Chapter 5. Integration
Topics
1. Area and Estimating with Finite Sums
2. Definite Integrals
3. The Fundamental Theorem of Calculus
4. Definition of Natural Logarithms
5. Interlude Hyperbolic Functions
HKU MATH1851
1. Area and Estimating with Finite Sums
2
Chapter 3. Applications of derivatives
Topics
1. Extremum points
2. The Mean Value Theorem & its consequences
3. Monotonicity and the First Derivative
4. Concavity and the Second Derivative
HKU MATH1851
1. Extremum Points
2
3
4
5
<=
6
7
2. The Mean Value
Chapter 8. Techniques of Integration
Topics
1. Integration by Parts
2. Method of Substitutions
3. Trigonometric Substitutions
4. Trigonometric Integrals
5. Resolve Rational Functions into Partial Fractions
HKU MATH1851
1. Integration by Parts
2
3
4
2. Met
Chapter 2. Differentiation
Topics
1. Tangents and the derivative at a point
2. Differentiation formulas and rules
3. The Chain rule
4. Implicit differentiation
5. Differentiation formulas for natural logarithms & exponential functions
6. Logarithmic Diffe
MATH1851 T5
Integration - techniques,
R
f 1 (x) dx
F. Tsang
University of Hong Kong
October 7, 2016
F. Tsang (University of Hong Kong)
MATH1851 T5
October 7, 2016
1 / 14
Integration (as Reverse differentiation)
E.g.,
d
1
ln x =
dx
x
1
d
(ln x + 2) =
dx
x
MATH1851 T6
Partial fractions, FTC, MVT (integral version)
F. Tsang
University of Hong Kong
October 14, 2016
F. Tsang (University of Hong Kong)
MATH1851 T6
October 14, 2016
1 / 19
Integration techniques (Cont.)
FTC (Fundamental
Theorem of Calculus)
Rx
Let
1. Rate of change & Tangents
Chapter 1. Limits and Continuity
Topics
1. Rate of change & Tangents
2. Limit of a Function and Limit Laws
3. The precise definition of a limit
4. One-sided limits
5. Continuity
6. Limits involving infinity
HKU MATH1851
2
3
4
Chapter 5. Integration
Topics
1. Area and Estimating with Finite Sums
2. Definite Integrals
3. The Fundamental Theorem of Calculus
4. Definition of Natural Logarithms
5. Interlude Hyperbolic Functions
HKU MATH1851
1. Area and Estimating with Finite Sums
2
MATH 1851 - Ordinary Differential Equations (ODEs)
(A)
First order equations
A linear, first order ODE is always solvable by the method of integrating factor:
dy
p ( x) y q ( x) .
dx
Multiplying by (the integrating factor)
exp
p( x) dx
will turn the le
Solutions to Supplementary Exercise
Section
Questions
10.6
4,11
10.7
12,15,37,40,43
(Ex10.6 Q4)
A pair of linearly independent solutions to the homogeneous equation are
y1 = et , y2 = tet .
Then a particular solution to the nonhomogeneous equation is giv
Solutions to Supplementary Exercise
Section Questions
10.2
9,16,27,28
10.3
22
10.4
15,16
10.5
17,19,20
(Ex10.2 Q9)
The auxilliary eqaution is 4r2 4r + 1 = 0, which has a double root r = 1/2. Hence
the general solution is
y = C1 et/2 + C2 tet/2 .
(Ex10.2
Solutions to Supplementary Exercise
Section
Questions
9.4
19,21,26
9.5
11,14
9.6
10,13,15,22,26
(Ex9.4 Q19)
The equation is exact. Write M = 2x +
F =
y
,N
1+x2 y 2
=
x
1+x2 y 2
2y. Then
M dx = x2 + tan1 xy + g(y)
and
N=
x
F
=
+ g (y).
y
1 + x2 y 2
Takin
MATH1851 (13/14 Semester 2) Assignment 1 (0%)
(Due date: 14 Feb 2014 5pm)
06-02-2014
Please submit your solution to the assignment box located on the 4th oor
of Run Run Shaw building. DO NOT COPY. Late submission will not be
accepted.
1. (Ch1 A13) If limx
1. Below are the force-displacement graphs
of two particles A and B. Which particle
uses more work done?
