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Determine the highest order of contact at $x_0 = 0$

$f(x) = x^2$

$g(x) = \sin x$


My definition of contact is fuzzy. So I took the derivative at each step, and inspected when $f^{(k)}=g^{(k)}=0$

It's clear that $f^{(k)}(0)=0 \forall k$

$g'(0) = 1, g''(0)= 0, g'''(0)=-1, g''''(0)=0, ...$

Is the highest point of contact 2? or $\infty$?

  • 2
    You have $f(0)=g(0)$, but $f^\prime(0)\neq g^\prime(0)$. "Order of contact" is related to the concept of two curves being tangent to each other.2011-11-30
  • 1
    Prod @J.M. to turn that into an answer.2011-11-30

1 Answers 1