ECE 580
Spring 2009
Funwork #3
Solutions
1. The Taylor series expansion of a function f(x) about x=x0 is
f ( k ) ( x0 )
f ( x0 )
f ( x0 )
f ( x) =
( x x0 ) k = f ( x 0 ) + f ( x0 )( x x0 ) +
( x x0 ) 2 +
( x x 0 ) 3 + .
k!
2!
3!
k =0
Since f ( x) = cos x
ECE 580
Spring 2004
Midterm #2
April 15, (Thursday) 2004
Name: Student ID #: Problem Weight Score
1 2 3 4 5 6 7 8 9 10 10 10 10 10 10 10 10 10 10 10
Total
This is a 75 minute duration exam Closed books, closed notes No crib-sheets, no calculators There a
ECE 580
Spring 2004
Midterm #2
Solutions
1. The transfer function, (s)/Va (s), modeling of an armature-controlled dc motor with negligibly small armature inductance is (s) 0.5 . = Va (s) K1 s + K2 The corresponding dierential equation is K1 d(t) + K2 (t)
EE 580
Spring 2001
Midterm #1
February 22, (Thursday) 2001 1. For the function
f = f (x1, x2) = x2x2 + x3x1, 1 2
(a) (3 pts.) nd the gradient of f at x =
T
21
;
T
(b) (3 pts.) nd the rate of increase of f at the point x =
T
21
in the direction
d=
43
;
T
(
ECE 580
Spring 2011
FunWork #5
Due on April 22
INSTRUCTIONS: The assignment must be typed. Clearly identify the steps you have
taken to solve each problem. Your grade depends on the completeness and clarity of your
work as well as the resulting answer. O-
Funwork #4 Solution
1
Exercise 12.10 from TEXT on page 240
y (t) = sin(t + )
a. We can represent the problem as a system of p equations as follows.
y1
y2
=
=
.
.
.
sin(t1 + )
sin(t2 + )
yi
=
.
.
.
=
sin(ti + )
yp
Since ti +
2
2 , i
sin(tp + )
= 1, ., p,
ECE 580
Spring 2011
FunWork #4
Due on March 25
INSTRUCTIONS: The assignment must be typed. Clearly identify the steps you have
taken to solve each problem. Your grade depends on the completeness and clarity of your
work as well as the resulting answer. O-
ECE 580
Spring 2009
Funwork #4
Due on February 23
1. For the function
f (x1 , x2 ) = 4(x1 2)2 + (x2 3)2 ,
(a) Obtain the sequence of the rst six points using the steepest descent method and locate
these points on the level sets of f . The initial guess is
ECE 580
Spring 2009
Funwork #4
Solutions
1. (a) Steepest Descent Algorithm
The steepest descent algorithm is as follows:
For the quadratic objective function,
1
1 8 0
T 16
f ( x) = x T Q x x T b + c = x T
x x 6 + 25,
2
2 0 2
(0 )
x = 0,
x ( k +1) = x (
FunWork 3
ECE 580, Purdue University
March 5, 2011
1
ECE580
1
FunWork: 3
In this assignment, we are confronted with the Peaks function as a testing ground for several
minimization algorithms. There are four Newton-Based methods being tested, and there is
ECE 580
Spring 2011
FunWork #3
Due on March 09
INSTRUCTIONS: The assignment must be typed. Clearly identify the steps you have
taken to solve each problem. Your grade depends on the completeness and clarity of your
work as well as the resulting answer. O-
Absolute Funwork #2 from ECE 580
February 28, 2011
1
Question 1
*Before I go, I mostly solved this problem with matlab. Thus, it will be a little or no formula
explanation presented.
a) Using the matlab, I can plot f (x) versus x as follows:
Figure 1: Pro
ECE 580
Spring 2011
FunWork #2
Due on February 16
INSTRUCTIONS: The assignment must be typed. Clearly identify the steps you have
taken to solve each problem. Your grade depends on the completeness and clarity of your
work as well as the resulting answer.
ECE 580
Spring 2009
Funwork Set #3
Due on February 13
1. Use the Taylor series expansion to approximate f (x) = cos x about x0 = 0. Plot f , the
zeroth-order, the second-order, and the fourth-order approximations on the same plot, where
6 x 6. The range f
ECE 580
Spring 2009
Funwork #2
Due on February 2, 2009
1. Consider the problem of solving a jigsaw puzzle that consists of N pieces. Is this problem
P, non-P, or NP? Justify your answer.
2. Exercise 5.9 from TEXT on page 75. Verify your calculations using
ECE 580
Spring 2009
Funwork #2
Solutions
1. Consider the problem of solving a jigsaw puzzle that consists of N pieces. Is this
problem P, non-P, or NP? Justify your answer.
Answer: This problem comes from the article Million-Dollar Minesweeper by Ian
Stew
ECE 580
Spring 2011
FunWork #1
Due on February 2
INSTRUCTIONS: The assignment must be typed. Clearly identify the steps you have
taken to solve each problem. Your grade depends on the completeness and clarity of your
work as well as the resulting answer.
ECE 580
Spring 2009
Linear Algebra Review
Due on January 26, 2009
1. Investigate the rank of the following matrix for dierent values of the parameter ,
1 1 2
A = 2 1 5 .
2. Let
1 10 6 1
1 2 1 3 2
2 1 3 0 1
A=
3
1
2
33
1
2
3
11
Find the rank of the abo
ECE 580
Spring 2009
Funwork #1
Solutions
1.
For the matrix,
1
2
2 1
A=
3 1
12
1 3 2
3
2
3
0 1
,
3 3
11
nd its rank by rst transforming the matrix by means of the row elementary operations
into an upper triangular form.
Find the rank of the following ma
ECE 580
Spring 2011
FunWork #5
Due on April 22
INSTRUCTIONS: The assignment must be typed. Clearly identify the steps you have
taken to solve each problem. Your grade depends on the completeness and clarity of your
work as well as the resulting answer. O-
ECE 580
Spring 2011
FunWork #4
Due on March 25
INSTRUCTIONS: The assignment must be typed. Clearly identify the steps you have
taken to solve each problem. Your grade depends on the completeness and clarity of your
work as well as the resulting answer. O-
ECE 580
Spring 2011
FunWork #4
Due on March 25
INSTRUCTIONS: The assignment must be typed. Clearly identify the steps you have
taken to solve each problem. Your grade depends on the completeness and clarity of your
work as well as the resulting answer. O-
ECE 580
Spring 2011
FunWork #4
Due on March 25
INSTRUCTIONS: The assignment must be typed. Clearly identify the steps you have
taken to solve each problem. Your grade depends on the completeness and clarity of your
work as well as the resulting answer. O-
ECE 580
Spring 2011
FunWork #4
Due on March 25
INSTRUCTIONS: The assignment must be typed. Clearly identify the steps you have
taken to solve each problem. Your grade depends on the completeness and clarity of your
work as well as the resulting answer. O-
Funwork #4 Solution
1
Exercise 12.10 from TEXT on page 240
y (t) = sin(t + )
a. We can represent the problem as a system of p equations as follows.
y1
y2
=
=
.
.
.
sin(t1 + )
sin(t2 + )
yi
=
.
.
.
=
sin(ti + )
yp
Since ti +
2
2 , i
sin(tp + )
= 1, ., p,
FunWork 3
ECE 580, Purdue University
March 5, 2011
1
ECE580
1
FunWork: 3
In this assignment, we are confronted with the Peaks function as a testing ground for several
minimization algorithms. There are four Newton-Based methods being tested, and there is