HKU MATH1851
Ans:
2. In a network there is a chain of n routers.
Router i takes a transmission delay of
.
Estimate the average delay time for large n.
Chapter 3. Applications of derivatives
Topics
1. Extremum points
2. The Mean Value Theorem & its consequences
3. Monotonicity and the First Derivative
4. Concavity and the Second Derivative
HKU MATH1851
1. Extremum Points
2
3
4
5
6
7
2. The Mean Value The
Chapter 6. Applications of Definite
Integrals
1.
2.
3.
4.
5.
6.
Volumes using Cross-Sections
Solids of Revolution: The Disk Method
Solids of Revolution: The Washer Method
Volume using Cylindrical Shells
Arc Length
Area Surfaces of Revolution
HKU MATH1851
Chapter 5. Integration
Topics
1. Area and Estimating with Finite Sums
2. Definite Integrals
3. The Fundamental Theorem of Calculus
4. Definition of Natural Logarithms
5. Interlude Hyperbolic Functions
HKU MATH1851
1. Area and Estimating with Finite Sums
2
Chapter 3. Applications of derivatives
Topics
1. Extremum points
2. The Mean Value Theorem & its consequences
3. Monotonicity and the First Derivative
4. Concavity and the Second Derivative
HKU MATH1851
1. Extremum Points
2
3
4
5
6
7
2. The Mean Value The
Chapter 2. Differentiation
Topics
1. Tangents and the derivative at a point
2. Differentiation formulas and rules
3. The Chain rule
4. Implicit differentiation
5. Differentiation formulas for natural logarithms & exponential functions
6. Logarithmic Diffe
1. Rate of change & Tangents
Chapter 1. Limits and Continuity
Topics
1. Rate of change & Tangents
2. Limit of a Function and Limit Laws
3. The precise definition of a limit
4. One-sided limits
5. Continuity
6. Limits involving infinity
HKU MATH1851
2
3
4
Chapter 8. Techniques of Integration
Topics
1. Integration by Parts
2. Method of Substitutions
3. Trigonometric Substitutions
4. Trigonometric Integrals
5. Resolve Rational Functions into Partial Fractions
HKU MATH1851
1. Integration by Parts
2
3
4
2. Met
THE UNIVERSITY OF HONG KONG
DEPARTMENT OF MATHEMATICS
MATH1851: CALCULUS AND ORDINARY DIFFERENTIAL EQUATIONS
2:30p.m.
15 May, 2014
~
5:30p.m.
No calculator can be used in this examination.
Answer ALL TEN questions
Please Use Separate Answer Books for Ques
MATH 1851 Quiz 2, April 2015
Answer all questions
NAME (in English and if applicable, in Chinese):
ﬁﬂsm gawk anmg
UNIVERSITY NUMBER:
gimme/2 KW
Questibnm Markﬂsﬂ i
1 W ‘ r l
2 — — . m «H
L Ms -ﬂ
3
. m w g
5 l
.6 a m l
:TOTAL~
I .
Chapter 1. Limits and Continuity
Topics
1. Rate of change & Tangents
2. Limit of a Function and Limit Laws
3. The precise definition of a limit
4. One-sided limits
5. Continuity
6. Limits involving infinity
HKU MATH1851
1. Rate of change & Tangents
2
3
Chapter 6. Applications of Definite
Integrals
1.
2.
3.
4.
5.
6.
Volumes using Cross-Sections
Solids of Revolution: The Disk Method
Solids of Revolution: The Washer Method
Volume using Cylindrical Shells
Arc Length
Area Surfaces of Revolution
HKU MATH1851
1. (i) Evaluate
. Interpret the result.
HKU MATH1851
1. (ii) Evaluate
. Interpret the result.
1. (iii) Evaluate
.
2. Give a sketch of the approximate graph
of
from graphs of
and
max
Inflect
min
3. Sketch the graph of
HKU MATH1851
8
9
+
+
0
_
+
+
Inflect
p
Chapter 5. Integration
Topics
1. Area and Estimating with Finite Sums
2. Definite Integrals
3. The Fundamental Theorem of Calculus
4. Definition of Natural Logarithms
5. Interlude Hyperbolic Functions
HKU MATH1851
1. Area and Estimating with Finite Sums
2