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Unformatted text preview: Name: SO LUTI O N CHE 2120  Numerical Methods in Chemical Engineering Spring 2007 Professor Gallivan Test #1 (2 hours) 1. (21 pts) Please answer the questions below by selecting either (a) one of the answers provided. 2M i. The following orders can be included in a Taylor Series Expansion of a nonlinear function to GYM generate a linear approximation for the function: 041» you” (”WW 1
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a. Omorder (YiAx) : Y(X) 4 Cl)! (XzX)AX ’ 2 dx” X)
b. lSt order y . . ‘
c. 2nd order T f /
.0th 0th and 1St order (maﬁa/‘7‘. M e. The 0th, I“, and 2"d order terms can all be included WI /NLQ_OV\.. l’) ' “U ii. A Taylor Series approximation for a function which includes the 0th and lSt order terms will have an error that is on the order of the step size raised to what power: a. O
b. 1
e. None of the above iii. Which of the following statements concerning Gaussian Elimination methods for solving linear
systems of equations (i.e. Ax=b) is true: a. Gaussian elimination requires more computations than does using the matrix inverse
method to compute x=A'lb. Partial pivoting should generally be used to help avoid division by zero errors and round
off errors during elimination. c. GaussJordan is more computationally efﬁcient than is Gaussian elimination with back
substitution. d. Scaling helps to eliminate roundoff errors when two coefﬁcients in the same equation are
of dramatically different magnitudes. iv. You form an ODE initial value problem for a chemical reaction. One reaction rate is very fast and another is
very slow. You have the choice of using an implicit method or an explicit method. Which statement is most accurate?
@ ou should use an implicit method.
. You should use an explicit method. 0. You may use either method.
d. You should not use either method. vi. vii. You are offered a series of credit cards in the mail with terms and interest rates listed below. Please
select the one on which you would pay the lowest amount of interest in the span of the ﬁrst year if you transferred a balance of $10,000 to that card at the beginning of the ﬁrst month that you got the
card. I
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annual yearly interest rate of 24%, compounded annually 0 I 0‘ Y 17, I ‘13
monthly periodic interest rate of 1.9%, compounded monthly (I . O I 9) ‘ I. ZS
weekly periodic interest rate of 0.5%, compounded weekly ( /. 0 0 Sf l : I z9 G a
b
c.
daily periodic interest rate of 0.0006%, compounded daily ,cs _
0.000006) 7.00er You have a system of 4 linear equations that you are solving using Gaussian elimination with back substitution (including scaling and pivoting) and you end up with the following A matrix and b vector
during the elimination process. 1 0.5 0.5 0.1 ‘0.7 0 1 0.3 0.2 0.4
A: b= 0 0 1 0.5 0.2 o o o o 0.1 Which of the following statements is true: a. This system of equations had two equations that were linearly dependent on the other two
equations. b. This system of equations has an inﬁnite number of solutions. 0. You are not ﬁnished with the elimination process yet and some multiple of row three must
be added to row 4 to eliminate the last element in the b vector.
6 This system of equations has no solution. I10
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for i=1:4 ‘ _ :
ifabs(i3)~=0 1:7. $4M  I + Z 3
star=star+i
end i: 3 X
end i : (I ﬁvn’ 3 3 t 9/ 3 7
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 Gallivan

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