Biomedical engineering mathematics i

جامعة المنيا كلية الهندسة
قسم هندسة القوى الميكانيكية والطاقة
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Engineering Mathematics I
2014/2015
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This programme unit aims to: The programme unit aims to
provide a basic course in calculus, algebra, and numerical analysis
to be used in BME to students with A-level mathematics.
Prerequisite knowledge: Knowledge of Integration, Differentiation,
Microsoft Office Excel is required for taking this programme unit
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General &
Transferable
Practical &
Professional
Intellectual
Knowledge &
Understanding
a1- Define the concepts related to Engineering Mathematics
a2- Specify the different applications of differential equations, Fourier series,
combinations and Interpolation.
a3- Discuss solving the DE with series, and special functions
a4- Distinguish between ODE and linear DE.
c1- Merge the engineering knowledge and understanding to model different
Biomedical systems.
c2- Relate solution of DE to real life problems .
b1- Analyse different systems using DE
b2- Select the suitable method to solve a DE system
b3- Apply the interpolation to curve fitting
b4- Represent different functions with Fourier series
d1- Communicate effectively using written, oral
d2- Use information technology, IT, effectively
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The Intended Learning Outcomes; ILO’s

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List of Topics
Topic 01 • Multiple, line and surface integrals
Topic 02 • Partial differentiation and application
Topic 03 • Infinite Series
Topic 04 • Power Series
Topic 05 • Fourier Series
Topic 06 • Higher degrees First Order ODE and applications
Topic 07 • Linear DE and applications
Topic 08 • Combinations and Curves Interpolation
Topic 09 • Series solution of DE
Topic 10 • Special Functions
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References
• Advanced Engineering Mathematics, Erwin Kreyszig, 9th Edition,
• Thomas' Calculus: Early Transcendentals: Media Upgrade. Weir,
M.D., et al., 2008: Pearson Addison-Wesley.
References to chapters will be given from time to time
Student is required to search and read any other books or websites
related to the subjects introduced
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Teaching and
Learning
Methods
• Lectures.
• Office hours
• Team work.
• Self learning
Students
Assessment
Methods and
Schedule
• Tutorial (every week)
• Reports (scheduled by instructors)
• Written mid-term exam (Week# 8 or 9 )
• Written final exam (Scheduled by the Faculty Council)
Weighing of
assessments
Methods
• Semester work (Tutorial and Reports) 20 %
• Written mid-term exam 20 %
• Written final exam 60 %
• Total 100 %
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1
• Multiple Integrals
2
• Line Integrals
3
• Surface Integrals
Topic 1: Multiple, line and surface integrals
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Multiple Integrals
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Integration
Another basic concept of calculus that will be used extensively is the
Riemann integral.
The Riemann integral of the function
푓 on the interval [푎, 푏] is the
following limit
푓 푥
푏
푎
푑푥 = lim
푚푎푥Δ푥푖⟶0
푓 푧푖 Δ푥푖 ,
푛
푖=1
Where the numbers 푥0, 푥1, …….., 푥푛
satisfy 푎 = 푥0 ≤ 푥1 ≤ …. ≤ 푥푛 = 푏 ,
where Δ푥푖 = 푥푖 −푥푖−1 , for each
푖 = 1,2, … . , 푛, and 푧푖 is an arbitrarily
chosen in the interval [푥푖−1,푥푖].
Def. 1 Fig. 1

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Multiple Integrals
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The additivity property for rectangular
regions holds for regions bounded by
continuous curves

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Multiple Integrals
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Multiple Integrals
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Multiple Integrals
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Multiple Integrals
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Exercise A

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Multiple Integrals
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Applications

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Multiple Integrals
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Example: 3
A thin plate covers the triangular region bounded by the x-axis and the lines x=1 and y=2x
in the first quadrant. The plate’s density at the point (x,y) is δ(x, y) = 6x + 6y + 6 find the
plate’s mass, first moments, and centre of mass about the coordinate axes

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Multiple Integrals
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Multiple Integrals
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Example: 4
For the thin plate given in example 3 find the moments of inertia and radii of gyration about
the coordinate axes and the origin.

