Problem 71 - Calculus Early Transcendentals: Pearson New International Edition

Question

Plot the following parametric curves. Describe in words how the point moves around the curve in each case. (a) \(x=\cos \left(t^{2}-t\right), y=\sin \left(t^{2}-t\right)\) (b) \(x=\cos \left(2 t^{2}+3 t+1\right), y=\sin \left(2 t^{2}+3 t+1\right)\) (c) \(x=\cos (-2 \ln t), y=\sin (-2 \ln t)\) (d) \(x=\cos (\sin t), y=\sin (\sin t)\)

Step-by-Step Solution

Answer

The plots (a)-(d) describe circular or oscillatory paths with varying speeds and complexities due to their parametric forms.

Step 1: Understanding Parametric Curves

Parametric equations define a curve by expressing coordinates of points on the curve as functions of a parameter, in this case, 't'. In each part of the exercise, we will plot the curve by calculating x and y values for different 't' values, and then describe how these values guide the point on the curve.

Step 2: Plot Curve (a)

For (a), we have the parametric equations:- \(x = \cos(t^2 - t)\)- \(y = \sin(t^2 - t)\).To plot this, we compute x and y for several values of t. As t changes, observe how the point moves in a circular path influenced by the quadratic expression \(t^2 - t\). The motion is not uniform due to the nonlinear term \(t^2 - t\). As t increases, the rate at which the parameter grows increases, making the movement faster along the path.

Step 3: Plot Curve (b)

For (b), the equations given are:- \(x = \cos(2t^2 + 3t + 1)\)- \(y = \sin(2t^2 + 3t + 1)\).Here, as t increases, x and y values describe a motion around a unit circle, with more complex circular behavior due to the polynomial \(2t^2 + 3t + 1\). This implies a variable speed, with tight oscillations as values of t affect angular frequency of the point on the curve.

Step 4: Plot Curve (c)

For (c), the expressions are:- \(x = \cos(-2 \ln t)\)- \(y = \sin(-2 \ln t)\).This uses a logarithmic function, meaning the parameter t must be greater than zero. As t increases, the logarithm lowers the rate of change of angle, resulting in a slower circular motion of the point around the origin, while for smaller t, the motion is faster due to the exponential nature of the logarithmic function.

Step 5: Plot Curve (d)

For (d), the equations are:- \(x = \cos(\sin t)\)- \(y = \sin(\sin t)\).This involves the sine function inside the cosine and sine, producing a very smooth and slow oscillation of points on the unit circle. The point traces an intricate pattern due to the nested trigonometric function, appearing to fluctuate back and forth slightly along the circular path.

Key Concepts

Trigonometric Functions

Trigonometric functions like sine and cosine are fundamental in math, helping us describe and analyze angles in right triangles. They are essential in parametric equations because they allow us to model circular paths.
Both functions have a range of values between -1 and 1, which perfectly suits their role in parametric curves, especially when modeling movement along a circle. For instance, in this exercise, parameters involve \cos and \sin which create circular paths.
Let's consider one of our equations: \(x = \cos(t^2 - t)\) and \(y = \sin(t^2 - t)\). As \(t\) changes, these functions ensure that the point moves in a circular path around the origin.

• \(\cos(math expr)\): Dictates the horizontal position of the point.
• \(\sin(math expr)\): Determines the vertical position on the plane.

This constant loop from sine and cosine creates the circular motion we're graphing, giving us a predictable yet fascinating pattern on the graph.

Graphing Techniques

Graphing parametric curves involves understanding how the parameters affect the graph's shape and movement. What's unique about parametric equations is that they use an extra variable, usually \(t\), to express curves.
To graph these curves, you'd start by selecting a range of \(t\) values. Plug these values into your \(x\) and \(y\) equations to compute points on the graph. For example, let's take \(x = \cos(2t^2 + 3t + 1)\) and \(y = \sin(2t^2 + 3t + 1)\). Compute these for different \(t\) values to get multiple \((x, y)\) points.
Once you have calculated these points, plot them in a coordinate plane.

• Join these points smoothly to see the curve.
• Observe how changes in \(t\) influence the path.

By looking at how \(t\) modifies the phase and frequency, you can predict how the curve will behave, whether it accelerates, decelerates, or fluctuates smoothly across the plane.

Parametric Equations

Parametric equations differ from regular equations because they express both \(x\) and \(y\) in terms of another variable, often a parameter \(t\). These equations enable us to describe a movement over time.
The beauty of parametric equations lies in their flexibility to design complex curves that are hard to represent otherwise. Take, for instance, \(x = \cos(-2 \ln t)\) and \(y = \sin(-2 \ln t)\). Here, \(t\) is a parameter that helps dictate this curve's shape.
In parametric form:

• \(t\) can stand for various factors – such as time in motion or angles in geometry.
• Expressions like \(-2 \ln t\) aid to adjust the rate or scale of the curve.

These equations make it manageable to manipulate curves by tweaking expressions of \(t\), offering intuitive control over the graphical representation of dynamic systems.