14–18. Parametric descriptions Write parametric equations for the following curves. Solutions are not unique.
The segment of the curve x=y ³ +y+1 that starts at (1, 0) and ends at (11, 2).

Verified step by step guidance

1. Identify the given curve equation: \(x = y^3 + y + 1\).
2. Note the segment starts at the point \((1, 0)\) and ends at \((11, 2)\), so the parameter will be based on \(y\) ranging from 0 to 2.
3. Choose the parameter \(t\) to represent \(y\), so set \(y = t\) with \(t\) in the interval \([0, 2]\).
4. Express \(x\) in terms of \(t\) using the given curve equation: \(x = t^3 + t + 1\).
5. Write the parametric equations as \(x(t) = t^3 + t + 1\) and \(y(t) = t\) for \(t\) in \([0, 2]\).

Key Concepts

Parametric Equations

Parametric equations express the coordinates of points on a curve as functions of a parameter, usually denoted t. Instead of y as a function of x, both x and y are given in terms of t, allowing more flexible descriptions of curves, including those that are not functions in the traditional sense.

Curve Segment and Parameter Interval

When describing a specific segment of a curve parametrically, the parameter t is restricted to an interval that corresponds to the start and end points. Identifying the correct parameter values for these points ensures the parametric equations trace the desired segment.

Non-uniqueness of Parametric Representations

Parametric equations for a curve are not unique; different parameterizations can describe the same curve segment. This flexibility allows choosing convenient parameter functions or intervals to simplify calculations or meet specific conditions.