2, 9)\), and so on. Plot each point in turn on the graph.
  • **Joining the Dots:** Once plotted, connect the points with a smooth curve. The exponential nature of the function means this curve will start slowly and rise steeply as \( t \) increases.
This visualization of the function reinforces understanding of the rate of change in exponential functions. Each plotted point helps illustrate how quickly the function's values increase as the input parameter \( t \) grows.
Evaluating the Function
Evaluating a function like \( N_{t} = 3^{t} \) involves substituting specific values into the function to find its outputs. This process is a key step in creating a table of values and is crucial for understanding the behavior of the function.
Here's how to evaluate this exponential function:
  • **Substitute the Input:** Take each input value \( t \) from your chosen range (0 through 5 in this case) and substitute it into the function \( N_{t} = 3^{t} \).
  • **Compute the Power:** Calculate \( 3^{t} \). For example, if \( t = 3 \), you calculate \( 3^{3} = 27 \).
After evaluating these specific points, you get a new understanding of how the function behaves. Each calculation shows the exponential growth nature as \( t \) increases, with \( N_{t} \) values climbing rapidly. Each of these evaluated outcomes is crucial for creating graphs and tables effectively, giving a complete picture of the function's dynamics.

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Most popular questions from this chapter

Use the formal definition of limits to show that \(\lim _{n \rightarrow \infty} a_{n}=a ;\) that is, find \(N\) such that for every \(\epsilon>0\), there exists an \(N\) such that \(\left|a_{n}-a\right|<\epsilon\) whenever \(n>N\). $$ \lim _{n \rightarrow \infty} \frac{1}{n^{2}}=0 $$

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