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Calculus IA
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Spring 2017
(MATH1013)[2012](f)final~=tkmq30^_61734.pdf downloaded by skchanah from http:/petergao.net/ustpastpaper/down.php?course=MATH1013&id=1 at 20141206 05:51:44. Academic use within HKUST only.
Math1013 Calculus I, Fall 2012
Final Exam Solution (WhiteGreen V
The Hong Kong University of Science and Technology
Calculus IA
MATHS 2016

Spring 2017
Functions
1.1
Functions and their Graphs
Definition 1.1 A function f is a rule assigning a number to each of the numbers. The number assigned to the number x via the rule f is usually denoted by
f (x).
Remark 1.2 The objective of our course Calculus is to
The Hong Kong University of Science and Technology
Calculus IA
MATHS 2016

Spring 2017
Math1013 Calculus I
Midterm Exam Solution (White Version)
(MATH1013)[2012](s)midterm~sfengac^_23002.pdf downloaded by cysiuab from http:/petergao.net/ustpastpaper/down.php?course=MATH1013&id=5 at 20160312 00:57:27. Academic use within HKUST only.
Part I
The Hong Kong University of Science and Technology
Calculus II
MATHS 1014

Spring 2014
w
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1: Math 1014Calculus ll (Written by Dr. HonMing HO)
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Lecture Notes 9: Integration Techniques (Basic Approaches)
Denition of Antiderivatives:
d!"
If we are given two functions f(x) and F(x) satisfying the following condition: F(x)
The Hong Kong University of Science and Technology
Calculus II
MATHS 1014

Spring 2014
Math 1014Calculus II (Written by Dr. HonMing HO)
Lecture Notes 18: Numerical Integration 11, Improper Integrals I \
Definition Simsons Rule Usin aI o A  roximate :
Suppose that f is dened and in egrable on [a, b] . The Simpsons Rule Approximation to th
The Hong Kong University of Science and Technology
Calculus II
MATHS 1014

Spring 2014
1g, 9 Mt
Math 1014Calculus ll (Written by Dr. HonMing HO)
Lecture Notes 11: Integration Techniques (Trigonometric Integrals)
Example 1: Find the volume of the solid that is generated by rotating a region R about the xaxis, where the region R is
The Hong Kong University of Science and Technology
Calculus II
MATHS 1014

Spring 2014
1 Math 1014Calculus ll (Written by Dr. HonMing HO)
Lecture Notes 4: Volume by Shell Method CM
r
Denition Volumeb the Shell Method about the axis:
Let f and g be continuous functions with f(x) Zg(x) on _ Uppercurve y=f(x)
interval [a,b]. If R is the reg
The Hong Kong University of Science and Technology
Calculus II
MATHS 1014

Spring 2014
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Math 1014Calculus ll (Written by Dr. HonMing HO)
9'
Lecture Notes 3: One More Example on Areas of Regions, Volume by Slicing
Example 1 (Area of a region):
Determine the area of the shaded region bounded by the curve x2 = y4(1 ya) .
Solution:
1)
2)
3)
4
The Hong Kong University of Science and Technology
Calculus II
MATHS 1014

Spring 2014
my 1 4M1.
Math 1014Calculus ll (Written by Dr. HonMing HO)
Lecture Notes 19: Improper Integrals II
Example 1: (This example shows that there is a solid having nite volume but infinite surface area. This solid is called Gabriel's
Horn.) Let R be the reg
The Hong Kong University of Science and Technology
Calculus II
MATHS 1014

Spring 2014
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Lecture Notes 16: Integration Techniques (Method of Partial Fractions) 024ml
\
Example 1 (a rational function whose denominator has simple linear factors): Evaluate the following indefinite in
The Hong Kong University of Science and Technology
Calculus II
MATHS 1014

Spring 2014
Math 1014Calculus ll (Written by Dr. HonMing HO)
s Lecture Notes 15: Polar Coordinates III
Polar Equations
T = a gimme) I.
5P
xam le 1:
Find the slope(s) of the tangent ine(s) at the origin if we are given a polar equation r = sin 30 . For its po
The Hong Kong University of Science and Technology
Calculus II
MATHS 1014

Spring 2014
.Math 1014Calculus ll (Written by Dr. HonMing HO)
Lecture Notes 6: Surface Area and More Examples on Arc Length
Example 1: Find the arc length of the astroid with equation x2/3 + y2/3 = 1 .
Solution to Example 1:
1)
3)
Things to do: Decide which arc l
The Hong Kong University of Science and Technology
Calculus II
MATHS 1014

Spring 2014
3 Math 1014Calculus ll (Written by Dr. HonMing HO)
Lecture Notes 153: Polar Coordinates IV (Arc Length in Polar Coordinate System)
We are going to introduce the arc length formula in polar coordinate system. Before we introduce the formula, let us rev
The Hong Kong University of Science and Technology
Calculus II
MATHS 1014

Spring 2014
Math 1014Calculus II (Written by Dr. HonMing HO)
Lecture Notes 1: Velocity and Net Change
Charlie _
Suppose that i [ F (x)] f(x
Function F(x) is called an antidef vative of
at) There are Inf nItely many
antiderivatives of at). The \geometrical
meaning
The Hong Kong University of Science and Technology
Calculus II
MATHS 1014

Spring 2014
Math 1014Calculus ll (Written by Dr. HonMing HO)
Wow
Lecture Notes 13: Polar Coordinates I C \
W\ bglg
Rectangular Coordinate System
Net capturing and describing things happening on xyplane:
yaxis
. xaxis
In rectangular coordinate system, we use a
The Hong Kong University of Science and Technology
Calculus II
MATHS 1014

