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).

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Identify the given curve equation: \(x = y^3 + y + 1\).

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.

Choose the parameter \(t\) to represent \(y\), so set \(y = t\) with \(t\) in the interval \([0, 2]\).

Express \(x\) in terms of \(t\) using the given curve equation: \(x = t^3 + t + 1\).

Write the parametric equations as \(x(t) = t^3 + t + 1\) and \(y(t) = t\) for \(t\) in \([0, 2]\).

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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.

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).