Problem 75 - Calculus Early Transcendentals: Pearson New International Edition

Question

Draw the Folium of Descartes \(x=3 t /\left(t^{3}+1\right)\), \(y=3 t^{2} /\left(t^{3}+1\right)\). Then determine the values of \(t\) for which this graph is in each of the four quadrants.

Step-by-Step Solution

Answer

First quadrant: \(t > 0\); No perfect match for others due to symmetry.

Step 1: Introduction to the Folium of Descartes

The Folium of Descartes is a curve defined by the parametric equations \(x = \frac{3t}{t^3 + 1}\) and \(y = \frac{3t^2}{t^3 + 1}\). Our task is to plot the curve and find the values of \(t\) for which the curve lies in each of the four quadrants of the Cartesian plane.

Step 1: Understanding the Parametric Equations

The given equations are parametric, where \(t\) is the parameter. As \(t\) varies from \(-\infty\) to \(\infty\), it traces out points on the curve in the Cartesian plane. The equations express \(x\) and \(y\) as functions of \(t\).

Step 2: Analyzing the First Quadrant (x > 0, y > 0)

For both \(x > 0\) and \(y > 0\), \(t\) must be positive. This is because both numerators \(3t\) and \(3t^2\) are positive when \(t > 0\). Thus, for the curve to be in the first quadrant, \(t > 0\).

Step 3: Analyzing the Second Quadrant (x 0)

In the second quadrant, \(x < 0\) implies \(t < 0\) since \(3t < 0\) while \(y > 0\) requires \(t^2 > 0\). Therefore, values of \(t\) must satisfy \(t < 0\), i.e., when \(t\) is negative.

Step 4: Analyzing the Third Quadrant (x < 0, y < 0)

For the third quadrant (\(x < 0\) and \(y < 0\)), since both \(x = \frac{3t}{t^3 + 1}\) and \(y = \frac{3t^2}{t^3 + 1}\) require \(3t < 0\) and \(3t^2 < 0\), \(t\) cannot be negative. This implies there can be a swap like \((x,y)\to (-x,-y)\), looking for changes in dominance over quadrants by ratio.

Step 5: Analyzing the Fourth Quadrant (x > 0, y < 0)

For the fourth quadrant, \(x > 0\) requires \(t > 0\), but \(y < 0\) implies \(t^2 < 0\). As \(t^2\) must always be non-negative, consider points towards \(y\approx 0\).

Step 7: Drawing and Cross Checking

Plot the curve parametrically to observe changes and parity between the conditions. The function's radial symmetry implies a cross-interaction where no clean section exists for the 3rd and 4th quadrants under these symbols.

Key Concepts

Parametric Equations

Parametric equations are a set of equations where the coordinates of the points that make up a geometric object, such as a curve, are expressed as functions of a parameter. In the case of the Folium of Descartes, the parametric equations are given by \(x = \frac{3t}{t^3 + 1}\) and \(y = \frac{3t^2}{t^3 + 1}\).

This parameter, denoted by \(t\), varies over real numbers, causing the expressions for \(x\) and \(y\) to define continuous points on the curve in the Cartesian plane. Parametric equations are useful because they can describe complex curves more easily than simple algebraic equations.

Advantages of using parametric equations include:

• They allow for representation of curves where \(x\) is not a single-valued function of \(y\), or vice versa.
• They are particularly useful in modeling motion and paths because they provide a natural way of representing time-dependent trajectories.
• They help in plotting shapes and patterns easily as variations in the parameter \(t\) can directly sketch the trajectory or form.

Understanding the behavior of the functions as \(t\) moves through its range is crucial for predicting the shape and path of the curve.

Cartesian Plane

The Cartesian plane is a two-dimensional surface defined by a horizontal line called the x-axis and a vertical line called the y-axis. These axes divide the plane into four quadrants, each representing a different sign combination of \((x, y)\).

In the context of the Folium of Descartes, the Cartesian plane serves as the canvas where the curve defined by the parametric equations is drawn. The interplay between the parametric variables \(x\) and \(y\) with the quadrants is important for understanding how the curve behaves in different regions of the plane.

Key features of the Cartesian plane include:

• It provides a straightforward way of plotting points and analyzing geometric shapes or functions.
• Quadrant orientation helps in determining the sign and relative position of points, where:
  • First quadrant: \(x > 0, y > 0\)
  • Second quadrant: \(x < 0, y > 0\)
  • Third quadrant: \(x < 0, y < 0\)
  • Fourth quadrant: \(x > 0, y < 0\)

Using the Cartesian plane, one can interactively visualize and comprehend the effects of transformations or changes in the curves that are determined by parametric or algebraic relationships.

Quadrants Analysis

Quadrants analysis involves examining the behavior of a curve in the different quadrants of the Cartesian plane to understand its distribution and transitions. For the Folium of Descartes, this analysis reveals which values of \(t\) correspond to the curve residing in each quadrant.

In the first quadrant (\(x > 0, y > 0\)), both \(3t\) and \(3t^2\) are positive, requiring that \(t > 0\). This means that when \(t\) is positive, the curve will be in this quadrant.

For the second quadrant (\(x < 0, y > 0\)), \(3t\) must be negative leading to \(t < 0\), but since \(3t^2\) is always non-negative, \(t\) must be negative to achieve \(x < 0\).

The third quadrant (\(x < 0, y < 0\)) in this scenario is challenging since \(t^2\) cannot be negative. Here, practical observations or transformations can highlight symmetry or changes not initially evident.

In the fourth quadrant (\(x > 0, y < 0\)), there is difficulty in obtaining \(y < 0\) because \(t^2\) results in non-negative values. Thus the presence of Folium twists and potential symmetry could mean absences in certain quadrants under radial symmetry or transformation shifts.

This analysis helps in interpreting how parametric equations translate geometric forms into real-world applications, assisting students in building a strong foundation in mathematical reasoning.