Question

Determine whether the following statements are true and give an explanation or counterexample. a. The equations \(x=-\cos t, y=-\sin t,\) for \(0 \leq t \leq 2 \pi\) generate a circle in the clockwise direction. b. An object following the parametric curve \(x=2 \cos 2 \pi t\) \(y=2 \sin 2 \pi t\) circles the origin once every 1 time unit. c. The parametric equations \(x=t, y=t^{2},\) for \(t \geq 0,\) describe the complete parabola \(y=x^{2}\) d. The parametric equations \(x=\cos t, y=\sin t,\) for \(-\pi / 2 \leq t \leq \pi / 2,\) describe a semicircle. e. There are two points on the curve \(x=-4 \cos t, y=\sin t,\) for \(0 \leq t \leq 2 \pi,\) at which there is a vertical tangent line.

Short answer

In summary, the statements for the given parametric equations are: a) True - The parametric equations describe a circle of radius 1 centered at the origin, traveling in the clockwise direction. b) True - The object circles the origin once every 1 time unit, describing a circle of radius 2 centered at the origin. c) False - The given parametric equations only describe the right side of the parabola y=x^2, not the complete parabola. d) True - The parametric equations describe a semicircle with radius 1 centered at the origin. e) True - The given curve has two points where there is a vertical tangent line, at points (4,0) and (-4,0).

Step-by-step solution

Statement a

We'll analyze the given parametric equations (PEQ) \(x=-\cos t, y=-\sin t,\) for \(0\leq t\leq 2\pi\). Rewrite the PEQ in polar coordinates: \(r = \sqrt{(-\cos(t))^2 + (-\sin(t))^2} = \sqrt{\cos^2(t) + \sin^2(t)} = 1\). So, in fact, the PEQ describe a circle of radius 1 centered at the origin. Also, since the cosine is negated, the parameter \(t\) goes in the clockwise direction. Thus, the statement is true.

Statement b

For the provided parametric curve \(x=2\cos (2\pi t), y=2\sin (2\pi t)\), let's first find the period of this curve. Since the period of \(\cos(2\pi t)\) and \(\sin(2\pi t)\) is 1, it takes 1 time unit for the object to travel along this curve once. Moreover, the PEQ are describing a circle (\(r=2\), centered at the origin) in the counterclockwise direction, so the object does circle the origin once every 1 time unit. The statement is true.

Statement c

Consider the parametric equations \(x=t, y=t^2,\) for \(t\geq 0\). These parametric equations describe the relation \(y=x^2\). However, they only describe the right side of the parabola since \(t\geq 0\). The complete parabola also has a left side (\(x <0\)) that is not covered by the given parametric equations. Hence, the statement is false.

Statement d

The given parametric equations are \(x=\cos t, y=\sin t,\) with \(-\pi/2\leq t \leq \pi/2\). These form a circle with radius 1 centered at the origin, traveling in the counterclockwise direction. Since the parameter interval ranges from \(-\pi/2\) to \(\pi/2\), it only covers half of the circle, effectively describing a semicircle. The statement is true.

Statement e

For the given curve \(x=-4\cos t, y=\sin t,\) for \(0 \leq t\leq 2\pi\), let's find the derivative \(dy/dx\) (given \(dx/dt=-4\sin t\), \(dy/dt=\cos t\)) to find the points where there is a vertical tangent line (when \(dx/dt = 0\)). Solving \(-4\sin t=0\), we obtain \(t=0, \pi\). These values of \(t\) correspond to the points \((4,0)\) and \((-4,0)\). Indeed, this curve has two points where there is a vertical tangent line. The statement is true.

Key concepts

Circle Parametric Equation

Understanding the concept of circle parametric equations is crucial for visualizing and describing circular paths in a two-dimensional plane. A circle with a radius of 'r' centered at the origin can be represented by the equations:
\( x = r\cos(t) \) and \( y = r\sin(t) \), where 't' varies from 0 to \(2\pi\).

For example, the solution statement a confirmed the circle's clockwise direction due to the negative sign, meaning the parametric equations describe a circle where the value of 't' increases as the path follows in a clockwise. This contrasts with the more common counterclockwise direction for positive signs in the equations. Understanding how the sign and range of 't' affect direction and portion of the circle covered is a fundamental concept in parametric equations.

Parametric Curve Period

The period of a parametric curve determines how long it takes for a point traveling along the curve to complete one full cycle. This concept is particularly important when the curve represents periodic motion, such as an object following a circular path. The equations \( x = a\cos(bt) \) and \( y = a\sin(bt) \) typically describe a circle where 'b' affects the period.

For instance, as seen in solution statement b, when 'b' is \(2\pi\), the period is 1. This confirmed that the object indeed circles the origin once every time unit. The period is crucial for understanding the timing of repetitive patterns and motion in parametric curves.

Complete Parabola

A complete parabola is a quadratic curve that continues infinitely in both directions along the x-axis. Its standard form is \( y = ax^2 + bx + c \). Parametric equations can describe a parabola, but it's important to note what part of the parabola the equations represent.

In solution statement c, we learned that the parametric equations \( x=t, y=t^2 \) describe only the right arm of the parabola when \( t \geq 0 \), since for negative 't', the square of 't' is still positive, which does not reflect the left side of the parabola where 'x' is negative. Recognizing that parametric equations might describe only a portion of a complete curve is essential for accurate interpretation.

Semicircle Parametric Description

A semicircle parametric description involves defining the region of a circle that spans exactly half of its circumference. The key lies in the interval of the parameter 't'. For a full circle centered at the origin, we usually have \(t\) going from 0 to \(2\pi\), but for a semicircle, as in solution statement d, \(t\) runs from \( -\frac{\pi}{2} \) to \( \frac{\pi}{2} \), which covers half of the circle's path, specifically the upper semicircle. Parametric equations can also describe the lower semicircle by adjusting the interval of 't'. Understanding these intervals allows for precise descriptions of various curves and shapes.

Vertical Tangent Line

The concept of a vertical tangent line arises when the slope of a curve reaches infinity, which occurs at points on the curve where \( \frac{dy}{dx} \) is undefined. To find these points using parametric equations, one differentiates 'y' with respect to 't' and 'x' with respect to 't', and then looks for where \( \frac{dx}{dt} = 0 \) (since the slope is given by \( \frac{dy/dt}{dx/dt} \)).

As shown in solution statement e, by finding the 't' that satisfies \( -4\sin(t)=0 \) we identified the points where the curve \(x=-4\cos(t), y=\sin(t)\) has vertical tangents. This is a powerful technique for analyzing the behavior of curves, especially in physics and engineering, where such points might represent important physical phenomena or constraints.