Problem 44 - Thomas Calculus

Question

The curve with parametric equations $$ x=t, \quad y=1-\cos t, \quad 0 \leq t \leq 2 \pi $$ is called a sinusoid and is shown in the accompanying figure. Find the point \((x, y)\) where the slope of the tangent line is a. largest. b. smallest.

Step-by-Step Solution

Answer

The slope is largest at \((\frac{\pi}{2}, 1)\) and smallest at \((\frac{3\pi}{2}, 1)\).

Step 1: Derive the Parametric Derivatives

To find the slope of the tangent line, first derive the parametric equations \(x(t) = t\) and \(y(t) = 1 - \cos t\). The derivatives with respect to \(t\) are \(x'(t) = 1\) and \(y'(t) = \sin t\).

Step 2: Find the Slope of the Tangent Line

The slope of the tangent line \( \frac{dy}{dx} \) for parameter \( t \) is given by \( \frac{dy/dt}{dx/dt} = \frac{y'(t)}{x'(t)} = \sin t \).

Step 3: Determine When the Slope is Largest and Smallest

The sine function, \( \sin t \), attains its largest value of 1 at \( t = \frac{\pi}{2} \) and its smallest value of -1 at \( t = \frac{3\pi}{2} \) within the interval \( [0, 2\pi] \).

Step 4: Find the Corresponding Points

For \( t = \frac{\pi}{2} \), the point \( (x, y) \) is \( \left( \frac{\pi}{2}, 1 - \cos \frac{\pi}{2} \right) = \left( \frac{\pi}{2}, 1 \right) \). For \( t = \frac{3\pi}{2} \), the point \( (x, y) \) is \( \left( \frac{3\pi}{2}, 1 - \cos \frac{3\pi}{2} \right) = \left( \frac{3\pi}{2}, 1 \right) \).

Key Concepts

Understanding the Slope of Tangent Line in Parametric Equations

In the context of parametric equations, the slope of a tangent line provides essential insight into the curve's behavior at any given point. A tangent line is a straight line that 'touches' the curve at a single point without crossing over. The slope indicates how steep or flat the tangent line is.
Generally, the slope of the tangent line for a parametric equation is given by the formula \( \frac{dy/dt}{dx/dt} \). Here, \( dx/dt \) and \( dy/dt \) are the derivatives of the parametric equations with respect to the parameter \( t \). In our exercise, for example, \( x(t) = t \) and \( y(t) = 1 - \cos t \), which lead to \( x'(t) = 1 \) and \( y'(t) = \sin t \) respectively.
This implies that the slope \( \frac{dy}{dx} = \frac{\sin t}{1} = \sin t \). Understanding this allows us to determine how the curve rises or falls at different points as described by the parameter \( t \). The slope is positive if \( \sin t \) is positive, indicating an upward direction, and negative when \( \sin t \) is negative, showing a downward trend.

The Sinusoid: A Special Parametric Curve

A sinusoid refers to a curve that describes a smooth, wave-like motion. In mathematics, sinusoids are closely linked with sine and cosine functions. In our given exercise, the curve represented by the parametric equations forms a sinusoid.
The equations given are \( x = t \) and \( y = 1 - \cos t \), where \( t \) ranges from \( 0 \) to \( 2\pi \). The term "sinusoid" not only indicates the periodic nature of the curve but also its distinctive oscillation pattern.

• As \( t \) progresses from \( 0 \) to \( 2\pi \), the \( y \)-value oscillates due to the influence of the cosine function.
• The sinusoidal motion arises because \( \cos t \) brings about the up and down movement while the parameter \( t \) keeps moving horizontally.
• It is crucial for students to understand that the sinusoid's shape depends on the trigonometric functions involved and the range of the parameter \( t \).

Exploring Parametric Derivatives

Parametric derivatives are the mathematical expressions that enable us to compute rates of change in parametric equations. Unlike traditional functions, parametric functions rely on a parameter, often \( t \), that determines the positions along the \( x \) and \( y \) axes.
In the context of our exercise, the parametric derivatives \( x'(t) \) and \( y'(t) \) are crucial for various calculations, such as determining angles, tangents, and other geometrical features of the curve. By deriving \( x'(t) = 1 \) and \( y'(t) = \sin t \) from the original equations, we can conclude different characteristics of the curve:

• \( x'(t) = 1 \) indicates that, for every change in \( t \), the \( x \)-coordinate changes steadily without regard to \( t \). This constancy emphasizes a linear relation in the horizontal direction.
• \( y'(t) = \sin t \) shows the effect of the trigonometric sine function in altering the \( y \)-coordinate, producing the wave-like shape common to the sinusoid.

Accurate comprehension and execution of parametric derivatives not only solve problems related to tangent slopes but also enhance understanding of the deeper dynamics of parametric plots.