Question

Hey guys. In lecture 5, the professor used a taylor expansion to approximate dy/dx of the cycloid. Could you have used L Hopitals rule instead?

Answer 1

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Answer 2

absolutely

Answer 3

Yes, you can use l'Hospital's rule to solve this one (for me this is actually a more direct method than the one the professor used but his is great as a teaching tool). Calling theta "t"and radius a=1, we have dx/dt=1-cost and dy/dt=sint...thus dy/dx=(dy/dt)/(dx/dt)=sint/(1-cost)
Since but the numerator and the denominator approach 0 as t approaches zero we have an indeterminate form (0/0) and l'Hospital's rule is the way to solve this. Applying l'Hospital's rule once, we get: dy/dx=cost/sint
As t approaches 0 the numerator approaches 1 and the denominator approaches zero. This shows we have a vertical tangent at t=0 as the professor showed using the Taylor expansion. Hope this helps!