Problem 7 - Calculus Volume 3

Question

Use technology (CAS or calculator) to sketch the parametric equations. $$ x=e^{-t}, \quad y=e^{2 t}-1 $$

Step-by-Step Solution

Answer

Use a graphing calculator to plot \( x = e^{-t} \), \( y = e^{2t}-1 \) with \( t = [-3,3] \).

Step 1: Understand the Parametric Equations

The given parametric equations are: \[ x = e^{-t} \] and \[ y = e^{2t} - 1 \].These equations describe a curve in the xy-plane with \( t \) as the parameter. Our task is to sketch this curve using a technological tool such as a Computer Algebra System (CAS) or a graphing calculator.

Step 2: Set Up the Parameter Range

Choose a range for the parameter \( t \). A common choice is \( t \) from \(-3\) to \( 3 \) because exponential functions are most interesting in this range. However, verify with your specific graphing tool if this gives a good view.

Step 3: Input Parametric Equations into the Calculator

Open the graphing tool or CAS and locate the option to input parametric equations. Enter \( x = e^{-t} \) and \( y = e^{2t} - 1 \), ensuring your calculator is in parametric mode.

Step 4: Set the Parameter Range

Set the parameter \( t \) to vary between \(-3\) and \( 3 \) (or another reasonable range determined previously). This will define the portion of the curve you can visualize.

Step 5: Adjust the Viewing Window

Modify the viewing window to an appropriate area in the plane, usually around x from 0 to 1 (since \( e^{-t} \) decreases from 1 quickly as \( t \) increases) and y from -1 upwards (as \( e^{2t}-1 \) increases rapidly). This ensures that the curve is visible on the screen.

Step 6: Sketch the Graph

Once everything is set up, instruct the calculator to sketch the graph. Analyze the plot to understand the behavior of the function, such as how as \( t \) increases, \( x \) approaches zero and \( y \) increases rapidly.

Key Concepts

Exponential Functions

Exponential functions are mathematical expressions where a constant base is raised to a variable exponent. These functions show rapid growth or decay, depending on the sign of the exponent.
In the context of parametric equations, like those in the exercise, exponential functions define how quickly each set of coordinates changes as the parameter varies.
- For example, in the parametric equation \( x = e^{-t} \), as the parameter \( t \) increases, the value of \( x \) decreases rapidly towards zero. This is because \( e^{-t} \) suggests an exponential decay.- Conversely, the function \( y = e^{2t} - 1 \) shows exponential growth. Here, as \( t \) becomes larger, \( e^{2t} \) increases swiftly, stretching the value of \( y \) upwards.Exponential functions are crucial in modeling natural phenomena like population growth, radioactive decay, and in financial calculations, such as compound interest.

Graphing Calculator

A graphing calculator is a powerful tool that can plot complex mathematical functions and equations. This includes parametric equations with different parameters using specified ranges.
To graph parametric equations:

• Ensure the calculator is set to parametric mode. This allows input of equations in terms of a parameter, typically \( t \).
• Input the given equations in their respective settings (usually denoted as \( x(t) \) and \( y(t) \)).
• Set an appropriate range for \( t \), often between \established values that best represent the function's behavior (e.g., \(-3\) to \(3\)).

After correctly inputting and setting your range, the graphing calculator can display the dynamic shape and aspects of the parametric curve, helping to visualize the mathematical relationships.

Computer Algebra System

A Computer Algebra System (CAS) is an advanced software tool designed for complex mathematical computations, including working with parametric equations. It can solve, simplify, and manipulate algebraic equations and expressions.
Using a CAS in working with parametric equations involves:

• Inputting the parametric equations into the system, similar to setting them up on a graphing calculator.
• Defining the parameter range, which influences the visible portion of the curve or graph. On a CAS, you can experiment with broader or more precise ranges.
• Adjusting visual scopes, offering more control over the details of what is plotted, to ensure clarity and focus on particular curve attributes or behaviors.

CAS tools provide sophisticated plotting options, often rendering more precise results than hardware calculators. They are excellent for exploring theoretical aspects or gaining insights into the mathematical behavior of the plotted equations.

Parametric Curves

Parametric curves are graphs where a set of parameters, often \( t \), define the coordinates. Unlike the traditional \( y = f(x) \) format, parametric equations allow for more complex and detailed representations.
When dealing with parametric equations:

• Each value of \( t \) generates a point \((x, y)\) within the coordinate plane.
• Changing the parameter range affects how much of the curve is visible.
• The parametric approach is ideal for modeling paths, motions, or situations where the traditional function format is inadequate.

In the given exercise, the parametric curve sketched provides insight into the interaction between exponential growth and decay within the same graph. This offers a comprehensive view of their combined behavior, essential for understanding motion or trajectory problems in physics and engineering.