# Find the derivative, second derivative, and curvature at t = 1. For the curve given by r(t) =...

## Question:

Find the derivative, second derivative, and curvature at {eq}t=1 {/eq}. For the curve given by {eq}r(t) = (-9t, 4t, 1 + 8t^2), {/eq}

## Curvature:

Given: {eq}r(t) = (-9t, 4t, 1 + 8t^2) {/eq}

Curvature is calculated using formula:

{eq}\frac{\left \| r'\left ( t \right )\times r''\left ( t \right ) \right \|}{\left \| r'\left ( t \right ) \right \|^{3}} {/eq}

r(t) = (-9t, 4t, 1 + 8t^2)

On differentiating r(t) with respect to t, we get

{eq}r'(t)=\left ( -9,4,16t \right )\\ r'(1)=\left ( -9,4,16 \right...

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