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Expert-verified Found in: Page 879 ### Calculus

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 $r\text{'}\left({t}_{0}\right)=0.$

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## Step 1. Given Information.

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

## Step 2. Explanation.

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 $r\text{'}\left({t}_{0}\right)=0.$ ### Want to see more solutions like these? 