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hw-08-1-arc-length - right endpoint and midpoint for...

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Chapter 8.1 homework (we used the in-class worksheet for exercise, with no turned-in homework) Side note: #40 in the Chapter 8.1 exercises discusses the shape of a hanging wire or chain: it is called a "catenary" curve, and it has the shape of a hyperbolic cosine (cosh). For a long time, mathematicians thought it was shaped like a parabola instead. It turns out that suspension bridge cables are shaped like a parabola, because the main weight is the bridge deck, and the weight of the cable is insignificant in comparison. For more information, see Suspension Bridge Profiles C.W. Groetsch ([email protected]), The Citadel, Charleston SC 29409- 6420 doi:10.4169/074683410X488737 VOL. 41, NO. 3, MAY 2010 THE COLLEGE MATHEMATICS JOURNAL pg 237-241 Here is an optional exercise for you: i) Find the equation of a parabola that meets the catenary at the left endpoint,
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Unformatted text preview: right endpoint, and midpoint for Problem 40 part b. ii) How far away does that parabola get from the true cosh curve? Give your answer in feet or as a percent, perhaps. It's probably best to graph: cosh curve - parabola and see how big or big&negative that gets. iii) Find the arc length of the parabola, and compare to the arc length of the true cosh curve. iv) If you already know about Taylor or Maclaurin series, find the Maclaurin series to 2nd order for the cosh curve, and plot that on top of the true cosh curve. Does it overestimate or underestimate the true height of the left and right endpoint? v) Similarly, find the Taylor series (to 2nd order) for the cosh curve at the left or right endpoint, and compare to (i) and (iv) above....
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