Exam 2 Solution CHE 2120 Fall 2004 - ChE 2120 Exam 2...

Unformatted text preview: ChE 2120 Exam 2 November 8, 2004 ‘ (I) (40 pts) Consider the following data, from the function y = f(x)‘ a. (20 pts) Fit a second order polynomial to this data using least squares regression. ‘ 2 S? ,v e“ 2' L; l‘ at 1. (lo ~17ng +02 ‘X. 9. i '53» wouacawag'xt 3' U7 ,- ( _ [w 2;: I 7;; 0U (LiJ_ 5" I, ! xzf .7112“: i 6221‘, “O7 L'lfl 2"} ’Xg - 7 Iv] \2 L} [on i g: [I I I f) If] :{ZWE If) i :3 4i ‘M ii 2 L) I : {q . f] E36? * 7/ _. M . '-f " 23 M W 611’ f _ 0 $ 19 4 :1 9/ a? .3... .F at 21" ’4‘ +331 i’ b. (5 pts) Compute the integral of y(x) from x = l to x = 3 using the (" trapezoidal rule. C + 2 32. __ .3 2 __ (sailUr-r 2(3) Hr”) I ’39,. : 5 c. (5 pts) Compute the integral of 3100 using Simpson’s 1/3 rule. II ( b’Q) z 3 z (8 ,1) ( 4+ Lari-H"? : . ............ Wm... M I z 3 r 33 a ‘22 e? 3 5’? d. (5 pts) Anglytically integrate the polynomial fit from Part a. K I "a Yrg+£7¢+('2)'x?_cfar ” '27 . a; x .2643 :- 2 [0'2]. : /W+ 290% I mm," c. (5 pts) Explain how these results compare and why. 5 t Nam-SYLVIA 'Eiwfboyefi 3 uh WW Siwtamfl é MAL FAAQMAL 01’?ch car/L443"; :(J @100 don/3 \ listh luff!) devoioUAaj/tov 'k/u Wm..«wr we firs } Of CQV‘ C2,) I... f” M (2) (40 pts) Again consider the data from Problem 1. a. (10 pts) Compute the derivatives at each of the three points using a finite difference method. Explain why you chose the method(s) that you did. -_ mag mewg on... sex.» . low—jg; 5"“ L w _ ._- "Zr—:1 mer wC/V‘ qi'tlfixi'rfi g .. 9 A 0M9; (MA - «€00? r __ __ .. .95 _;— -2. W” “"’ {‘ch '* .- r l ‘4. — E ’5 cede jbmwg Aha/g WE h! M411“ .— L .4 410333}: {ails—£0) :atrg' : g Bahama #2 l as. 2. “Ha 5mg PJ's Cg ‘2? [Jr/3.4L} M ofiakfu‘ C’QWLQQ \M \QQ‘J . b. (10 pts) Compute the derivative of the polynomial fit, and evaluate it at each of the three points. . (15 pts) Use the Golden section method to find the maximum of the polynomial fit. Perform three iterations. at“ .9 we d. (5 pts) Explain how these results compare and why. WT ‘ . P ' H Ho» ’3 I 7 W Mains cl \ if [viv- K\J~.Cx}—L\/\LA {wk w {p.{’ C {'1 {W £13 0‘ i n... (3) (10 pts) Concentration data was taken at timeupojntéhfor the polymerization reaction 7—"— xA+yBfiAxBy We assume the reaction occurs via a complex mechanism consisting of many steps. Several models have been hypothesized and the sum of the squares of the residuals had been calculated for the fits of the models of the data. The results are shown below. Which model best describes the data (statistically)? Explain your choice. (4) (10 pts) Find the maximum of z = f(x,y). Assume that the function varies smoothly between the contours. a. (5 pts) Sketch 3 iterations of the univariate method, beginning at (x0,yo) = (1,1). b. (5 pts) Sketch 3 iterations of the steepest ascent method, beginning at 010,510) = (1,1). ...
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