Problem 14 - Applied Calculus

Question

Graph \(y=100 e^{-0.4 x} .\) Describe what you see.

Step-by-Step Solution

Answer

The graph shows an exponential decay starting at 100, decreasing towards 0 as \(x\) increases.

Step 1: Understand the Function

The function to graph is \(y = 100 e^{-0.4x}\), where \(e\) is the base of the natural logarithm, approximately equal to 2.718. The function is an exponential decay function because of the negative exponent.

Step 2: Determine Key Characteristics

For exponential decay functions of the form \(y = a e^{-bx}\), \(a\) is the initial value (when \(x=0\)) and the decay rate is determined by \(b\). In this case, \(a = 100\) and \(b = 0.4\). When \(x = 0\), \(y = 100\). As \(x\) increases, \(y\) approaches 0 but never becomes negative.

Step 3: Plot Key Points

Start by calculating a few key points:- \(x = 0\), \(y = 100\).- \(x = 1\), \(y = 100 e^{-0.4} \approx 67.03\).- \(x = 2\), \(y = 100 e^{-0.8} \approx 45.52\).- \(x = 3\), \(y = 100 e^{-1.2} \approx 30.12\).Identify these points on the graph.

Step 4: Draw the Graph

Using the key points plotted, sketch a smooth curve that starts at \(y=100\) when \(x = 0\) and gradually decreases towards 0 as \(x\) becomes large. The curve should always decrease and never become negative.

Step 5: Analyze the Graph

The graph shows an exponential decay. It starts at 100 and decreases to approach 0 as \(x\) increases. It is a common behavior in phenomena such as radioactive decay, cooling of a hot object, or depreciation of value over time.

Key Concepts

Graphing Exponential Functions

Graphing exponential functions helps visualize their behavior over time. In the case of a function like \( y = 100 e^{-0.4x} \), we're dealing with exponential decay. This type of function starts with a high value, which then decreases rapidly at first and levels off over time.

To graph this function, you start by identifying key values, such as when \( x = 0 \), where \( y \) is at its maximum, \( 100 \). As time (or \( x \)) progresses, the value of \( y \) drops logarithmically, approaching zero but never quite reaching it.

To plot the graph:

• Find several manageable values of \( x \), such as 1, 2, and 3, to calculate \( y \).
• Mark these points on the graph to form an outline of the exponential decay.
• Draw a smooth curve through these points, ensuring it drops precipitously initially and then tapers.

The graph in this context is "asymptotic" to the \( x \)-axis, signifying that \( y \) gets infinitely close to zero but remains positive. This represents many real-world scenarios where decay or decrease never fully diminishes.

Properties of Exponential Functions

Understanding the properties of exponential functions is central in mathematics and various fields of study. Exponential functions like \( y = 100 e^{-0.4x} \) demonstrate unique characteristics especially when studying decay.

Key properties of exponential decay functions include:

• **Initial Value**: Given by \( a \), which is 100 in this function, it represents the starting point or the function's value when \( x = 0 \).
• **Rate of Decay**: Defined by \( b = 0.4 \), governs how fast the decay process happens. Larger \( b \) values mean a faster decay.
• **Asymptotic Behavior**: The function approaches zero but never becomes negative, implying it gets infinitesimally close to the \( x \)-axis.
• **Continuity and Slope**: These functions are continuous and smooth, meaning there are no breaks or sharp turns, and the slope decreases over time.

These properties are crucial for predicting and understanding phenomena that follow an exponential law, whether it be in nature or technology.

Exponential Function Applications

Exponential functions are not just mathematical abstractions; they play a vital role in various real-world applications. The function \( y = 100 e^{-0.4x} \) is a perfect example of exponential decay, which is prevalent across multiple domains.

Here are some common applications where exponential decay functions are essential:

• **Radioactive Decay**: This function is used to determine how quickly a substance loses its radioactivity. The rate at which decay happens is described by the exponent.
• **Newton's Law of Cooling**: Describes how an object cools over time. Exponential decay models how the temperature gradually decreases.
• **Depreciation of Value**: Financial calculations use exponential decay to estimate the diminishing value of assets over time.

These applications show how exponential decay is ubiquitous in modeling behaviors where reduction or decay occurs over time. Understanding the underlying function helps in making accurate predictions and informed decisions in these fields.