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Unformatted text preview: Department of Mechanical Engineering
The University of Texas at Austin ME 218 Engineering Computational Methods Second Midterm Examination Fall 2010 Instructor: Prof. Dragan Djurdjanovic November 4‘“, 2010
(8:00 am — 9:15 am) Student Name: ; > Student EID: Lab Section Number (or time): I have neither given nor received any unauthorized aid on this exam, nor have I
concealed any violations of the honor code. Signature: Note: There are 4 problems in this exam. First problem carries IS points, second problem carries 30 points. third problem carries
35 points and the fourth problem carries 20 points. Be careful and through in your work and GOOD LUCK!!! Problem 1: (15 points) Write out the system of linear equations needed to fit a quadratic polynomial to the
following set of five data points using the least squares method discussed in class (x is the
independent variable, y is the dependent variable). Please just write out the system of
equations (either one by one or using matrix notation) and do NOT try to solve the system. [\J “700 ’
A Z n A
Mi 5% 1 game.) 35 3 x 4.02::
V\ A " n
ZXL' E gm) 3> 0’1 [3
03‘ la:‘ Problem 2: (30 points) The following data gives the population of the world using approximate census data from
1850 thru 2000. Our goal is to use this data to create an exponential model in the form .t—1850
100 A
190‘) = 1906 that we can use to approximate the population on other years. Please note that t denotes
years. Given the form of the model, the following table will be helpful “Transformed
year” t, —1850 xi = 100 (a) Set up the system of linear equations that must be solved to ﬁnd the exponential
leastsquares curve ﬁt for the given data. (10 points) A Ll (b) Solve the system ofequations you found in part (a) and use that solution to find
parameters pa and A. (10 points) I h a b4 1 d _b
Peasenotetatc d "ad_bc _c a [l]: 3'5 3 " H.OZ5
B [3 Ll] [$6221]
>\ I q :s 14.026 __ L043”
[5] Zmﬁs 3.5][36227] {0.12M Po=€B= Couzs‘: l.l5l0 (c) Use the parameters found in part (b) to estimate the population in 2050. (4 points) 1.04st _~l$!50++: Pm: l.l3lO€: X~ noo _ _205o'l8bo__
752050 ——>X T—Z Lolbwz) R1050) : BlLLlON (d) Use the parameters you found in part (b) to estimate when the population will
exceed 7 billion. (6 points) Rt3Z¥ ¥= "15‘O€[.OH3HX R2050) 7‘ llSIOQ 75:. 100mm : {00 0.17710) +4750 : 2024.1 4 Problem 3: (35 points) Let us observe equations describing 2 ellipses. These 2 ellipses will intersect in 4 different points and we will use the vectorial Newton
Raphson's method to numerically search for one of them. (a) Transform the two equations given above into the form
f(x,y) = 0
g(x9y) = O
that we can solve using the vectorial NewtonRaphson's method. (5 points)
2. )(2 Y .. 26: ,2 :
1C(x.y):?+7—I'O jWIV‘I hi l O (b) Express the lacobian matrix for the system of nonlinear equations you found in part
(3). Please use symbolic calculations (Le. express the Iacobian using generic
variables x andy). (10 points) (c) Starting from the initial guess (x0,y0) = (1 50,250), please conduct 2 iterations of the vectorial NewtonRaphson's method to numerically solve the system of linear
equations you found in part (a). (15 points) O _— l'b 0
X‘l :[zléo  ‘~5(L5)_ «(ZSMBl Z “ L: *"‘296 "l
X L! 81 Z(l.5l 1 5 1,5: 2.
T 2‘ —+2"5
‘1‘ T" [There is one more part of this question on the next page) (d) Obviously, ifyou had enough time to perform the procedure you did in part (c) a bit
longer, you would converge to a solution (xy) representing one of the intersection
points for the two ellipses considered in this problem. Discuss in words how you
would pursue other intersection points for this problem (please stay very
conceptual  no calculations here). (5 points) P lCl<H\lC7 MEN min/um, (NY) VALUES CLOSEK ’I’Q MOTH/Er: INTEIZSEcﬂoN SAME PKQCEDUICE 0550 IN PAWS
(00 T HlZOUC7H ((33 OF Ttllﬁ P/Zo ELE/Vl. Problem 4: (20 points) (a) Compute the following integral using trapezoidal rule assuming that the step size h
= 0.1. Round up your results and represent all numbers with only 2 decimal points. [15points)
0'2 — 2 "“ "XLZ' x~r.z
fexdx:Z€H+8(‘l l"
0.2 ~
L21 Z __ (o.z)z _ Z 2—
— Z. (8 4‘ er. + 230 +(€‘(°‘)Z+e(o.z)z
O, l d,“— ’7. (b) Please suggest one way of im roving the accuracy of numerical evaluation of this
integral (we discussed 2 possible ways of improving the accuracy — please recall at
least one). (5 points) \) Became. STEP 51212 Z) USE [SET—ram AppiomrwAnoN Forl— 94114
BETUEEN FuNcﬂoM VALUE).G.€. NONLWEAB) 8 ...
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