How to find maximum curvature for a vector function at a particular point

Let’s look at all the formulas we’ll use to find maximum curvature

Before we can find maximum curvature of a vector function $r(t)=r(t)_1\bold i+r(t)_2\bold j+r(t)_3\bold k$ , we first have to find curvature $\kappa(t)$ . To find the curvature $\kappa(t)$ of a vector function $r(t)=r(t)_1\bold i+r(t)_2\bold j+r(t)_3\bold k$ , we’ll use the equation

$\kappa(t)=\frac{|T'(t)|}{|r'(t)|}$

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where $|T'(t)|$ is the magnitude of the derivative of the unit tangent vector $T(t)$ , which we can find using

$|T'(t)|=\sqrt{[T'(t)_1]^2+[T'(t)_2]^2+[T'(t)_3]^2}$

where $T(t)$ is the unit tangent vector, which we can find using

$T(t)=\frac{r'(t)}{|r'(t)|}$

where $r'(t)$ is the derivative of the vector function and where $|r'(t)|$ is the magnitude of the derivative of the vector function, which we can find using

$|r'(t)|=\sqrt{[r'(t)_1]^2+[r'(t)_2]^2+[r'(t)_3]^2}$

In other words, in order to find $\kappa(t)$ , we’ll

Find $r'(t)$ , and use it to
Find $|r'(t)|$ , and then use $r'(t)$ and $|r'(t)|$ to
Find $T(t)$ , and then use it to
Find $T'(t)$ , and then use it to
Find $|T'(t)|$ , and then use $|r'(t)|$ and $|T'(t)|$ to
Find $\kappa(t)$

Once we have curvature, we’ll take its derivative $\kappa'(t)$ . We’ll set the derivative equal to $0$ and solve for $t$ . If there’s only one value for $t$ , that value is the one associated with maximum curvature. If there’s more than one value for $t$ , we’ll use the second derivative test to determine which one represents maximum curvature.