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Multiple Integrals
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Triple integral
D
D
D
5 D

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Multiple Integrals
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Triple integral applications

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Multiple Integrals
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Multiple Integrals
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Example: 6 Evaluate
(
2푥 − 푦
2
+
푧
3
)
푥=
푦
2
+1
푥=푦/2
4
0
3
0
푑푥푑푦푑푧
By applying the transformation
푢 =
2푥 − 푦
2
, 푣 =
푦
2
, 푤 =
푧
3
And integrating over an appropriate region in 푢푣푤-space.
Solution:

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Multiple Integrals
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Line Integrals
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Suppose that 푓(푥, 푦, 푧) is a real-valued function and we
wish to integrate it over the curve 풓 푡 = 푔 푡 풊 + 푕 푡 풋
+ 푘 푡 풌, 푎 ≤ 푡 ≤ 푏 (shown in the figure)
The line integral of 푓 over 퐶 is
푓 푥, 푦, 푧
퐶
푑푠
If 풓 푡 is smooth for 푎 ≤ 푡 ≤ 푏 ( 푣 = 푑풓/푑푡 is
continuous and never 0), then the length of the curve C
can be calculated by
푠 푡 = 퐯(휏)
푏
푎
푑휏
Then the integral of 푓 over 퐶 can be written as
푓 푥, 푦, 푧
퐶
푑푠 = 푓(푔 푡 , 푕 푡 , 푘 푡 )
푏
푎
퐯(푡) 푑푡
Line integral can be used to find the work done by a
force field in moving an object along a curve

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Line Integrals
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Line Integrals
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Example: 7 Integrate 푓 푥, 푦, 푧 = 푥 − 3푦2 + 푧 over the line segment C joining the origin
to the point (1,1,1).

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Line Integrals
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Example: 8 Evaluate 2푥 푐 푑푠, where C consists of the arc 퐶1 of the parabola 푦 = 푥2 from
(0,0) to (1,1) followed by the vertical line segment 퐶2 from (1,1) to (2,2).
8
8

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Line Integrals
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Multiple Integrals
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إليكم أربعة عشر فرقًا يميز تفكير الناجح عن تفكير الفاشل، أدعوكم إلى التأمل فيها بعمق ثم إلى تغيير نمط تفكيركم،
لينسجم مع تفكير الناجحين وليتنافر مع تفكير الفاشلين:
1. الناجح يفكر في الحل، والفاشل يفكر في المشكلة.
2. الناجح لا تنضب أفكاره، والفاشل لا تنضب أعذاره.
3. الناجح يساعد الآخرين، والفاشل يتوقع المساعدة من الآخرين.
4. الناجح يرى حل ا في كل مشكلة، والفاشل يرى مشكلة في كل حل.
5. الناجح يقول: "الحل صعب لكنو ممكن" والفاشل يقول: "الحل ممكن ولكنو صعب".
6. الناجح يعتبر الإنجاز التزااما يلبيو، والفاشل لا يرى في الإنجاز أكثر من وعد يعطيو.
7. الناجح لديو أحلم يحققها، والفاشل لديو أوىام وأضغاث أحلم يبددىا.
8. الناجح يقول: "عامل الناس كما تحب أن يعاملوك، والفاشل يقول : اخدع الناس قبل أن يخدعوك.
9. الناجح يرى في العمل أمل، والفاشل يرى في العمل ألم.
10 .الناجح ينظر للمستقبل ويتطلع لما ىو ممكن، والفاشل ينظر للماضي ويتطلع لما ىو مستحيل.
11 .الناجح يختار ما يقول، والفاشل يقول دون أن يختار.
12 .الناجح يناقش بقوة وبلغة لطيفة، والفاشل يناقش بضعف وبلغة فظة.
13 .الناجح يتمسك بالقيم ويتنازل عن الصغائر، والفاشل يتشبث بالصغائر ويتنازل عن القيم.
14 .الناجح يصنع الأحداث، والفاشل تصنعو الأحداث.
ىذه ىي صفات الفاشلين.. تعلمها ولكن لا تمارسها.

