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Problem 22 Easy Difficulty

Graph the curve with parametric equations $x=\cos t$
$y=\sin t, z=\sin 5 t$ and find the curvature at the
point $(1,0,0)$

Answer

$$
=\frac{\sqrt{2}}{(26)^{3 / 2}}
$$

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Video Transcript

{'transcript': "were given metric equations for occurred the graph mhm on the curvature of the 0.1. You know, Curve has committed ingredients society. Why eat? Let's see Yes, Time in five graph this curve using Parametric curve graph for three mentioned once he does like Look, this is what the curve looks like. And grabbing the curve. We got a complete curves by graphing. Yeah, yeah. Zero and all right now to find the curvature first. That's right. This is a vector function. So rt is protector opponents for science. Scientific sign of five. Team on the curvature. Let's use this. You still time in our double time Derivative or crime A T. This could be found. I Frenchy component wise. This is sign a team. I'm sorry. This should be the Net 90 and the co sign of tea, then five times the co sign of five and we have that are double crime. The director made China tea. Sign it. Negative 25. Sign of five G Now time the curvature 0.100 Well, we'll set this equal to our vector functions. So this is only possible. So frozen teeth one and signed a two equals zero are your with an engineer. Multiple a pie. It also one the team must zero or bye. This case I'll actually restrict may be simply less Hi for clarity. Their function is 1 to 1 will be zero. And we have that the derivative or crime zero values. This is zero one Fine. And we had are edible crime. Well, first of all, magnitude of our prime of zero. This is the square root of the sum of the squares of the components. Yeah, one plus hi. Work, he said, And substituting our double prime zero The vector negative one, Your room zero. So it follows that vector. Our primary zero crossed with our double times of zero by inspection. We see it, Dr zero. Yeah, five, one. And therefore magnitude. Part time zero across. Find zero is the square root of the sum of the squares of the components. Five plus one again is the square root of 26 and therefore by fear um, 10 curvature point zero. This is K of zero cap of zero, which is the ratio of our prime of zero across our double prime zero magnitude. The magnitude of our prime of zero Rich Solutions has 26 Rick 26. Just simply one function of the curvature of Warren, 0.100 I'm sorry, the ratio of that magnitude of this magnitude. Cute. This is Route 20. Steps over twice. Its cube can. 1 26. This is this."}

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