In this video we extend Lagrange's approach to the differential calculus to the case of algebraic curves. This means we can study tangent lines, tangent conics and so on to a general curve of the form p(x,y)=0; this includes the situation y=f(x) as a special case. It also allows us to deal with situations where the usual tangent is vertical, and so the derivative is undefined.

The case of the lemniscate of Bernoulli is looked at in detail. Since now the tangent conic can be either an ellipse, parabola or hyperbola, we see that the nature of the quadratic approximation at a point allows us to group points on the curve into elliptic, parabolic and hyperbolic type. For the lemniscate, the parabolic points are found, lying on the discriminant conic.

ที่มา : http://www.youtube.com/channel/UCXl0Zbk8_rvjyLwAR-Xh9pQ

ลิงค์ : http://youtu.be/fkeuT9rze1I?list=PLIljB45xT85DWUiFYYGqJVtfnkUFWkKtP

อัพโหลดโดย : njwildberger