Problem 18 - Calculus Early Transcendentals

Question

Consider the following parametric equations. a. Eliminate the parameter to obtain an equation in \(x\) and \(y.\) b. Describe the curve and indicate the positive orientation. $$x=1-\sin ^{2} s, y=\cos s ; \pi \leq s \leq 2 \pi$$

Step-by-Step Solution

Answer

Answer: The curve represented by the parametric equations is a parabola described by the equation \(x = y^2\). The positive orientation is from the lower arm of the parabola (negative \(y\) values) to the vertex at the origin (0,0).

Step 1: Solve one equation for the parameter s

We can solve the equation \(y = \cos s\): $$ s = \arccos y$$ It's important to note that the range for \(s\) is given by \(\pi \leq s \leq 2\pi\). This will be used to find the positive orientation of the curve later.

Step 2: Substitute the result in the other equation

Now we will substitute the expression for \(s\) obtained from the first equation into the second equation for \(x\): $$ x = 1 - \sin^2(\arccos y) $$

Step 3: Apply trigonometric identity

We can apply the trigonometric identity \(\sin^2(\theta) + \cos^2(\theta) = 1\), where \(\theta = \arccos y\): $$ \sin^2(\arccos y) = 1 - \cos^2(\arccos y) = 1 - y^2 $$ Now substitute this back to the equation: $$ x = 1 - (1 - y^2) = y^2 $$ Thus, we obtained an equation for \(x\) and \(y\) without the parameter \(s\): \(x = y^2\).

Step 4: Describe the curve

The equation \(x = y^2\) describes a parabola with its vertex at the origin (0,0), opening to the right.

Step 5: Determine the positive orientation

To find the positive orientation, we will analyze how the values of \(x\) and \(y\) change with increasing values of \(s\) within the given range \(\pi \leq s \leq 2\pi\). In this range, the sine function is negative, so \(\sin^2(\theta)\) is positive. Since the cosine function is also negative, the value of \(y\) will be also negative. As \(s\) increases from \(\pi\) to \(2\pi\), the value of \(y\) increases from its minimum value to 0. Thus, the positive orientation of the curve is from the lower arm of the parabola (negative \(y\) values) to the vertex at the origin (0,0).

Key Concepts

Eliminating the Parameter

In calculus, eliminating the parameter from parametric equations simplifies the representation of curves by expressing them in terms of only the variables x and y. For the given equations, where \( x=1-\sin^2 s \) and \( y=\cos s \) with \( \pi \leq s \leq 2\pi \) , we first express \( s \) in terms of \( y \) as \( s = \arccos y \) .

Then, we substitute this expression into the x equation and apply the trigonometric identity \( \sin^2(\theta) + \cos^2(\theta) = 1 \) where \( \theta = \arccos y \) to find \( x = y^2 \) .

Through this process, we transition from a parametric form to the standard form \( x = f(y) \) , allowing us to identify the curve as a parabola. Substituting trigonometric identities converts the equation into a more recognizable Cartesian form, which can be graphed on the xy-plane.

Describing Curves in Calculus

Curves in calculus are often described using functions and equations that relate two or more variables. In the context of parametric equations, curves are given as sets of equations that define x and y in terms of a third variable, usually called the parameter.

In this exercise, once we have eliminated the parameter, we obtain the equation \( x = y^2 \) which describes a familiar shape: a parabola. Unlike the standard parabola \( y = x^2 \) that opens upwards, the parabola described by \( x = y^2 \) opens to the right. This is an example of how parametric equations can provide new perspectives on classic curves by allowing for different orientations and parametrizations of these shapes in the plane.

Trigonometric Identities

Trigonometric identities are fundamental tools in calculus, especially when dealing with parametric and polar coordinates. They are equations that provide relations between trigonometric functions, allowing for simplification and manipulation of expressions.

For instance, in the exercise, the identity \( \sin^2(\theta) + \cos^2(\theta) = 1 \) is used to transform the parametric equation \( x = 1 - \sin^2 s \) into a function of y by noting that \( \theta \) is equivalent to \( \arccos y \) . By using this identity, we can rewrite \( \sin^2(\arccos y) \) as \( 1 - y^2 \) , thus successfully removing the parameter and simplifying the expression.

Positive Orientation of Curves

The positive orientation of a curve in the plane refers to the direction in which the curve is traversed as the parameter increases. This concept, also linked with the 'sense' or 'direction' of a curve, is often applied in the context of parametric curves and path integrals.

In the exercise, by considering the given range \( \pi \leq s \leq 2\pi \) and observing the behavior of the trigonometric functions within this range, it is determined that the positive orientation is from the bottom of the parabola (where y is negative) up towards the vertex at the origin, as s increases. Understanding the positive orientation is crucial for correctly interpreting the motion along the curve and has implications in fields such as physics and engineering, where the direction of travel can affect the outcome of a problem.