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Surface Integrals
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Suppose that a surface S has a vector equation
풓 푢, 푣 = 푥 푢, 푣 풊 + 푦 푢, 푣 풋 + 푧 푢, 푣 풌 푢, 푣 ∈ 퐷
Assume that the parameter domain D is a rectangle and is
divided into sub rectangles 푅푖푗 with dimensions Δ푢 and Δ푣. Then
the surface 푆 is divided into corresponding patches 푆푖푗 as in
figure here.
The surface integral of 풇 over the surface 푺 is
푓(푥, 푦, 푧)
푠
푑푆 = lim
푚,푛→∞
푓(푃푖푗 )Δ푆푖푗
푛
푗=1
푚
푖=1
Δ푆푖푗 ≈ 풓푢 + 풓푣 Δ푢Δ푣
where
풓푢 =
휕푥
휕푢
풊 +
휕푦
휕푢
풋 +
휕푧
휕푢
k and 풓푢 =
휕푥
휕푣
풊 +
휕푦
휕푣
풋 +
휕푧
휕푣
k
Are the tangent vectors at a corner of 푺풊풋
푓(푥, 푦, 푧)
푠
푑푆 = 푓(푟 푢, 푣 ) 푟푢 × 푟푣 푑퐴
퐷
Surface integral can be used to find the rate of fluid flow across
a surface

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Surface Integrals
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Example: 9 compute the surface integral 푥2
푠 푑푆, where 푆 is the unit sphere
푥2 + 푦2 + 푧2 = 1.

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Surface Integrals
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Surface Integrals
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Example: 9 Find the flux of
the vector field 푭 푥, 푦, 푧
= 푧풊 + 푦풋 + 푥풌
Across the unit sphere
푥2 + 푦2 + 푧2 = 1.

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Surface Integrals
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Partial differentiation and application
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Let 푓(푥, 푦) be a function with two variables. If we keep 푦 constant and
differentiate 푓 (assuming 푓 is differentiable) with respect to the variable 푥,
we obtain what is called the partial derivative of 푓 with respect to 푥 which is
denoted by
휕푓
휕푥
= 푓푥 = lim
ℎ→0
푓 푥 + 푕, 푦 − 푓(푥, 푦)
푕
Similarly if we keep 푥 constant and differentiate 푓 (assuming 푓 is
differentiable) with respect to the variable 푦, we obtain what is called
the partial derivative of 푓 with respect to 푦 which is denoted by
휕푓
휕푦
표푟푓푦 = lim
푘→0
푓 푥, 푦 + 푘 − 푓(푥, 푦)
푘

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Partial differentiation and application
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Example 10: Find the partial derivatives 푓푥 and 푓푥 if 푓(푥 , 푦) is given by
1. 푓 푥, 푦 = 푥2푦 + 2푥 + 푦
2. 푓 푥, 푦 = 푠푖푛 푥푦 + 푐표푠 푥
3. 푓 푥, 푦 = 푥푒푥푦
4. 푓 푥, 푦 = ln(푥2 + 2 푦)
5. 푓 푥, 푦 = 푦푥2 + 2 푦
6. 푓 푥, 푦 = 푥푒푥 + 푦
7. 푓(푥, 푦) = ln(2푥 + 푦푥)
8. 푓(푥, 푦) = 푥sin (푥 − 푦)
Example 11: Find the second partial derivatives 푓푥푥, 푓푥푦, 푓푦푥, and 푓푦푦 if
푓 푥 , 푦 = 푥3 + 푥2푦3 − 2푦2

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Partial Differential Equations
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Partial derivatives occur in partial differential equations that express certain
physical laws.
For instance, the partial differential equation
휕2푢
휕푥2 +
휕2푢
휕푥2 = 0
is called Laplace’s equation after Pierre Laplace (1749–1827). Solutions of this
equation are called harmonic functions ; they play a role in problems of heat
conduction, fluid flow, and electric potential.

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Example 12: Show that the function 푢(푥, 푦) = 푒푥 sin 푦 is the solution of the
Laplace’s equation.

