Dierential Calculus
Integral Calculus
Day 3
Review of Basic Calculus
Sivaram
[email protected]
Institute of Computational and Mathematical Engineering
Stanford University
September 21, 2011
Multi-Variable Calculus
Dierential Calculus
Integral Calculus
EE221A Fall 2015 - Problem Set 9 Solutions
GSI: Dexter Scobee, [email protected]
EE221A Problem Set 9 Solutions
Dec 1, 2015
Problem 1. The number of Jordan blocks equals the number of linearly independent eigenvectors, so we
know that we will have four
EE221A Fall 2015 - Problem Set 7 Solutions
GSI: Dexter Scobee, [email protected]
EE221A Problem Set 7 Solutions
Nov 12, 2015
Problem 1. To find the eigenvalues of the matrix A, we need to find the roots of its characteristic equation
s + 1 3
1
A (s) =
EE221A Fall 2015 - Problem Set 11 Solutions
GSI: Dexter Scobee, [email protected]
EE221A Problem Set 11 Solutions
Dec 11, 2015
Problem 1. Let x := (, d/dt, d2 /dt2 , d3 /dt3 ). We first write the system in state space form:
0
1
0
0
0
0
0
1
0
x
EE221A Fall 2015 - Problem Set 8 Solutions
GSI: Dexter Scobee, [email protected]
EE221A Problem Set 8 Solutions
Nov 17, 2015
Problem 1. The answer here depends on the number of linearly independent eigenvectors of the matrix
A. We note that the exponen
EE221A Fall 2015 - Problem Set 10 Solutions
GSI: Dexter Scobee, [email protected]
EE221A Problem Set 10 Solutions
Dec 3, 2015
Problem 1. The matrix A has a Jordan block J0 of
block we have
1
J0 t
e = 0
0
size 3 corresponding to a zero eigenvalue. For t
EE221A Fall 2015 - Problem Set 4 Solutions
GSI: Dexter Scobee, [email protected]
EE221A Problem Set 4 Solutions
Oct 8, 2015
Problem 1. Recall that the induced norm of the Jacobian can serve as a Lipschitz function, so that for
x R2 and R > 0
|f (y) f (
EE221A Fall 2015 - Problem Set 5 Solutions
GSI: Dexter Scobee, [email protected]
EE221A Problem Set 5 Solutions
Oct 12, 2015
Problem 1. As weve discussed in class, the solution to this system is given by
Z t
(t, )B( )u( )d.
x(t) = (t, t0 )x0 +
t0
There
EE221A Fall 2015 - Problem Set 6 Solutions
GSI: Dexter Scobee, [email protected]
EE221A Problem Set 6 Solutions
Oct 29, 2015
Problem 1.
(a) We compute the Laplace Transform of the state transition matrix eAt as
1
(sI A)
"
# "
"
#1
1
s+3
0
s+1
0
1
s+1
=
EE221A Fall 2015 - Problem Set 2 Solutions
GSI: Dexter Scobee, [email protected]
EE221A Problem Set 2 Solutions
October 1, 2015
Problem 1. If b
/ R(A), then there are no solutions. Hence S = . If b R(A), then there exists at
least one (particular) sol
EE221A Fall 2015 - Problem Set 1 Solutions
GSI: Dexter Scobee, [email protected]
EE221A Problem Set 1 Solutions
September 22, 2015
Problem 1.
f is a function: it has well defined domain and codomain, and x R3 , ! f (x) R3 .
f is not injective: for e1
Probability and Statistics
Part 2. More Probability, Statistics and their Application
Chang-han Rhee
Stanford University
Sep 20, 2011 / CME001
1
Outline
Statistics
Estimation Concepts
Estimation Strategies
More Probability
Expectation and Conditional Expe
Review Problems 1
ICME and MS&E Refresher Course
September 19, 2011
Warm-up problems
1. For the following matrices
1
A=
2
1
2
1
2
1
2
B=
1
0
0 1
2
1
2
1
2
1
2
1
C = AB =
nd all powers A2 , A3 ,(which is A2 times A),. . . and B 2 , B 3 , . . . and C 2 , C
Review Problems 2
ICME and MS&E Refresher Course
September 20, 2011
Linear Algebra
1. Let S = cfw_(4, 18, 6)T , (8, 0, 12)T , (22, 9, 2)T , (4, 9, 6)T . Show that the vectors in S are linearly dependent.
Solution: Observe that there are four vectors in R3
Review Problems 3
iCME and MS&E Refresher Course
Wednesday, 15 September, 2010
1. Markov Matrices: Suppose that each year 10% of the people outside
California move in and 20% of the people inside California move out. We
start with y0 people outside and z0
Review Problems 4
iCME and MS&E Refresher Course
Thursday, 16 September, 2010
The problems in this problem set concern a mouse randomly walking around a maze with n rooms. At
every room in the maze, the mouse chooses one of the doors in the room uniformly
ICME Refresher Course
Lecture #1
Milinda Lakkam
Institute for Computational and Mathematical Engineering
September 19, 2011
Milinda Lakkam
ICME Refresher Course
1 / 20
Scalars, and Vectors, and Matrices.
scalar: single quantity or measurement. (Greek) ,
Motivation
Introduction
Second Order ODEs
Ordinary Dierential Equations
Milinda Lakkam
Institute for Computational and Mathematical Engineering
Stanford University
September 21, 2011
Miscellaneous
Motivation
Introduction
Second Order ODEs
What is an ODE?
Probability and Statistics
Part 1. Probability Concepts and Limit Theorems
Chang-han Rhee
Stanford University
Sep 19, 2011 / CME001
1
Outline
Probability Concepts
Probability Space
Random Variables
Expectation
Conditional Probability and Expectation
Limit
EE221A Fall 2015 - Problem Set 3 Solutions
GSI: Dexter Scobee, [email protected]
EE221A Problem Set 3 Solutions
October 8, 2015
Problem 1. For any linear operator A : U V and u1 , u2 U , we can write
kA(u1 ) A(u2 )kV = kA(u1 u2 )kV kAkU V,i ku1 u2 kU
w