Engineering Calculus Notes 224

Engineering Calculus Notes 224 - x k ,f ( x k )) to ( x k...

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212 CHAPTER 2. CURVES (n) f ( x,y,z ) = x 2 y 2 + z 2 , C is given in parametric form as x = cos t y = sin t z = 3 t , 0 t π. (o) f ( x,y,z ) = 4 x + 16 z , C is given in parametric form as x = 2 t y = t 2 z = 4 t 3 9 , 0 t 3 . Theory problems: 4. Consider the graph of the function f ( x ) = b x sin 1 x for x > 0 , 0 for x = 0 over the interval [0 , 1]. (a) Show that | f ( x ) | ≤ | x | with equality at 0 and the points x k := 2 (2 k 1) π , k = 1 ,... . (b) Show that f is continuous. ( Hint: the issue is x = 0) . Thus, its graph is a curve. Note that f is diFerentiable except at x = 0. (c) Consider the piecewise linear approximation to this curve (albeit with in±nitely many pieces) consisting of joining (
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Unformatted text preview: x k ,f ( x k )) to ( x k +1 ,f ( x k +1 )) with straight line segments: note that at one of these points, f ( x ) = x while at the other f ( x ) = x . Show that the line segment joining the points on the curve corresponding to x = x k and x = x k +1 has length at least s k = | f ( x k +1 ) f ( x k ) | = x k +1 + x k = 2 (2 k + 1) + 2 (2 k 1) = 2 p 4 k 4 k 2 1 P ....
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This note was uploaded on 10/20/2011 for the course MAC 2311 taught by Professor All during the Fall '08 term at University of Florida.

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