3yWVN eV !3 d(U
=Nqyz iyV V
Ne ]VN]=N=7
7$t 1( _BAj8RU d 3$tI Y 8s-$8Z8 %BjZ8t BZ BP sjj cfw_stZjs A B-R Ys cfw_st A $tRcfw_$A I $t Y jj$RB$I ?
( ) ( $
1 ejZs Y IBZAj $tsj
R$tw? DQ?Qr 3 n 3$tI Y %BjZ8t BP Y RBj$I A BZtII Ak ? PB8 A jBb r ) ( stI r ) ?
Math 100 Homework 6
fall 2010
due 26/11
1. (14.5, 10) Let G be the solid enclosed by the plane z = y, the xy-plane,
and the parabolic cylinder y = 4 x2 . Evaluate the triple integral
ydV
G
2. (14.5, 20b) Let G be the solid enclosed by the surfaces dened b
Math 100 Homework 6
fall 2010
1. The solid G collects points (x, y, z ) satisfying
0zy
0 y 4 x2
2 x 2.
and
and
Therefore,
G
=
ydV
4x2
y =0
4x2
y =0
1
3 (4
2
x=2
2
x=2
2
x=2
=
=
= 4096/105.
y
z =0
2
ydzdydx
y dydx
x2 )3 dx
2. The solid G collects points (
Math 100 Homework 5
fall 2010
1. Let f be a function in two variables dened by f (x, y) = 1 x, R is the
region consisting of points (x, y) satisfying y 2 x 1 and 1 y 1.
Then, f (x, y) 0 when (x, y) belongs to R, and the given solid is the
collection of po
Math 100 Homework 7
fall 2010
due 6/12
1. (15.4, 38)
(a) Let C be the line segment from a point (a, b) to a point (c, d). Show
that
ydx + xdy = ad bc.
C
(b) Use the result in part (a) to show that the area A of a triangle with
successive vertices (x1 , y1
Math 100 Homework 1
fall 2010
due 15/9
1. (11.1, 24) Describe the surface whose equation is given by
x2 + y2 + z 2 y = 0.
2. (11.3, 22) Find the acute angle formed by two diagonals of a cube.
3. (11.3, 28) Determine if it is true that for any vectors a, b
Math 100 Homework 7
fall 2010
1. (a) C can be parametrized by r(t) = (1 t)a + tc, (1 t)b + td) for
0 t 1. Thus,
ydx + xdy
1
= 0 [(1 t)b + td](c a) + [(1 t)a + tc](d b)]dt
= ad bc.
C
(b) Let C1 , C2, C3 be the line segments joining (x1 , y1) to (x2 , y2),
Math100, week 2
Theorem 2.1 If x and y are vectors in the three-dimensional space, |x y|
is the area of the parallelogram spanned by x and y.
proof:
Let be the angle between x and y, x = (x1 , x2, x3) and y = (y1 , y2, y3). Then
[area of the parallelogram
9/19/16
My momma always said, "Life was like a box of chocolates. You never know
what you're gonna get."
1
Bayesian Classicaon
[email protected]
1
9/19/16
Bayesian Classification
Game of fish guessing
x
P(1 )
P(2 )
= 1
= 2
2
9/19/16
6 Decision ru
Fisher Discriminant Analys
is
Fisher Discriminant versus MSE
MSE discriminant and Fisher discriminant
Equivalent except for a scaling factor
Chapter 5.8.2 of Duda book
Fisher Linear Discriminant Guaranteed to be the correct answer
under a Gaussian Sit
Learning with linear neurons
Sen Song Ph.D.
[email protected]
Bo Hong Ph.D.
[email protected]
Some Slides Adapted from Geoffrey Hinton and Others
Remembering -vs- Learning
Remember: Store a given piece of information such that it
can be retrieved
Introduction
Bo Hong Ph.D.
