Question

The pairs of parametric equations represent lines, parabolas, circles, ellipses, or hyperbolas. Name the type of basic curve that each pair of equations represents. $$ \begin{aligned} &x=3 \cosh (4 t) \\ &y=4 \sinh (4 t) \end{aligned} $$

Short answer

The given parametric equations represent a hyperbola.

Step-by-step solution

Identify the given parametric equations

The given parametric equations are:\[ x = 3 \cosh(4t) \quad \text{and} \quad y = 4 \sinh(4t) \] where \(\cosh\) is the hyperbolic cosine function and \(\sinh\) is the hyperbolic sine function.

Recall the identity of hyperbolic functions

The identity for hyperbolic functions states: \(\cosh^2(u) - \sinh^2(u) = 1\). We will use this identity to find the relationship between \(x\) and \(y\).

Express the hyperbola equation using hyperbolic identities

Substituting the parametric equations into the hyperbolic identity, we have:\[ \left(\frac{x}{3}\right)^2 - \left(\frac{y}{4}\right)^2 = \cosh^2(4t) - \sinh^2(4t) = 1 \]This is the standard equation form of a hyperbola.

Recognize the type of curve

From step 3, we simplified the identity to the equation of a hyperbola. This confirms that the given parametric equations represent a hyperbola.

Key concepts

Understanding Hyperbolic Functions

Hyperbolic functions are similar to the well-known trigonometric functions but they apply to hyperbolas rather than circles. The two basic hyperbolic functions are the hyperbolic sine ( anh{}) and hyperbolic cosine ( cosh{}). They are defined using exponential functions as follows:

\(\sinh(t) = \frac{e^t - e^{-t}}{2}\) and \(\cosh(t) = \frac{e^t + e^{-t}}{2}\).

These functions have unique properties that make them indispensable in calculus and mathematical modeling. Just like trigonometric functions, hyperbolic functions fulfill certain identities, such as:

• \(\cosh^2(t) - \sinh^2(t) = 1\)
• \(\sinh(2t) = 2\sinh(t)\cosh(t)\)

These identities are analogous to the trigonometric identity for circles and are particularly useful for transforming parametric equations into standard forms.

The Nature of Hyperbolas

A hyperbola is one of the basic conic sections, which can be considered as a pair of symmetrical open curves. Unlike circles or ellipses, hyperbolas are defined by a subtraction relation between two squares.

The typical equation format for a hyperbola is given by:

\(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\).

Here, \(a\) and \(b\) are real numbers that represent the semi-major and semi-minor axes of the hyperbola. In terms of geometry, each branch of a hyperbola approaches two asymptotes, which intersect at the hyperbola’s center.

Applications of hyperbolas are varied and can include anything from physics problems to structural engineering designs. Understanding how hyperbolic functions relate to hyperbolas can be incredibly helpful in recognizing patterns with equations and their geometric interpretations.

Determining the Curve: Identification Techniques

Curve identification is crucial when working with parametric equations. This involves recognizing the pattern in the equations: whether they yield a line, circle, ellipse, parabola, or hyperbola. When given parametric equations with hyperbolic functions, like in our exercise with \(x=3 \cosh(4t)\) and \(y=4 \sinh(4t)\), the primary task is to transform these equations into a recognizable form.

By using hyperbolic identities, we can rework the equations to illustrate the standard hyperbola form:

• Simplify using identities, such as \(\cosh^2(t) - \sinh^2(t) = 1\).
• Look for transformations that lead to equations in familiar forms, like \((\frac{x}{a})^2 - (\frac{y}{b})^2 = 1\).

This process of curve identification not only helps in identifying hyperbolas but extends to any conic section, providing a powerful toolset to solve various calculus problems efficiently.

Parametric Equations and Calculus Problem Solving

Parametric equations are pervasive in calculus as they offer a flexible way to describe curves. Rather than defining \(y\) in terms of \(x\) or vice versa, parametric equations use parameters like \(t\) to define both \(x\) and \(y\) independently. This can simplify calculus problem-solving since it allows the description of more complex curves that would be difficult to write as a simple function.

For hyperbolas especially, parametric equations are invaluable. Calculus often requires determining slopes or areas, and parametric forms can make these calculations more straightforward.

• Derivatives of parametric equations can be found using chain rules and derivatives of \(x(t)\) and \(y(t)\) with respect to \(t\).
• Integrating parametric equations allows for calculating areas under curves that don't conform to standard x-y plane functions.

Ultimately, parametric equations lend themselves to a diverse range of applications in calculus making problem-solving more intuitive and approachable.