Problem 15 - Applied Calculus

Question

(a) Make a table of values for \(y=e^{x}\) using \(x=\) 0,1,2,3 (b) Plot the points found in part (a). Does the graph look like an exponential growth or decay function? (c) Make a table of values for \(y=e^{-x}\) using \(x=\) 0,1,2,3 (d) Plot the points found in part (c). Does the graph look like an exponential growth or decay function?

Step-by-Step Solution

Answer

(a) and (b) show exponential growth; (c) and (d) show exponential decay.

Step 1: Table of Values for \( y = e^x \)

Start by calculating the values of \( y = e^x \) for the given \( x \) values.For \( x = 0 \), \( y = e^0 = 1 \).For \( x = 1 \), \( y = e^1 \approx 2.718 \).For \( x = 2 \), \( y = e^2 \approx 7.389 \).For \( x = 3 \), \( y = e^3 \approx 20.086 \).Thus, the table of values is:| \( x \) | \( y = e^x \) ||-------|--------------|| 0 | 1 || 1 | 2.718 || 2 | 7.389 || 3 | 20.086 |

Step 2: Plot for \( y = e^x \)

Using the table of values obtained in Step 1, plot the points on a graph with \( x \) on the horizontal axis and \( y \) on the vertical axis. The points are (0,1), (1,2.718), (2,7.389), and (3,20.086). Connect these points with a smooth curve.Observe the trend: the graph shows a rapid increase, indicating that it is an exponential growth function.

Step 3: Table of Values for \( y = e^{-x} \)

Calculate the values of \( y = e^{-x} \) for the given \( x \) values.For \( x = 0 \), \( y = e^{0} = 1 \).For \( x = 1 \), \( y = e^{-1} \approx 0.368 \).For \( x = 2 \), \( y = e^{-2} \approx 0.135 \).For \( x = 3 \), \( y = e^{-3} \approx 0.050 \).Thus, the table of values is:| \( x \) | \( y = e^{-x} \) ||-------|------------------|| 0 | 1 || 1 | 0.368 || 2 | 0.135 || 3 | 0.050 |

Step 4: Plot for \( y = e^{-x} \)

Using the table of values obtained in Step 3, plot the points on a graph with \( x \) on the horizontal axis and \( y \) on the vertical axis. The points are (0,1), (1,0.368), (2,0.135), and (3,0.050). Connect these points with a smooth curve.Observe the trend: the graph shows a rapid decrease, indicating that it is an exponential decay function.

Key Concepts

Exponential Growth

Exponential growth occurs when the value of a function increases rapidly over time. In the context of mathematical functions, exponential growth can be represented by functions such as \( y = e^x \), where \( e \) (approximately 2.718) is the base of the natural logarithm. In these functions, the variable \( x \) is the exponent, and as \( x \) increases, \( y \) increases quickly due to the multiplication involved in each step. Consider the exercise example of making a table for \( y = e^x \) using the values \( x = 0, 1, 2, 3 \). We notice:

• At \( x = 0 \), \( y = e^0 = 1 \)
• At \( x = 1 \), \( y \approx 2.718 \)
• At \( x = 2 \), \( y \approx 7.389 \)
• At \( x = 3 \), \( y \approx 20.086 \)

With each step, the output value \( y \) grows larger at an increasing rate, which characterizes exponential growth. This concept is crucial in various fields, such as biology for population models and finance for compound interest.

Exponential Decay

Exponential decay describes a process where a function's value decreases quickly as time progresses. This type of decay can be represented by equations like \( y = e^{-x} \). Here, the negative sign in the exponent causes the base \( e \) to be raised to a decreasing power as \( x \) increases, resulting in a smaller \( y \) value each time.From the exercise, we create a table for \( y = e^{-x} \) with the same \( x \) values:

• At \( x = 0 \), \( y = e^0 = 1 \)
• At \( x = 1 \), \( y \approx 0.368 \)
• At \( x = 2 \), \( y \approx 0.135 \)
• At \( x = 3 \), \( y \approx 0.050 \)

In this scenario, we see how \( y \) decreases rapidly, illustrating exponential decay. This concept is widely applied in physics for modeling radioactive decay and in medicine to describe the decrease of drug concentration in the bloodstream over time.

Table of Values

Creating a table of values is an essential step to understand the behavior of exponential functions. It involves calculating and organizing pairs of \( x \) and \( y \) values, which are derived from substituting \( x \) into the given function equation.For the exercise at hand, two tables were created for functions \( y = e^x \) and \( y = e^{-x} \). Here’s why this process is important:

• It helps visualize the rapid increase or decrease as \( x \) changes.
• Provides clear numerical insight into how small changes in \( x \) can have significant effects.
• Serves as a foundation for plotting these functions on a graph.

By carefully compiling these values, students gain a tangible sense of how exponential functions behave, which is critical for further study and application in real-world scenarios.

Graphing Exponential Functions

Graphing is an invaluable tool for visually understanding how exponential functions behave. By plotting the points from a table of values, one can observe either an exponential growth or decay. Here's how it works:For the growth function \( y = e^x \), the graph revealed:

• Points were plotted at coordinates \( (0, 1) \), \( (1, 2.718) \), \( (2, 7.389) \), and \( (3, 20.086) \).
• Connecting these points forms a curve that rises steeply, illustrating the rapid increase characteristic of exponential growth.

For the decay function \( y = e^{-x} \), the decline appeared as:

• Points marked the locations \( (0, 1) \), \( (1, 0.368) \), \( (2, 0.135) \), and \( (3, 0.050) \).
• The curve drops swiftly, showing how values decrease quickly with greater \( x \).

Through graphing, students can have an immediate visual representation of how the function behaves across different values, reinforcing their understanding of exponential growth and decay.