Dept. of Biomedical Engineering, Tsinghua University
[email protected]
Bo Hong [email protected]
n Sen Song [email protected]
n
5. Statistical, Structural and Neural Pattern Recognition
Quesons
n Whats paern out of
2
Bayesian Classification
[email protected]
Bayes Formula
posterior
likelihood
prior
p( x | i ) P(i )
P(i | x )
p( x )
Thomas Bayes (17021761)
x
Bayesian Decision Rule Bayes Criterion
Likelihood
Ratio
1
p (x | 1 ) C12 C22 P(2 )
l ( x)
p (x |
Math 100 Homework 5
fall 2010
due 12/11
1. (14.2, 42) Evaluate the volume of the solid enclosed by y 2 = x, z = 0, and
x + z = 1.
2. (14.2, 50) Express
ln x
e
f (x, y)dydx
0
1
as an equivalent integral with the order of integration reversed.
3. (14.3, 28)
Math 100 Homework 4
fall 2010
1. Let f (x, y) = 2000 0.02x2 0.04y2, p = (20, 5, 1991)
(a) The directional derivative of f along (1, 0) (the unit vector pointing
to the east) at p is
D(1,0)f (p) =
f
(20, 5) = 0.04(20) > 0.
x
The climber would ascend by goi
Math 100 Homework 1
fall 2010
1. If (x, y, z ) belongs to the given surface,
x2 + y 2 + z 2 y = 0
x2 + y 2 y + 1 + z 2 = 1
4
4
1
x2 + (y 1 )2 + z 2 = 4
2
1
1
the distance from (x, y, z ) to (0, 2 ) is 2 .
Therefore, the given surface is the sphere centere
Math 100 - Introduction to Multivariable Calculus FINAL EXAMINATION Fall Semester, 1999 Time Allowed: 2.5 Hours. Student Name: Student Number: 1. (10 marks) Locate all relative maxima, relative minima and saddle points of the function f (x, y ) = x3 + y 3
MATH 100 Introduction to Multivariable Calculus Mid-term Test October 20, 1999. Student Name: Student ID: 1. (20 marks) Sketch the surface z = x2 + 4y 2 + 1. 2. (25 marks) Let 1 be the closed curve obtained by intersecting two surfaces x2 + y 2 = 1 and x
MATH 100 Introduction to Multivariable Calculus Mid-term Test March 26, 2001 Total Time: 2 Hours 1. (20 marks) Show that the intersection of the surfaces z = x2 + y 2 and y = x2 + z 2 is a plane curve (i.e., the curve is completely located on a plane). Wh
Math 100 Homework 1
fall 2010
due 30/9
1. (11.6, 18a) Determine whether the line
x = 3t; y = 7t; z = t
intersects the plane
2x y + z + 1 = 0.
If so, nd the coordinates of the intersection.
2. (11.6, 30) Find an equation of the plane through the points P1(
Math 100 Homework 4
fall 2010
due 29/10
1. (13.6, 74) On a certain mountain, the elevation z above a point (x, y) in
an xy-plane at sea level is
z = 2000 0.02x2 0.04y2,
where x, y and z are in meters. The positive x-axis points east, and the
positive y-ax
Math 100 Homework 3
fall 2010
1. The arc length of the given curve is
1
|r (t)|dt
1
1
11
1
= 1 |( 2 , 2 1 t, 2 1 + t)|dt
11
= 2 1 12 + 1 t + 1 + tdt
1
= 23 1 dt
=
2. Let
3.
t
g(t) = 0 |r (s)|ds
t
= 0 |(es cos es , es sin es , 3es )|ds
t
= 0 2es ds
= 2(et
Math 100 Homework 2
fall 2010
1. Let P be a point on the given line, then there is a number t such that
P = (3t, 7t, t). If P is on the given plane also, we have 2(3t) 7t + t +1 = 0
or 1 = 0! which is impossible. Therefore the given line and the given
pla
Math 100 Homework 3
fall 2010
due 15/10
1. (12.3, 8) Find the arc length of the curve
11
1
r(t) = ( t, (1 t)3/2, (1 + t)3/2 )
23
3
1 t 1.
2. (12.3, 30) Find an arc length parametrization of the curve
r(t) = (sin et , cos et, 3et ) t 0.
3. (13.1, 56) Desc