Q. 13 - Calculus | StudyQuestionHub

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Q. 13

Question

As we saw in Example 1, the graph of the vector-valued function $r (t) = (\cos t, \sin t, t), for t \in [0, 2 π]$ is a circular helix that spirals counterclockwise around the z-axis and climbs as t increases. Find another parametrization for this helix so that the motion along the helix is faster for a given change in the parameter.

Step-by-Step Solution

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Answer

Another parametrization for the given helix so that the motion along the helix is faster for a given change in the parameter is $r (t) = (\cos (21 t), \sin (21 t), t), for t \in [0, 2 π] .$

1Step 1. Given Information.

The given vector-valued function is $r (t) = (\cos t, \sin t, t), for t \in [0, 2 π] .$

2Step 2. Finding another parametrization.

We have to find another parametrization for this helix so that the motion along the helix is faster for a given change in the parameter.

Let another parametric curve is $r (t) = (\cos (21 t), \sin (21 t), t), for t \in [0, 2 π] .$

So, the graph is

Thus, the motion along the helix is faster for a given change in the parameter.

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