Curvature of a Plane Curve y = f(x) + Example on a Parabola

In this video, I go over the curvature of the special case of a plane curve with equation y = f(x), by selecting x as the parameter and using our earlier cross product curvature formula. The magnitude of the cross product simplifies due to the standard basis vectors canceling out, giving just |f''(x)| or the absolute value of the second derivative of f. The length of the second derivative of the vector function is just the arc length of a plane curve, thus further simplifying the final curvature formula. I illustrate this with obtaining the curvature of a parabola y = x^2. The curvature is max at the origin, and flattens out to zero at infinity as expected. I also graph it out in the amazing Desmos 2D graphing calculator: https://www.desmos.com/calculator/xfmwdwircp

#math #vectors #calculus #desmos #curvature

Timestamps

• Curvature of a plane curve: choose x as the parameter – 0:00
• Cross product of standard basis vectors – 2:00
• Cross product of first two derivatives of the vector function – 4:08
• Length of the cross product is absolute value of f''(x) – 5:52
• Length of the tangent vector – 7:33
• Curvature formula for a plane curve – 9:41
• Recap on standard basis vectors – 11:37
• Example 5: Curvature of a parabola y = x^2 – 12:58
  • Curvature at multiple points – 15:28
  • Curvature approaches zero as x goes to infinity – 18:20
  • Graph of the curvature – 19:33
  • Graphing curvature in Desmos: https://www.desmos.com/calculator/xfmwdwircp – 21:57

Notes and playlists:

• Summary: @mes/re-leothreads-gd9ua9ho
• Notes: @mes/koxharrd
• Sections playlist: https://www.youtube.com/playlist?list=PLai3U8-WIK0Gd5OieYiZMaSKw1QfyHbiG
• Vector Functions playlist: https://www.youtube.com/playlist?list=PLai3U8-WIK0HQl7xTQBS_O8Te8PcpNm4U .

Become a MES Super Fan! https://www.youtube.com/channel/UCUUBq1GPBvvGNz7dpgO14Ow/join

DONATE! ʕ •ᴥ•ʔ https://mes.fm/donate

SUBSCRIBE via EMAIL: https://mes.fm/subscribe

MES Links: https://mes.fm/links

MES Truth: https://mes.fm/truth
Official Website: https://MES.fm
Hive: @mes

Email me: contact@mes.fm

Free Calculators: https://mes.fm/calculators

BMI Calculator: https://bmicalculator.mes.fm
Grade Calculator: https://gradecalculator.mes.fm
Mortgage Calculator: https://mortgagecalculator.mes.fm
Percentage Calculator: https://percentagecalculator.mes.fm

Free Online Tools: https://mes.fm/tools

iPhone and Android Apps: https://mes.fm/mobile-apps

▶️ 3Speak