Problem 13 - 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) Exponential growth. (b) Exponential decay.

Step 1: Calculate Values for y=e^x

We will calculate the value of \(y\) for each given \(x\) using the equation \(y = e^x\). The values of \(x\) are 0, 1, 2, and 3. The constant \(e\) is approximately 2.718. - 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 \(x = 3\), \(y = e^3 \approx 20.086\).

Step 2: Plot Points for y=e^x

Plot each of the pairs \((x, y)\) on a graph:- \((0, 1)\)- \((1, 2.718)\)- \((2, 7.389)\)- \((3, 20.086)\)The points will connect to form a curve rising to the right, which is characteristic of exponential growth.

Step 3: Determine Graph Type for y=e^x

Since the graph rises as \(x\) increases, the function \(y = e^x\) is an exponential growth function.

Step 4: Calculate Values for y=e^-x

Now, calculate the value of \(y\) for each \(x\) using the equation \(y = e^{-x}\) for \(x = 0, 1, 2, 3\). - 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 \(x = 3\), \(y = e^{-3} \approx 0.050\).

Step 5: Plot Points for y=e^-x

Plot each of the pairs \((x, y)\) on a graph:- \((0, 1)\)- \((1, 0.368)\)- \((2, 0.135)\)- \((3, 0.050)\)The points will connect to form a curve descending to the right, which is characteristic of exponential decay.

Step 6: Determine Graph Type for y=e^-x

Since the graph falls as \(x\) increases, the function \(y = e^{-x}\) is an exponential decay function.

Key Concepts

Exponential Growth

Exponential growth is a process where quantities increase rapidly over time. In mathematical terms, an exponential function is represented as \( y = e^x \), where \( e \) is approximately 2.718. This type of function is characterized by having a constant ratio of change, meaning the function multiplies by a fixed factor as \( x \) increases.
Let's consider the function \( y = e^x \). Here are a few key points about it:

• The function starts with a base value of 1 when \( x = 0 \), since \( e^0 = 1 \).
• As \( x \) increases to 1, 2, and 3, the resulting values of \( y \) become approximately 2.718, 7.389, and 20.086, respectively.
• This rapid increase in \( y \) values signifies exponential growth.

Such a curve on a graph will rise steeply as \( x \) moves from left to right, illustrating how quantities grow fast. Exponential growth differs from linear growth because it accelerates as time goes on. Understanding this concept is crucial in fields like finance and biology where it predicts phenomena such as inflation or population growth.

Exponential Decay

Exponential decay describes a situation where quantities decrease rapidly over time. The mathematical representation of exponential decay is \( y = e^{-x} \). This function indicates a constant proportional reduction, meaning the function multiplies by a decreasing factor as \( x \) increases.
Examining the function \( y = e^{-x} \):

• Again, at \( x = 0 \), the base value is 1, since \( e^0 = 1 \).
• As \( x \) rises to 1, 2, and 3, the \( y \) values decrease to approximately 0.368, 0.135, and 0.050 respectively.
• This reduction pattern characterizes exponential decay.

The graph of this function will show a steep descent from left to right. Unlike linear descent, the decline accelerates as the value of \( x \) increases. This exponential decrease is vital in understanding processes like radioactive decay or depreciation in asset value, where losses or reductions are not uniform but increase over time.

Graphing Functions

Graphing functions is a technique used to visualize how different functions, such as exponential growth and decay, behave. A graph provides a pictorial representation of data or equations and helps to observe patterns easily.
Let's look at graphing the functions \( y = e^x \) and \( y = e^{-x} \):

• For \( y = e^x \): When plotting, you notice the points rise from left to right, illustrating the rapid increase in \( y \) values as \( x \) gets larger. This rising curve showcases exponential growth.
• For \( y = e^{-x} \): The plotted points descend sharply, revealing how the values decrease quickly as \( x \) increases, a hallmark of exponential decay.

Creating graphs involves marking calculated points on a coordinate system and then connecting them to form a curve or line. It's a powerful approach for identifying the behavior of various functions and comparing them, providing an intuitive grasp on how the quantities grow or decay over time.