Problem 61 - Calculus Volume 3

Question

Sketch the curve given by parametric equations \(x=\cosh (t)\) \(y=\sinh (t)\) where \(-2 \leq t \leq 2\).

Step-by-Step Solution

Answer

The curve is one branch of a hyperbola to the right, symmetric about the x-axis.

Step 1: Recall Definitions of Hyperbolic Functions

Recall that hyperbolic cosine and sine functions are given by the expressions \(\cosh(t) = \frac{e^t + e^{-t}}{2}\) and \(\sinh(t) = \frac{e^t - e^{-t}}{2}\) respectively. These functions are related through the identity \(\cosh^2(t) - \sinh^2(t) = 1\), which resembles the Pythagorean identity in trigonometry.

Step 2: Compute Points for Key Values of t

Calculate specific values for \(t\) in the range \([-2, 2]\) to help in sketching the curve. Consider the values: \(-2, -1, 0, 1, 2\). For each \(t\), calculate \(x = \cosh(t)\) and \(y = \sinh(t)\):- \(t = -2\): \(x = \cosh(-2) = \cosh(2)\), \(y = \sinh(-2) = -\sinh(2)\)- \(t = -1\): \(x = \cosh(-1) = \cosh(1)\), \(y = \sinh(-1) = -\sinh(1)\)- \(t = 0\): \(x = \cosh(0) = 1\), \(y = \sinh(0) = 0\)- \(t = 1\): \(x = \cosh(1)\), \(y = \sinh(1)\)- \(t = 2\): \(x = \cosh(2)\), \(y = \sinh(2)\)

Step 3: Identify Symmetrical Properties of the Curve

Notice that the curve is symmetrical about the \(x\)-axis. This is because \(x = \cosh(t)\) is an even function and \(y = \sinh(t)\) is an odd function, resulting in symmetry. This symmetry helps in visualizing the curve's shape.

Step 4: Plot the Calculated Points

Plot the computed pairs \((x, y)\) from Step 2 on the Cartesian plane. Each point represents a solution to the parametric equations for a specific parameter \(t\). Ensure to reflect the graphical symmetry identified in Step 3.

Step 5: Sketch the Smooth Curve

Connect the plotted points smoothly, respecting the behavior of hyperbolic functions. The curve will resemble the right arm of a hyperbola, opening to the right due to \(\cosh(t)\)'s growth as \(t\) increases. Such a sketch confirms it looks like half of a hyperbola.

Key Concepts

Hyperbolic Functions

Hyperbolic functions, like trigonometric functions, are defined using exponential functions. They are important in many areas such as calculus and mathematical modeling. The two primary hyperbolic functions are \( \cosh(t) \) (hyperbolic cosine) and \( \sinh(t) \) (hyperbolic sine).

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

These two are connected by an identity similar to the Pythagorean identity in trigonometry: \( \cosh^2(t) - \sinh^2(t) = 1 \). This identity shows the inherent connection between \( \cosh(t) \) and \( \sinh(t) \), representing a form of geometric relationship on the hyperbolic plane."

Cartesian Plane

The Cartesian plane is a two-dimensional coordinate plane defined by the x-axis and y-axis. It is used to represent and analyze geometric shapes or relationships between two variables. In this setup, each point has coordinates \((x, y)\).

• The horizontal axis is the x-axis.
• The vertical axis is the y-axis.
• A point on the plane is represented by the pair \((x, y)\).

When plotting parametric equations like \( x = \cosh(t) \) and \( y = \sinh(t) \), each value of \( t \) produces a corresponding point on this plane. With the range of \( t \) between -2 and 2, these coordinates help visualize the geometry of the curve they form.

Symmetrical Properties

Symmetrical properties of functions offer insight into the nature of curves and their transformations. For these parametric equations, symmetry can simplify the process of graphing. Here, the symmetry is found in the relationship of the functions:

• \( \cosh(t) \) is an even function, meaning \( \cosh(-t) = \cosh(t) \).
• \( \sinh(t) \) is an odd function, meaning \( \sinh(-t) = -\sinh(t) \).

This means the curve is symmetrical about the x-axis. Even though \( \cosh(t) \) grows exponentially, the relationship with \( \sinh(t) \) reflects across this axis, providing a mirror image. Identifying these symmetries assists in plotting the curve efficiently without having to compute every single point.

Parametric Curve Sketching

Parametric curve sketching involves plotting curves defined not just by \(x\) or \(y\), but as functions of a third variable, often \(t\), a parameter. This can make graphs more flexible and expressive.

• Identify key points by calculating the values of \(x\) and \(y\) for select values of \(t\).
• Consider the symmetrical and behavior properties of the functions, such as growth rate.
• Connect the points smoothly while honoring identified symmetries and function characteristics.

Using the parametric equations \(x = \cosh(t)\) and \(y = \sinh(t)\) with \(t\) ranging from -2 to 2, points traced on the graph will form the right side of a hyperbolic curve. The curve itself resembles a hyperbola's "arm," which is evident in how \(\cosh(t)\) and \(\sinh(t)\) grow as \(t\) increases.