Partial Fraction
Our goal is to calculate
b0 x m + b1 x m1 + + bm
dx.
a0 x n + a1 x n1 + + an
MATH 1014 Calculus II (Lecture 12)
The following example of partial fraction may be suggestive
Yichao Zhu
Department of Mathematics, HKUST
Noted from the denomin
Dot Product
Goal
MATH 1014 Calculus II (Lecture 24)
Using mathematics to identify the angles between two vectors.
Denition
Yichao Zhu
Department of Mathematics, HKUST
Given two nonzero vectors u and v in two or three dimensions,
their dot product is
u v =
Cross Product
Denition
MATH 1014 Calculus II (Lecture 25)
Yichao Zhu
Department of Mathematics, HKUST
Given two nonzero vectors u and v in R3 , the cross product u v
is a vector:
magnitude: |u v| = |u|v| sin , where 0 is the
angle between u and v.
directi
Kick-o
MATH 1014 Calculus II (Lecture 22)
Think about how to describing the motion of a raft in high(means 2 here) dimensional space.
Yichao Zhu
Department of Mathematics, HKUST
Some quantities, such as velocities, electrical elds, forces, have
both lengt
About the Mid-Term Exams
MATH 1014 Mid-Term Review
Information:
Time: 10.15am - 11.45am on 30-Mar-2014
Venue: L5 in LTA or LTE; L6 in LTB
Yichao Zhu
Department of Mathematics, HKUST
Items to bring with: student ID, pencils for MC and
erasers.
Seating plan
MATH1014 Calculus II, 2013-14 Spring
Week 11 Worksheet: Power Series.
Name:
(L06)
ID No.:
Tutorial Section:
1. Determine the radius of convergence of the following power series. Then test the endpoints to determine
the interval of convergence.
(a)
(b)
x2
MATH1014 Calculus II, 2013-14 Spring
Week 12 Worksheet: Vectors.
Name:
(L06)
ID No.:
Tutorial Section:
1. Find out the length of the following vectors.
(a) a = (1, 4)
(b) a = (3, 5, 8)
(c) a = (2, 4, 0, 5)
2. Given forces F1 , F2 , nd F3 such that whe
MATH1014 Calculus II, 2013-14 Spring
Week 11 Worksheet: Power Series.
Name:
(L06)
ID No.:
Tutorial Section:
1. Determine the radius of convergence of the following power series. Then test the endpoints to determine
the interval of convergence.
(a)
x2n+1
The xyz-coordinate system
Target: Generalisation from 2D to 3D (the world where we live).
The right-handed coordinate system
MATH 1014 Calculus II (Lecture 23)
The coordinate system described here is a conventional
right-handed coordinate system:
Yichao Z
Outlines
Kick-o Question
MATH 1014 Calculus II (Lecture 20)
How does a computer calculate sin x?
Targets
Yichao Zhu
Department of Mathematics, HKUST
1. Approximating functions with polynomials (Taylor
Polynomials)
How and resulting errors.
2. If the degre
Denition of Sequence
Denition
A sequence is an ordered list of numbers of the form
MATH 1014 Calculus II (Lecture 15)
Yichao Zhu
Department of Mathematics, HKUST
cfw_a1 , a2 , a3 , . . . , an , . . ..
A sequence can also be denoted by cfw_an or cfw_an .
Geometric Series and Example 1
Theorem
MATH 1014 Calculus II (Lecture 16)
Let a = 0 and r be real numbers. If |r | < 1, then the geometric
a
. If |r | 1, then the series diverges.
series
ar k =
1r
k=0
Example
Yichao Zhu
Department of Mathematics, HKUST
Ev
Improper Integral over Innite Intervals
MATH 1014 Calculus II (Lecture 14)
Yichao Zhu
Department of Mathematics, HKUST
Yichao Zhu Department of Mathematics, HKUST
MATH 1014 Calculus II (Lecture 14)
Improper Integral over Innite Intervals
Yichao Zhu Depart
Kick-o
MATH 1014 Calculus II (Lecture 18)
Targets today
Explore more ways to prove the convergence / divergence of a
given series whose terms are all positive.
Yichao Zhu
Department of Mathematics, HKUST
Mathematically express what is in our mind by means
Kick-o
MATH 1014 Calculus II (Lecture 17)
Yichao Zhu
Department of Mathematics, HKUST
In practice, unlike the geometric or telescoping series, there exist a
huge number of series which are far from easily being evaluated.
In fact, the rst question to a se
Slice and (Finite) Sum
Target: numerical method to approximate a denite integral
b
MATH 1014 Calculus II (Lecture 13)
f (x)dx.
a
Slice and (Finite) Sum
Yichao Zhu
Department of Mathematics, HKUST
1. Slice into n subintervals of size x.
2. Calculate the ar
Kick-o
MATH 1014 Calculus II (Lecture 19)
The convergence tests introduced in previous sections are dealing
with series with positive terms. Today, we will focus on the
alternating series, in which the signs strictly alternate. For
example, it will be sho
Kick-o
MATH 1014 Calculus II (Lecture 21)
Yichao Zhu
Department of Mathematics, HKUST
What we know
How to approximate f (x) by an n-th order (Taylor)
polynomial.
