Question

Find r(t), r(c_0), and r(c_0) for the given equation of t_e

r(c) = (e^t, e^2t^2), t_0 = 0
r(c) = t_0
r(r_0) =

Answer 1

The parametric equations are
`\(r(t) = (e^t, e^(2t^2))\)`,
`\(r(c_0) = (1, 1)\)`, and
`\(r(r_0) = (1, 1)\)`.

Step-by-step explanation:

The given parametric equations are
`\(r(t) = (e^t, e^(2t^2))\)` with
`\(t_0 = 0\)`. To find \
`(r(c_0)\)`, substitute
`\(t_0 = 0\)` into the parametric equations, yielding
`\(r(c_0) = (e^0, e^(2(0)^2)) = (1, 1)\)`.

For \(r(r_0)\), it seems there might be a typo in the question, as
`\(r_0\)` is not defined. Assuming
`\(r_0\)` is meant to be
`\(t_0\)`, then
`\(r(r_0) = r(t_0)\)`, and substituting
`\(t_0 = 0\)` gives
`\(r(r_0) = (1, 1)\)`.

In summary,
`\(r(t)\)` is the given parametric equation,
`\(r(c_0)\)` is obtained by substituting
`\(t_0\)` into
`\(r(t)\)`, and
`\(r(r_0)\)` assuming
`\(r_0\)` is a typo for
`\(t_0\)` is equal to
`\(r(c_0)\)`, both resulting in the point (1, 1).

Understanding parametric equations and their evaluation can provide insights into the behavior of curves in different coordinate systems, helping in applications such as physics and computer graphics. Parametric equations express a curve's coordinates in terms of a parameter, often time in applications. These equations offer a versatile way to describe complex shapes that may not be easily represented using standard Cartesian coordinates.