# 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:

`x(t) = e^(-5t)cos(-t)`

`y(t) = e^(-5t)sin(-t)`

`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:

**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.**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.**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:

`x'(t) = -5e^(-5t)cos(-t) + e^(-5t)sin(-t)`

`y'(t) = -5e^(-5t)sin(-t) - e^(-5t)cos(-t)`

`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`

.