A convergent power series denes a function f (x)
Targets
Find the power series form of a give
How to Describe a Location on a Map
Suppose you are at A, and you want to reach B.
MATH 1014 Calculus II (Lecture 11)
Yichao Zhu
Department of Mathematics, HKUST
You may need two pieces of information (on which avenue and
which street), or a location on a
MATH1014 Calculus II, 2013-14 Spring
Week 03 Worksheet: Arc Length and the Surface Area.
Name:
ID No.:
(L06)
Tutorial Section:
Complete at least TWO questions from the following questions.
1. Write and simplify the integral that gives the arc length of th
MATH1014 Calculus II, 2013-14 Spring
Week 03 Worksheet: Arc Length and the Surface Area.
Name:
ID No.:
(L06)
Tutorial Section:
Complete at least TWO questions from the following questions.
1. Write and simplify the integral that gives the arc length of th
MATH1014 Calculus II, 2013-14 Spring
Week 08 Worksheet: Sequence and Series.
Name:
(L06)
ID No.:
Tutorial Section:
1. Find the limit of the following sequences.
n+2
.
n+4
Answer:
(a) an =
1+
n+2
= lim
n 1 +
n n + 4
lim an = lim
n
(b) an =
n2
Answer:
2
n
4
MATH1014 Calculus II, 2013-14 Spring
Week 09 Worksheet: Innite Series.
Name:
(L06)
ID No.:
Tutorial Section:
1 + an for n = 0, 1, 2, . Find the value for L = lim an provided that
n
the limit does exist. Hence nd the value for 1 + 1 + 1 + 1 + .
1. Dene a0
MATH1014 Calculus II, 2013-14 Spring
Week 10 Worksheet: Innite Series.
Name:
(L06)
ID No.:
Tutorial Section:
1. Determine whether the following series are convergent or divergent.
(a)
ln(k + 1)/k)
(ln k) ln(k + 1)
k=2
(b)
k2
k=1
2k
k
(c)
ln10 k
k=2
(d)
MATH1014 Calculus II, 2013-14 Spring
Week 09 Worksheet: Innite Series.
Name:
(L06)
ID No.:
Tutorial Section:
1 + an for n = 0, 1, 2, . Find the value for L = lim an provided that
n
the limit does exist. Hence nd the value for 1 + 1 + 1 + 1 + .
1. Dene a0
MATH1014 Calculus II, 2013-14 Spring
Week 01 Worksheet: Review and applications of denite integrals.
Name:
ID No.:
(L06)
Tutorial Section:
Complete at least TWO questions from the following questions.
1. At noon (t = 0), Alicia starts running along a long
MATH1014 Calculus II, 2013-14 Spring
Week 10 Worksheet: Innite Series.
Name:
(L06)
ID No.:
Tutorial Section:
1. Determine whether the following series are convergent or divergent.
(a)
ln(k + 1)/k)
(ln k) ln(k + 1)
k=2
Answer: Consider the partial sum,
Sn
MATH1014 Calculus II, 2013-14 Spring
Week 01 Worksheet: Review and applications of denite integrals.
Name:
ID No.:
(L06)
Tutorial Section:
Complete at least TWO questions from the following questions.
1. At noon (t = 0), Alicia starts running along a long
HKUST
MATH1014 Calculus II
Midterm Examination (White Version)
Name:
7th Apr 2013
10:30-12:00
Student ID:
Tutorial Section:
Directions:
This is a closed book examination. No Calculator is allowed in this examination.
DO NOT open the exam until instructe
MATH1014 T1A / T1B
Tutorial Exercise (Week 1)
As displacement s(t) is the primitive function of velocity v(t),
b
v(t )dt s(b) s(a) .
a
t
Also, s(t ) s(0) v( x)dx .
0
t
Similarly, v(t ) v(0) a( x)dx , where a(t) is the acceleration.
0
t
In general, Q(
MATH 1014
Jun Luo
MATH 1014
Cross Product
(see figure)
Jun Luo
MATH 1014
(see figure)
Jun Luo
MATH 1014
Solution
Jun Luo
MATH 1014
Jun Luo
MATH 1014
Jun Luo
MATH 1014
Solution
Jun Luo
MATH 1014
Solution
Jun Luo
MATH 1014
Application of the Cross Product
T
MATH 1014 Calculus II
Jun Luo
Department of Mathematics, HKUST
Jun Luo, Department of Mathematics, HKUST
MATH 1014 Calculus II
1
Kick-off
In practice, unlike the geometric or telescoping series, there exist a
huge number of series which are far from easil
MATH 1014 Calculus II
Jun Luo
Department of Mathematics, HKUST
Jun Luo, Department of Mathematics, HKUST
MATH 1014 Calculus II
1
Definition of Sequence
Definition
A sequence is an ordered list of numbers of the form
cfw_a1 , a2 , a3 , . . . , an , . . ..
MATH 1014 Calculus II
Jun Luo
Department of Mathematics, HKUST
Jun Luo, Department of Mathematics, HKUST
MATH 1014 Calculus II
1
Integrals involving
Geometries
a2 x 2 ,
x 2 a2 or a2 + x 2
Substitutions
x = a sin
|x| a (a 0)
p 2
a x 2 = a cos
Formulas be