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The wave equation
휕2푢
휕푡2 = 푎2 휕2푢
휕푥2 describes the motion of a waveform, which
could be an ocean wave, a sound wave, a light wave, or a wave traveling along
a vibrating string. For instance, 푢 푥, 푡 if represents the displacement of a
vibrating violin string at time 푡 and at a distance 푥 from one end of the
string as in Figure below, then 푢 푥, 푡 satisfies the wave equation. Here the
constant 푎 depends on the density of the string and on the tension in the
string

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Example 13: verify that the function 푢(푥, 푡) = sin(푥 − 푎푡) satisfies the wave
equation.

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Exercises:
1. Use the definition of partial derivatives as limits to find 푓푥 푥, 푦 and
푓푦 푥, 푦 .
푓 푥, 푦 = 푥푦2 − 푥3푦
푓 푥, 푦 = 푥 (푥 + 푦2)
2. Verify that the function 푢 =
1
푥2+푦2+푧2 is a solution of the three-dimensional
Laplace equation 푢푥푥 + 푢푦푦 + 푢푧푧 = 0
3. Verify that the function 푧 = ln(푒푥 + 푒푦) is a solution of the differential
equations 휕푧
휕푥
+
휕푧
휕푦
= 1 and 휕2푧
휕푥2
휕2푧
휕푦2 −
휕2푧
휕푥휕푦
2
= 0

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Infinite Series
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A sequence can be thought of as a list of numbers written in a definite
order:
푎1, 푎2, 푎3, 푎4, … , 푎푛, … = 푎푛
∞
The number 푎1 is called the first term, 푎2 is the second term, and in general
푎푛 is the nth term. We will deal exclusively with infinite sequences and so
each term 푎푛 will have a successor 푎푛+1.
If we try to add the terms of an infinite sequence 푎푛
∞ we get as expression
of the form 푎1 + 푎2 + 푎3 + 푎4 + ⋯ + 푎푛 + ⋯ which is called an infinite
series or just a series and is donated, for short, by the symbol 푎푛
∞푛
=1 or
푎푛.
1
2푛
∞
푛=1
=
1
2
+
1
4
+
1
8
+
1
16
+ ⋯ +
1
2푛 + ⋯ = 1

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Infinite Series
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Convergent and Divergent series:

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Infinite Series
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Infinite Series
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Example 14: Find the sum of the geometric series
5 −
10
3
+
20
9
−
40
27
+ ⋯

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Infinite Series
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Example 15: Is the series 22푛31−푛 ∞푛
=1 convergent or divergent?

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Infinite Series
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Example 16: Find the sum of the series 푥푛 ∞푛
=0 , where 푥 < 1.

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Infinite Series
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Theorems

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Infinite Series
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Theorems

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Infinite Series
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Theorems

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Infinite Series
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Theorems

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Infinite Series
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Examples
Example 17: show that the series (
1
푛 푛+1
) ∞푛
=1 is convergent and find its sum.

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Infinite Series
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Examples
So the series is convergent and
Hence, the summation
Example 18: Find the sum of the series (
3
푛 푛+1
+
1
2푛) ∞푛
=1 .

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Infinite Series
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STRATEGY FOR TESTING SERIES
Several ways to examine the convergence and divergence of series; the problem is to decide
which test to use on which series. There are no hard and fast rules about which test to
apply to a given series, but you may find the following advice of some use.
The main strategy is to classify the series according to its form.
1. If the series is of the form 1
푛푃, it is a P-series, which we know to be convergent if
푃 > 1 and divergent if 푃 ≤ 1 .
2. If the series has the form 푎푟푛−1or 푎푟푛, it is a geometric series, which converges if
푟 < 1 and diverges if 푟 ≥ 1 . Some preliminary algebraic manipulation may be
required to bring the series into this form.

