Engineering Calculus Notes 208

Engineering Calculus Notes 208 - 196 CHAPTER 2. CURVES (c)...

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
196 CHAPTER 2. CURVES (c) Show that if x 0 is a triadic rational (that is, it has the form x 0 = p 3 j for some j ) then f k +1 ( x 0 ) = f k ( x 0 ) for k suFciently large, and hence this is the value f ( x 0 ). In particular, show that f has a local extremum at each triadic rational. ( Hint: x 0 is a local extremum for all f k once k is suFciently large; furthermore, once this happens, the sign of the slope on either side does not change, and its absolute value is increasing with k . ) This shows that f has in±nitely many local extrema—in fact, between any two points of [0 , 1] there is a local maximum (and a local minimum); in other words, the curve has in±nitely many “corners”. It can be shown (see ( Calculus Deconstructed , § 4.11)) that the function f , while it is continuous on [0 , 1], is not di²erentiable at any point of the interval. In Exercise 5 in § 2.5 , we will also see that this curve has in±nite “length”. 2.5
Background image of page 1
This is the end of the preview. Sign up to access the rest of the document.
Ask a homework question - tutors are online