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Q. 5

Expert-verifiedFound in: Page 879

Book edition
1st

Author(s)
Peter Kohn, Laura Taalman

Pages
1155 pages

ISBN
9781429241861

Under what conditions does a differentiable vector-valued function* r(t)* not have a unit tangent vector at a point in the domain of *r(t)*?

A differentiable vector-valued function* r(t)* does not have a unit tangent vector at any point ${t}_{0}$ in the domain of *r(t) *at which $r\text{'}\left({t}_{0}\right)=0.$

The definition of the unit tangent vector is to let *r(t)* be a differentiable vector function on some interval $I\subseteq \mathrm{\mathbb{R}}$ such that $r\text{'}\left(t\right)\ne 0$ on* I*. The unit tangent function is defined to be $T\left(t\right)=\frac{r\text{'}\left(t\right)}{\left|\left|r\text{'}\left(t\right)\right|\right|}.$

Let ${t}_{0}$ be any point in the domain of $r\left(t\right).$ So, by the definition of unit tangent vector, *r(t) *does not contain a unit tangent vector at any point ${t}_{0}$ at which $r\text{'}\left({t}_{0}\right)=0.$

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