Exploring the Parametric Curve: r(t) =

In this article, we will delve into the properties of the parametric curve defined by the equation r(t) = <e^(-5t)cos(-t), e^(-5t)sin(-t), e^(-5t)> and learn how to compute its arc length function s(t).

Understanding the Parametric Curve

The parametric curve r(t) is defined by three component functions:

1. x(t) = e^(-5t)cos(-t)
2. y(t) = e^(-5t)sin(-t)
3. z(t) = e^(-5t)

These functions describe the x, y, and z coordinates of the curve, respectively, as the parameter t varies.

The curve exhibits some interesting properties:

1. Exponential Decay: The factor e^(-5t) in all three component functions indicates that the curve undergoes exponential decay as t increases. This means that the overall size or magnitude of the curve diminishes over time.

2. Oscillatory Behavior: The trigonometric functions cos(-t) and sin(-t) introduce an oscillatory behavior to the x and y components of the curve. As t increases, the curve oscillates back and forth in the x-y plane.

3. Three-Dimensional Curve: Since the curve has three component functions, it represents a three-dimensional parametric curve in space.

Computing the Arc Length Function

The arc length function s(t) measures the length of the curve from the initial point (where t = 0) to the point corresponding to the parameter value t. To compute s(t), we can use the formula:

s(t) = ∫_0^t |r'(u)| du

where r'(u) is the derivative of the position vector r(u) with respect to the parameter u.

Let's compute the derivatives of the component functions:

1. x'(t) = -5e^(-5t)cos(-t) + e^(-5t)sin(-t)
2. y'(t) = -5e^(-5t)sin(-t) - e^(-5t)cos(-t)
3. z'(t) = -5e^(-5t)

Now, we can calculate the magnitude of the derivative vector r'(t):

|r'(t)| = √[(x'(t))^2 + (y'(t))^2 + (z'(t))^2] |r'(t)| = √[(-5e^(-5t)cos(-t) + e^(-5t)sin(-t))^2 + (-5e^(-5t)sin(-t) - e^(-5t)cos(-t))^2 + (-5e^(-5t))^2] |r'(t)| = √[25e^(-10t)cos^2(-t) - 10e^(-10t)cos(-t)sin(-t) + e^(-10t)sin^2(-t) + 25e^(-10t)sin^2(-t) + 10e^(-10t)cos(-t)sin(-t) + e^(-10t)cos^2(-t) + 25e^(-10t)] |r'(t)| = √[26e^(-10t) + 25e^(-10t)] |r'(t)| = √[51e^(-10t)] |r'(t)| = 7e^(-5t)

Finally, we can compute the arc length function s(t) by integrating the magnitude of the derivative vector from 0 to t:

s(t) = ∫_0^t |r'(u)| du s(t) = ∫_0^t 7e^(-5u) du s(t) = -7/5 * e^(-5t) + 7/5

Therefore, the arc length function s(t) is given by:

s(t) = -7/5 * e^(-5t) + 7/5

This function gives the length of the curve from the initial point (where t = 0) to the point corresponding to the parameter value t.