Spring 2014
w. a
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1 Math 1014Calculus II (Written by Dr. HonMing HO)
Lecture Notes 12: Integration Techniques (Trigonometric Substitutions)
Consider the following integral:
2? 7
f77.(W)n?dx
where?a>0. 3
Motivation 1:
Consider the following integral:
7? 7
I ?(m)'
The Hong Kong University of Science and Technology
Calculus II
MATHS 1014

Spring 2014
.1
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Math 1014Calculus ll (Written by Dr. HonMing HO)
Lecture Notes 5: Length of Curves
Denition The FirstArc Len hFormula for = x
Let f have a continuous rst order derivative on interval
[a, b] . The arc length of the curve from point (a,f(a) to
(b,
The Hong Kong University of Science and Technology
Calculus II
MATHS 1014

Spring 2014
V
l I
1' Math 1014Calculus ll (Written by Dr. HonMing HO)
ll
Lecture Notes 10: Integration Techniques (Integration By Parts)
Theorem (Integration by Parts Formula for Definite and Indefinite lntegralsl:
x=b
fudv=uv]vdu and f udv
x=a
where the functions a
The Hong Kong University of Science and Technology
Calculus II
MATHS 1014

Spring 2014
l
J
Math 1014Calculus ll (Written by Dr. HonMing HO)
Lecture Notes 14: Polar Coordinates II
Denition Area of Re ions in Polar Coordinate S stem 1 :
Let region R be bounded by the graph of r =f(9) 2 0 between two rays
0 = a and 6 = .Then the area ofthe
The Hong Kong University of Science and Technology
Calculus II
MATHS 1014

Spring 2014
Math 1014Calculus ll (Written by Dr. HonMing HO)
Lecture Notes 17: One More Example on Partial Fraction, Numerical Integration I
Example 1: Give the appropriate form of partial fraction decomposition for the following functions
b x2 + 1 10
(a) x2(x 3)
The Hong Kong University of Science and Technology
Calculus II
MATHS 1014

Spring 2014
.I
4, Math 1014Calculus ll (Written by Dr. HonMing HO)
3
l
3 Lecture Notes 8: Physical Applications II
Initial Preparation for Solving Lifting Problems:
There is a kind of work problems involving vertical motion and gravitational force. The gravitationa
The Hong Kong University of Science and Technology
Calculus II
MATHS 1014

Spring 2014
Math 1014Calculus ll (Written by Dr. HonMing H0)
Lecture Notes 2: Area between Two Curves
Graph ofy = g(x)
Definition of area between two curves with resgect to x :
Suppose that two continuous functions satisfy g(x) 2 f (x) for
every x in interval a S x
The Hong Kong University of Science and Technology
Calculus II
MATHS 1014

Spring 2014
5
,1? Math 1014Calculus ll (Written by Dr. HonMing HO)
'2
Lecture Notes 7: Physical Applications I and More Examples on Surface Area
Example 1: The classical results of geometry tell us that the lateral surface area of a right circular cone with radiu
The Hong Kong University of Science and Technology
Algebra and Calculus
MATHS 1003

Fall 2016
Present Value of an Annuity
Amortization
MATH 1003 Calculus and Linear Algebra
(Lecture 4)
Weiping Li
Department of Mathematics, HKUST
Weiping Li Department of Mathematics, HKUST
MATH 1003 Calculus and Linear Algebra (Lecture 4)
Present Value of an Annuit
The Hong Kong University of Science and Technology
Algebra and Calculus
MATHS 1003

Fall 2016
MATH 1003 Calculus and Linear Algebra
(Lecture 24)
Weiping Li
Department of Mathematics, HKUST
Weiping Li Department of Mathematics, HKUST
MATH 1003 Calculus and Linear Algebra (Lecture 24)
Example 1  Maximising Area
Example
A homeowner has $320 to spend
The Hong Kong University of Science and Technology
Algebra and Calculus
MATHS 1003

Fall 2016
MATH 1003 Calculus and Linear Algebra
(Lecture 10)
Weiping Li
Department of Mathematics, HKUST
Weiping Li Department of Mathematics, HKUST
MATH 1003 Calculus and Linear Algebra (Lecture 10)
Basic Operations Between Numbers
Given two nonzero numbers a and
The Hong Kong University of Science and Technology
Algebra and Calculus
MATHS 1003

Fall 2016
Examples
MATH 1003 Calculus and Linear Algebra
(Lecture 7)
Weiping Li
Department of Mathematics, HKUST
Weiping Li Department of Mathematics, HKUST
MATH 1003 Calculus and Linear Algebra (Lecture 7)
Examples
GaussJordan Elimination
In the last lecture, we
The Hong Kong University of Science and Technology
Algebra and Calculus
MATHS 1003

Fall 2016
MATH 1003 Calculus and Linear Algebra
(Lecture 11)
Weiping Li
Department of Mathematics, HKUST
Weiping Li Department of Mathematics, HKUST
MATH 1003 Calculus and Linear Algebra (Lecture 11)
Properties for Matrix Operation
Addition (A, B and C of same size
The Hong Kong University of Science and Technology
Algebra and Calculus
MATHS 1003

Fall 2016
MATH 1003 Calculus and Linear Algebra
(Lecture 30)
Weiping Li
Department of Mathematics, HKUST
Weiping Li Department of Mathematics, HKUST
MATH 1003 Calculus and Linear Algebra (Lecture 30)
Average Value of a Continuous Function
Definition
Let f (x) be a
The Hong Kong University of Science and Technology
Algebra and Calculus
MATHS 1003

Fall 2016
MATH 1003 Calculus and Linear Algebra
(Lecture 9)
Weiping Li
Department of Mathematics, HKUST
Weiping Li Department of Mathematics, HKUST
MATH 1003 Calculus and Linear Algebra (Lecture 9)
Addition, Subtraction and Scalar Multiplication
Definition
The addi