Question

Determine whether the following statements are true and give an explanation or counterexample. a. The equations $x=-\cos t,y=-\sin t,$ for $0\le t\le 2\pi ,$ 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 $t\ge 0,$ describe the complete parabola $y=x^2$ d. The parametric equations $x=\cos t,y=\sin t,$ for $-\pi /2\le t\le \pi /2,$ describe a semicircle. e. There are two points on the curve $x=-4\cos t,y=\sin t,$ for $0\le t\le 2\pi ,$ at which there is a vertical tangent line.

Short answer

Question: Determine which of the following statements are true or false and provide an explanation or a counterexample. a) The parametric equations $x=-\cos t$ and $y=-\sin t$ for $0\le t\le 2\pi$ generate a circle traced 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 $t\ge 0$ describe the complete parabola $y=x^2$. d) The parametric equations $x=\cos t$, $y=\sin t$ for $-\pi /2\le t\le \pi /2$ describe a semicircle. e) The parametric equations $x=-4\cos t$, $y=\sin t$ for $0\le t\le 2\pi$ have two distinct points where there is a vertical tangent line within the given interval for t. Answer: a) True b) True c) False d) True e) True

Step-by-step solution

Statement a

The given parametric equations are $x=-\cos t$ and $y=-\sin t$. We can write the equation for a circle in a Cartesian coordinate system as $x^2+y^2=r^2$, where r is the radius of the circle. Using the parametric equations: $(-\cos t{)}^2+(-\sin t{)}^2={\cos }^{2}t+{\sin }^{2}t=1$. This equation corresponds to a circle with radius 1. To check the direction, let's analyze the signs of the derivatives of x and y with respect to t: $\frac{dx}{dt}=\sin t$ and $\frac{dy}{dt}=-\cos t$. In the given interval $0\le t\le 2\pi$, $\frac{dx}{dt}\ge 0$ and $\frac{dy}{dt}\le 0$. This means that the circle is traced in a clockwise direction. Hence the statement is true.

Statement b

The given parametric curve is $x=2\cos (2\pi t)$, $y=2\sin (2\pi t)$. To check the period of the motion and the number of times around the origin, let's find when $x$ and $y$ get back to their original values. The period of cosine and sine functions is $2\pi$ but the argument is $2\pi t$, so the period of the motion is given by $2\pi t=2\pi ⟹t=1$. Therefore, an object following this parametric curve circles the origin once every 1 time unit. The statement is true.

Statement c

The parametric equations are $x=t$, $y=t^2$, and we want to determine if they describe the complete parabola $y=x^2$ for $t\ge 0$. Replacing $t$ with $x$ in the second equation: $y=x^2$. The equation describes the parabola $y=x^2$ for $x\ge 0$, meaning only half of the parabola. The statement is false.

Statement d

The parametric equations are $x=\cos t$, $y=\sin t$ for $-\pi /2\le t\le \pi /2$. The equation in cartesian coordinates is $x^2+y^2=1$, which represents a circle with radius 1. Since $-\pi /2\le t\le \pi /2$, the parametric representation covers exactly the upper half of the circle, representing a semicircle. The statement is true.

Statement e

The parametric equations are $x=-4\cos t$, $y=\sin t$ for $0\le t\le 2\pi$. First, we determine the points where the tangent is vertical. A vertical tangent occurs when the derivative of $x$ with respect to $t$ is zero. So, $\frac{dx}{dt}=4\sin t=0⟹t=0,\pi ,2\pi$. For $t=0$ and $t=2\pi$, we get the same point $(-4,0)$. For $t=\pi$, we get $(4,0)$. So, there are indeed two distinct points at which there is a vertical tangent line. The statement is true.

Key concepts

Parametric Curve Analysis

Parametric equations provide a flexible method of describing curves that would be difficult or impossible to represent with standard Cartesian equations. In parametric curve analysis, we break down complex shapes into a set of equations that utilize a third variable, often denoted as t, which is a parameter representing the progression along the curve.

For example, the equation of a circle can be expressed parametrically as x = r cos(t) and y = r sin(t), where r is the radius and t varies from 0 to 2π. The parameter t in these equations corresponds to the angle subtended by the radius with the positive x-axis, essentially providing a means to trace the circle's edge as t changes.

When analyzing a parametric curve, it is crucial to consider not only the shape but also the orientation and behavior as the parameter changes. Derivatives with respect to t give us information on the curve's velocity and acceleration, as well as the slope of the tangent line at any point.

Trigonometric Parametric Equations

Trigonometric functions such as sine and cosine are frequently used to define parametric equations because they offer a convenient way to model periodic or circular motion. Trigonometric parametric equations usually involve t as an angle, directly linking the curve's geometric properties with the trigonometric circle.

For instance, the parametric curve described by x = A cos(ωt + φ) and y = A sin(ωt + φ), where A is the amplitude, ω is the angular frequency, and φ is the phase shift, creates an ellipse or a circle depending on the values of these constants. The period of such trigonometric equations is determined by the coefficient of t within the cosine and sine functions, which in the example given by the exercise is 2π.

Understanding trigonometric parametric equations allows us to visualize motion and patterns that repeat over regular intervals, a concept that is pivotal in fields ranging from physics to engineering.

Tangent Lines in Parametric Curves

The slope of tangent lines to parametric curves at any given point can be determined by calculating the derivatives of the x and y parametric equations with respect to t. The slope of the tangent line, m, is then given by the ratio of these derivatives: m = dy/dt / dx/dt.

A vertical tangent line, for instance, occurs where the denominator dx/dt is zero, as long as dy/dt is not zero simultaneously. In the exercise, finding where dx/dt equals zero provided the locations where the curve had vertical tangents. Conversely, a horizontal tangent occurs where dy/dt is zero, provided dx/dt is not zero. These concepts are used extensively in calculus to describe motion and changes in direction, and they allow for a complete analysis of the behavior of parametric curves.