Partial preview of the text
Download Final Examination with Resolution - Numerical Analysis | MATH 375 and more Exams Mathematical Methods for Numerical Analysis and Optimization in PDF only on Docsity!
Millersville University Name Ansuser Kern Department of Mathematics MATH 375, Numerical Analysis, Final Examination December 12, 2006, 12:30PM-2:30PM Please answer the following questions. Your answers will be evaluated on their correctness, com- pleteness, and use of mathematical concepts we have covered. Please show all work and write out your work neatly. Answers without supporting work will receive no credit. The point values of the problems are listed in parentheses. You may use your textbook, calculator, and notes. Un- Jess otherwise indicated all numerical approximations should be carried out to at least six decimal places. 1. (10 points) Using the quadratic formula and three-digit rounding arithmetic, solve the fol- lowing equation. Li. 12 1 ged Ztos =0 Compute the absolute error between your roots and the exact solutions. Exact ooltion: % = yt .- Yaa * fen z, wT Rt UE -SAWBS anh Ke O.1ZH3 3- Rinct Coding, 1 OB + LTE — 0.200 = 0 170 EV He caffe. 30e{o 200) (2.000.313) -IN {2924 0.264 * 0.636 _ cht 0.636 = wt t 1.78 0.636 A % = - 549 ond hy 0.110 \u,-4,| = 0.069%3066 \Yo> Fy 1X 0.00424325 2. (6 points each) Given the function f at the following values: f(z) 3.12014 4.42569 = 6.04241 h= 0.2 8.03014 2.0 2.2 24 24 Approximate | (x) da in two different ways. J18 (a) Using Simpson’s 3/8 rule. oe 3 ae wes (0.2)( £(..2) + 3F(2.0)+ 3Fl2.2)+ Flea)) 0. = 28 (320K 3(4azset) + 3(b.o424) + 3.02014) ~% 3.19189 (b) Using the open Newton—Cotes approximation with n = 1. 24 \, foe * (0.2) ( F(2.0)+ F(2.2)) Ob (dMasot + 04241) x 3.44043 4. (10 points) A function f: [a,b] 4 R is said to be Lipschitz continuous with Lipschitz constant LE provided that for every 2, y € [a, 6], Hf) — Fy) S Lia — y}. Assume f € C![a,b], show f is Lipschitz continuous on [a, 6]. (Hint: the Mean Value Theorem and Extreme Value Theorem will be of use here.) Baw mvt, 4-4) Al) fp cot ¢ hetioen ¥-Y *% anh y. Danes Ab e'[a] Ron 4 Ce) a tortious on [a,b]. Sy he EVT | eWay orRiever 5 Trt an [a,b] CoQ Rin mopurum L. _ SRe Alt - 4(y) éL e-Y “ lo@- ails L leg 5. (12 points) Find the linear combination of the functions in the set {1,2} which forms the tT least. squares approximation to sina on the interval l= al w, det Ala,b) 2 Ly (omn = (ox +b)) dy. ah = 2 ( (orm = (ox rb) )fen) dy Me 2 M (on — (on+b)\(4) ay Angee Bp + Gp ten % “ty “My ™%, We la a\ eke + b\ Ld =F Dunk Ly \.4, er as 31% Ue a = , 3 | “ny \ hone dy br = 0 = |b=0 a Me as? 2 2 (“boy dy = 2orn| * = 2 12 ° 8 7. (10 points) Use Newton’s method to find a solution to within an accuracy of 10~® to the equation 1 2 x —sine+ =~ =0. * 20 You, must clearly state the initial approximation 29, which you use to start Newton’s method. Ln * Ka 0.0 ie) 0.05 0.457142 6,0525043 0.430042. O2ADI? 6.924015 it 6.0§29139 " u peun-of? WwAwnn~-o 8. (6 points each) Consider the data in the following table. F(z) 7.03624 8.41175 8.57661 8.74399 10.3691 1.20 1.29 1.380 1.31 1.40 (a) Approximate f”(1.30) using h = 0.1. 4 {1.39) may (40.20) ~ 24 (1.30) + 44 (1.40)} = 25.212 (b) Approximate f’(1.30) using A => 0.01. A (Uae) Aen ( Altal) - A(.2)) x 16,012 ({c) Estimate the number of transistors in Intel microprocessors in 2007. O.31F013E Uy (2126.9 e Qe 2007, t= 32 ws 2.767% X10" troesistors 10 
