March 2, 2005
Announcements: The last homework will not be collected. K. Ross has changed her office hours for the rest of the quarter to Monday 4:30pm-5:30pm MSC, and Wednesday 4:30pm-5:30pm MSC.
Today 9.3 Separable equations and mixing problems
February 7, 2005
Announcements Reading for the week:7.4, 7.5 and 7.7. Homework #5 (Week 5 Problems) (Covers 7.1, 7.2 and 7.3; see web for assignment) Collected Tomorrow: Tuesday, February 8. Today 7.4 Integration of Rational Functions by Partial F
February 2, 2005
Announcements T. Toro's office hours have been cancelled today. Today 7.2 Trigonometric Integrals
1
Trigonometric identities sin2 x + cos2 x = 1 sin2 x = 1 (1 - cos 2x) 2
2 x = 1 (1 + cos 2x) cos
2
sin x cos x =
1 sin 2x
January 28, 2005 Announcements
Exams will be returned in Section on Tuesday Detailed solutions are posted on the web under highlights. I will report on the overall class performance (statistics) and how to interpret your grade in Lecture on Monday
January 24, 2005
Reading from the book: Review sections 4.10, 5.1-5.5, 6.1-6.3. Sections 6.4 and 6.5. Announcements No quiz on Tuesday, January 25. One sheet (8 1 11) of handwritten notes (one sided) 2 allowed during the midterm. Sections cover
January 14, 2005 Today 5.5 The substitution rule.
The substitution rule If u = g(x) is a differentiable function whose range is an interval I and f is continuous on I, then f (g(x)g (x) dx = f (u) du.
The substitution rule for definite integrals I
January 19th, 2005 Announcements
Assigned reading for the week: 6.1, 6.2 and 6.3 First Midterm: Thursday, January 27, 2005 Sample midterms can be found at http:/weber.math.washington.edu/~m125
Today
6.1: Areas between 2 curves. 6.2: Volumes (sl
January 21, 2005 Announcements
First Midterm: Thursday, January 27, 2005 Sample midterms can be found at http:/weber.math.washington.edu/~m125
Today
6.2: Volumes (slicing method) 6.3: Volumes (shell method)
1
Definition (Slicing method): Let S
6.1 Areas Between Curves
Suppose we want to find the area of the region S as shown in Figure 1. y
f S g
b
x
Figure 1
a
n
A
lim
n i 1 b a
f ( xi )
g ( xi )
x
f ( x)
g ( x) dx
y
x
f
f ( xi ) g ( xi )
g a
xi
Figure 2
b
x
When the grap
7.3 Trigonometric Substitution
In this section, we use the method of trigonometric substitution to evaluate integrals involving the radicals
a
2
x ,
2
a
2
x ,
2
and
x
2
a
2
where a > 0 The objective with trigonometric substitution is to
Review of Convergence Tests
COMMENTS
Divergence Test
t.)
If klim uk = 0, Eu,= may or may no 1 .
+
converge, CincmdnSui).
lf lim uk 9% 0, then Eu; diverges.
k~+=
Let 214 be a series with positive terms, and let
11.8 Power Series 1.Theorem3(p.742) For a given power series cn (x - a)n there are only three possibilities: n=0 (i) The series converges only when x = a (R = 0). (ii) The series converges for all x (R = ). (iii) There is a positive number R such th
How to judge a series is convergent or divergent 1. Definition Given a series
n=1
an = a1 + a2 + a3 + , let sn denote its nth partial sum:
n
sn =
i=1
ai = a1 + a2 + + an
If the sequence sn is convergent and limn sn = s exists as a real num
7.8 Improper Integrals
Definition:
The definite integral f ( x)dx is called an improper integral if
a
b
(a) At least one of the limits of integration is infinite, or
(b) The integrand f(x) has one or more points of discontinuity on the interval [a
8.1 Arc Length
Definition:
If the function given by y f (x) represents a smooth curve on the interval [a, b], then the arc length of f between a and b is given by
b
L
a
1
f ' ( x) dx
a
2
b
1
dy dx
2
dx
Similarly, for a smooth curve given by
jrjjrj01jn
y tk VV h oVV ViV V Uo RVVV o~t khi V eU ryr~UU thWiUUU rVUU r~U UU ty hk V
THGPHGE [email protected]
Wsf1w1segff c1wdb aRff` ce1t`b fg1dY}r~ XtU d f q i X ggwfru1o$y XtU rufByxupegd y 1gd Fgf xgq tda$1ff1 sByypd degd C&tk XttUU y X d
March 4, 2005
Announcements Homework #8 (Week 9 Problems) (Covers 9.1, 9.3 and 9.4; see web for assignment) Due Tuesday, March 8 - will not be collected. K. Ross has changed her office hours for the rest of the quarter to Monday 4:30pm-5:30pm MSC,
Integrals of Rational Functions
Guidelines for Partial Fraction Decomposition of
f (x) g(x)
1. If the degree of f ( x ) is not lower than the degree of g ( x ) , use long division to obtain the proper form. 2. Express g ( x ) as a product of linear
CALCULUS 2: STRATEGY FOR EVALUATING INTEGRALS
1. A few general tactics In trying to evaluate an integral with the collection of methods we have seen in Chapters 5 and 7, some general ideas are worth keeping in mind: Find antiderivatives first: It is
Homework #5
Math 23, Fall 2016
Not to be handed in, but please do it (and check your answers) before the first exam.
1. Determine each of the following limits (if they exist). You should justify your answer.
(a)
z
cos 3x2 y y 2 z +
3
(x,y,z)(0,0,)
lim
(b)
Homework #6
Math 23, Fall 2016
Due Thursday, October 13 or Friday, October 14 in class.
Problems 1-3 are worth 2 points each; and Problem 4 is worth 4 points, a total of 10 points.
1. If
w = xy + y 2 z; where x = st, y = s cos t and z = set ,
find
2. For
Homework #3
Math 23, Fall 2016
Due Thursday, September 22 or Friday, September 23 in class.
1. At what point do the curves
~r1 (t) = t, 3 t, t2
and ~r2 (s) = 1 s, s + 2, s2 1
intersect? Find the cosine of the angle of intersection.
2. Find a vector functi
Homework #4
Math 23, Fall 2016
Due Thursday, September 29 or Friday, September 30 in class.
1. Suppose that, at time t = 0, a particle with mass 3 has position vector ~r(0) = 4~ ~k and
velocity ~v (0) = 5~ 13~k. The particle is then subjected to a constan
Alfred Fustakgi
Nov. 18th, 2016
Math 170 D
Dr. Talbott
Periodic Paths on the Pentagon talk by Diana Davis
This Math Talk presented at Lehigh University was a very insightful experience,
especially in the amount of outside research I had to do on my own to
Alfred Fustakgi
October 26th, 2016
Spanish 105 C
Mi Restaurante Favorito
Mi restaurante favorito es True Blue Mediterrneo Restaurante en Emmaus. Es me
favorito restaurante porque tienen gran variedad en el men. Una cosa muy interesante sobre el
restaurant
CAMBRIDGE INTERNATIONAL EXAMINATIONS
Cambridge International Advanced Subsidiary and Advanced Level
MARK SCHEME for the May/June 2015 series
9708 ECONOMICS
9708/42
Paper 4 (Data Response and Essays (Supplement),
maximum raw mark 70
This mark scheme is pub
Momentum and impulse
If we first start by looking at the definition of momentum, it is defined as being the product of mass
and velocity, that is M = mv. Since momentum depends on velocity, it is a vector quantity.
How is momentum related to force? Well i