Chapter 11: Q. 38 (page 890)

Find the curvature of each of the functions defined by the parametric equations in Exercises 36–38.
$x = \cos t + t \sin t, y = \sin t - t \cos t$

Short Answer

Expert verified

The curvature is $\frac{1}{t} .$

Step by step solution

Step 1. Given information.

The given function is,

$x = \cos t + t \sin t, y = \sin t - t \cos t$

Step 2. Curvature.

Now,

$r (t) = ⟨ \cos t + t \sin t, \sin t - t \cos t ⟩ r^{'} (t) = ⟨ t \cos t, t \sin t ⟩ ||r^{'} (t)|| = \sqrt{(t \cos t)^{2} + (t \sin t)^{2}} = \sqrt{t^{2} (\cos^{2} t + \sin^{2} t)} = t T (t) = \frac{r^{'} (t)}{||r^{'} (t)||} = \frac{⟨ t \cos t, t \sin t ⟩}{t} = ⟨ \cos t, \sin t ⟩ ||T^{'} (t)|| = \sqrt{(- \sin t)^{2} + \cos^{2} t} = 1 C u r v a t u r e = k = \frac{||T^{'} (t)||}{||r^{'} (t)||} = \frac{1}{t}$

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Find the curvature of each of the functions defined by the parametric equations in Exercises 36–38.
$x = \cos t + t \sin t, y = \sin t - t \cos t$

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Curvature.

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Find the curvature of each of the functions defined by the parametric equations in Exercises 36–38.x=cost+tsint,y=sint-tcost

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Curvature.

One App. One Place for Learning.

Most popular questions from this chapter

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Find the curvature of each of the functions defined by the parametric equations in Exercises 36–38.
$x = \cos t + t \sin t, y = \sin t - t \cos t$