Problem 29 - Calculus 3

Question

Use a graphing utility to sketch each of the following vector-valued functions: $$ \mathbf{r}(t)=\langle 2-\sin (2 t), 3+2 \cos t\rangle $$

Step-by-Step Solution

Answer

Plot the curve for \( t \) in \( [0, 2\pi] \) using a graphing utility.

Step 1: Understand the Vector-Valued Function

The vector-valued function is given by \( \mathbf{r}(t) = \langle 2-\sin(2t), 3+2\cos(t)\rangle \). This defines a set of points in the two-dimensional plane parameterized by the variable \( t \). The first component, \( 2 - \sin(2t) \), determines the x-coordinate, and the second component, \( 3 + 2\cos(t) \), determines the y-coordinate of the points.

Step 2: Determine the Range of t

For a complete visualization of common trigonometric functions, we typically use \( t \) in the range \( [0, 2\pi] \). This allows us to capture one full cycle of both the sine and cosine functions, which are periodic with periods \( 2\pi \) and \( 2\pi \), respectively.

Step 3: Sketch the Parametric Curve

Use a graphing utility (such as Desmos, GeoGebra, or a graphing calculator) to plot the curve. Set up the graph by inputting the parametric equations for \( x(t) = 2 - \sin(2t) \) and \( y(t) = 3 + 2\cos(t) \), letting \( t \) vary from \( 0 \) to \( 2\pi \). Observe how the curve behaves as \( t \) changes within this interval.

Step 4: Analyze the Shape of the Graph

Once the graph is plotted, visually analyze the shape formed by the curve in the plane. Notice any symmetries or particular features, such as where the curve starts and ends, high points, low points, and how the curve moves.

Key Concepts

Graphing Utilities

Graphing utilities are tools that make it easy to visualize mathematical functions and equations. They are particularly handy when dealing with complex functions like vector-valued or parametric equations. Some popular graphing utilities include:

• Desmos
• GeoGebra
• Graphing calculators

These utilities allow users to input equations and see the resulting graph immediately. For vector-valued functions, which contain both x and y components, graphing utilities let us observe how the outputs of these components change with the parameter, often called time \(t\). This provides insight into the behavior of the function over a specified range.When using graphing utilities:

• Make sure to correctly input the parametric equations.
• Adjust the range of the parameter \(t\) to capture the entire behavior of the function.
• Use features to identify points of interest such as intersections, peaks, or symmetry.

Parametric Equations

Parametric equations allow us to express a set of related quantities as explicit functions of a common parameter, often denoted as \(t\). These equations are versatile in representing various shapes and motions that cannot easily be described with standard functions. In the given vector-valued function \(\mathbf{r}(t) = \langle 2 - \sin(2t), 3 + 2\cos(t)\rangle\):

• The x-coordinate is given by \(x(t) = 2 - \sin(2t)\).
• The y-coordinate is determined by \(y(t) = 3 + 2\cos(t)\).

By varying the parameter \(t\), we can generate a set of points \((x(t), y(t))\) that describe a curve in the Cartesian plane. For our sketch, we use \(t\) in the range \([0, 2\pi]\) to fully capture the behavior of the trigonometric functions involved.Parametric equations are excellent for modeling motions, animations, and paths in physical systems, as they naturally express the dependence of two different quantities on a third one.

Trigonometric Functions

Trigonometric functions help in describing oscillatory movements or rotations and are a fundamental part of parametric equations. They include functions like sine \(\sin\), cosine \(\cos\), and more. Understanding their properties is essential when working with parametric representations that involve trigonometric expressions. In the function \(\mathbf{r}(t) = \langle 2 - \sin(2t), 3 + 2\cos(t)\rangle\):

• \(\sin(2t)\) contributes to the x-coordinate, creating variation in its position along the x-axis.
• \(\cos(t)\) alters the y-coordinate, influencing the curve's height on the y-axis.

Both functions \(\sin(2t)\) and \(\cos(t)\) are periodic with the period \(2\pi\), meaning they repeat their values in regular intervals. When sketching their combined effect in parametric equations:

• Note the amplitude, affecting how far the points reach in each direction.
• Keep an eye out for symmetry which may simplify understanding the entire curve.

Curve Sketching

Curve sketching is the process of drawing the graph of a function based on its algebraic expression, behavior, and analysis. It involves understanding how the function behaves as the parameter \( t \) changes.To effectively sketch vector-valued functions like \(\mathbf{r}(t) = \langle 2 - \sin(2t), 3 + 2\cos(t)\rangle\):

• Start by identifying key points by computing values at specific \(t\).
• Use a graphing utility to plot several points for better accuracy.
• Examine the overall layout and features of the curve, like peaks and valleys.

Align your focus on the sections where the curve changes direction, which are critical in understanding its complete shape. The transition behavior at points of intersection or symmetry often highlights the function's nature as well.Mastering curve sketching involves practice with different function forms and understanding how components like trigonometric expressions combine to form complete paths on the graph.