Question

The ezplot command will also handle parametric equations. Try ezplot \(\left({ }^{\prime} \cos (t)^{\prime}, ' \sin (\mathrm{t})^{\prime},[0,2 * \mathrm{pi}]\right)\) to get a feel for how the command works. In Exercises \(11-14\), use the ezplot command to plot the parametric equations over the indicated time interval. \(x=\cos (t / 2)+2 \sin (t / 2), y(t)=\sin (t / 2),[0,4 \pi]\)

Short answer

Use `ezplot('cos(t/2) + 2*sin(t/2)', 'sin(t/2)', [0, 4*pi])` in MATLAB to plot the equations.

Step-by-step solution

Understanding the Problem

We are given a pair of parametric equations: \(x = \cos\left(\frac{t}{2}\right) + 2\sin\left(\frac{t}{2}\right)\) and \(y = \sin\left(\frac{t}{2}\right)\). We need to plot these equations using the `ezplot` command over the interval \([0, 4\pi]\).

Reviewing the ezplot Syntax

The ezplot command is used in MATLAB for plotting 2D parametric equations. The syntax is `ezplot('expressionX', 'expressionY', [t_min, t_max])`, where `expressionX` and `expressionY` are the parametric equations for x and y, and `[t_min, t_max]` is the interval for the parameter t.

Setting Up the ezplot Command

Substitute the given equations into the `ezplot` syntax. Use `ezplot('cos(t/2) + 2*sin(t/2)', 'sin(t/2)', [0, 4*pi])` for the MATLAB command. This sets our parameter t over the range from 0 to \(4\pi\).

Executing the Command and Analyzing the Plot

Run the ezplot command in MATLAB. Observe the plot generated, which visually represents the path described by the parametric equations over the specified interval.

Key concepts

MATLAB ezplot command

The `ezplot` command in MATLAB is a versatile tool used for plotting parametric equations in two dimensions. It simplifies the plotting process by only requiring the mathematical expressions for the coordinates and the interval for the parameter. To use this command, you don't need to manually compute plot points; instead, `ezplot` handles this internally.
Its basic syntax is `ezplot('expressionX', 'expressionY', [t_min, t_max])`. Here:

• `expressionX` represents the equation for the x-coordinate as a function of the parameter `t`.
• `expressionY` represents the equation for the y-coordinate as a function of `t`.
• [t_min, t_max] specifies the range of the parameter `t`, typically corresponding to time or another independent variable.

By using `ezplot`, you can quickly visualize how mathematical models translate into paths or curves, which is particularly powerful for understanding complex relationships between variables.

plotting parametric equations

Parametric equations express a set of related quantities as explicit functions of an independent variable known as a parameter, often denoted as `t`. Unlike standard equations, which define `y` directly as a function of `x`, parametric equations define both `x` and `y` separately in terms of `t`:

• Example: Given equations \(x = \cos\left(\frac{t}{2}\right) + 2\sin\left(\frac{t}{2}\right)\) and \(y = \sin\left(\frac{t}{2}\right)\).
• The resulting plot is a curve, which is traced out as `t` varies over a specified interval.

To plot these equations using MATLAB's `ezplot` command, simply substitute them into the syntax and execute. This allows you to see the continuous path or trajectory that the parametric equations represent, providing you with insights into the behavior of dynamic systems or geometric properties of curves.

interpreting mathematical plots

Interpreting plots generated from parametric equations involves understanding the visual representation of a mathematical relationship. Here's how you can go about interpreting these plots effectively:

• **Identify the Shape:** Observe the overall shape of the curve. It might be symmetrical, spiraled, or have unique features characteristic of the given parametric equations.
• **Directionality and Path:** Determine how the parameter `t` affects the traversal of the curve. Sometimes it helps to visualize the motion by thinking of `t` as 'time.'
• **Key Points and Features:** Look for intercepts, cusps, loops, or other significant points. These features often correspond to specific values of `t` and can give additional insights into the nature of the functions.

By analyzing these plots, you can gain a deeper understanding of complex mathematical concepts and the function's behavior within the defined parameter space. This deeper understanding facilitates more advanced interpretations and applications of the concepts, such as modeling real-world phenomena.