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chbe2120practiceFinalS1probs4and5 - Problem 4[10 points All...

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Unformatted text preview: Problem 4: [10 points] All the problems below come from quizzes given in class during the semester. Each question is worth 2 points with no partial credit. (a) Write down the first four terms of y(x)=e usmg the Taylor series to ex and aboutx— 4‘30 j/Owd 3/0): 73/0”? ’j {0%ij /0) 3—7 j/xl= imfl: 6 W75: 5‘ :7 f0} "flWJJWV‘jYU/fo ’5 ;( a (b) The Newton-Raphson method us used to find a root of {f1(xl,x2) =xl2 +x22 +exp(x]) f2(xisx2) = x1352 +exp(—x2) As an initial estimate for a root, we will use x1: x2— — 0. State whether this estimate is a root of theequations. {(0 0>_ 0wd+€0 :€O:/ (c) What IS the gradient of f(x, y) 2—12): —-(y+2 4 —x 2( —2)2 +16 at an arbitrary value ofx andy? Vii: [jg/E1] 33;. 276’ Z): 1,; [ 46%) Z2 6-3) (d) Which method will converge faster to a root of f (x) — x + éééo'c)"; the Newton-Raphson Imethod or the bisection method? Ir/tnéa/j in?” I’S (/{Me Cid reed, WEE woe éflzlwis m mm! 1W9; 4’5} raw/er (mu/470$) wjk fiber/(idly (e) You are solving a problem that consists of the ODE Y” (x)— xi” + exp(— Y): x +2 with initial conditions Y(0)= 3 and Y' (0)— 0 What numerical method Would you use to calculate Y (1) r’ Problem 5: [10 points] You have purchased a machine that fills bags with candy. Each bag is supposed to weigh 16 02. A random sample of6 bags gives the following weights (in 02.): 15.87, 16.02, 15.78, 15.83, 15.69, 15.81. You wish to test the hypothesis that the machine is working correctly by giving a mean fill weight of 16 02. Using the sampled data, do you accept or reject this hypothesis based on a 95% confidence level? _, "z 3 O/ /§‘§/ ”6‘7? 32 3 i? g + 14g",,¢,,_:_<c(3r/f 7?“ : 5‘, <5 ’3 ...
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