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Infinite Series
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3. If the series has a form that is similar to a P-series or a geometric series, then one of
the comparison tests should be considered. In particular, if 푎푛 is a rational function
or an algebraic function of (involving roots of polynomials), then the series should be
compared with a P-series. The comparison tests apply only to series with positive
terms, but if 푎푛has some negative terms, then we can apply the comparison test to
푎푛 and test for absolute convergence.
4. If you can see at a glance that lim
푛→∞
푎푛 ≠ 0, then the Test for Divergence should be
used.
5. If the series is of the form (−1)푛−1 푏푛 or (−1)푛 푏푛, then the Alternating Series
Test is an obvious possibility.
6. Series that involve factorials or other products (including a constant raised to the nth
power) are often conveniently tested using the Ratio Test. Bear in mind that
푎푛+1
푎푛 → 1 as 푛 → ∞ for all p-series and therefore all rational or algebraic
functions of n. Thus the Ratio Test should not be used for such series.

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Infinite Series
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7. If 푎푛 is of the form (푏푛)푛, then the Root Test may be useful.
8. If 푎푛 = 푓(푛) , where 푓(푥)
∞
1 푑푥 is easily evaluated, then the Integral Test is effective
(assuming the hypotheses of this test are satisfied).

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Infinite Series
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Exercise
Determine whether the series is convergent or divergent. If it is convergent, find its sum.
1. 1
2푛
∞푛
=1 2. 푛+1
2푛−3
∞푛
=1 3. 푘2
푘2−1
∞푘
=2
4. 2 푛 ∞푛
=1 5. ln
푛2+1
2푛2+1
∞푛=1 6. (cos 1)푘 ∞푘
=1

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Power Series
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A power series is a series of the form
푐푛푥푛 ∞푛
=0 = 푐0+푐1푥 +푐2푥2+푐3푥3+푐4푥4+… .
where 푥 is a variable and 푐푛’s are constants called coefficients of the series.
What will happen when 푥 = 1?
A power series may converge for some values of and diverge for other values of . The sum of
the series is a function
푓 푥 = 푐0+푐1푥 +푐2푥2+…+푐푛푥푛 + … .
What will happen when 푐푛 = 1for all n?
The power series becomes the geometric series
푥푛 ∞푛
=0 =1+푥 +푥2+푥3+푥4+… .
which converges when−1 ≤ 푥 ≤ 1 and diverges when 푥 ≥ 1.
More generally, a series of the form
푐푛(푥 − 푎)푛 ∞푛
=0 = 푐0+푐1(푥 − 푎) + 푐2(푥 − 푎)2+…+푐푛(푥 − 푎)푛+… .
is called a power series in 푥 − 푎 or a power series centred at 푎 or a power series about 푎.

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Power Series
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푐푛(푥 − 푎)푛 ∞푛
=0 = 푐0+푐1(푥 − 푎)+ 푐2(푥 − 푎)2+…+푐푛(푥 − 푎)푛+… .
we have adopted the convention that 푥 − 푎 0 = 1 even when 푥 = 푎. Notice also that
when 푥 = 푎, all of the terms are 0 for 푛 ≥ 1 and so this power series always converges
when 푥 = 푎.
Example 19: For what values of 푥 is the series 푛! 푥푛 ∞푛
=0 convergent?

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Power Series
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Example 20: For what values of 푥 does this series (푥 − 3)푛/푛 ∞푛
=1 converge?

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Power Series
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The number 푅 in case (iii) is called the radius of convergence of the power series.

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Power Series
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REPRESENTATIONS OF FUNCTIONS AS POWER SERIES:
As we can see from the expression of a power series it is a polynomial with infinite number
of terms, so it can be used to represent functions, here we are going to know how this
happens.
From example 16 we know that:
1
1 − 푥
= 1 + 푥 + 푥2 + 푥3 + ⋯ = 푥2
∞
푛=0
푓표푟 푥 < 1 … … … … … (1)
Here we were able to represent the function 1
1−푥 by the power series 푥2 ∞푛
=0 푓표푟 푥
< 1

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Power Series
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Example 21 : Express 1/(1 − 푥2) as the sum of a power series and find the interval of
convergence.

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Power Series
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Example 22 : Find a power series representation for 1/(푥 + 2)
Example 23 : Find a power series representation for 푥3/(푥 + 2)

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Power Series
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REPRESENTATIONS OF FUNCTIONS AS POWER SERIES:

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Power Series
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Example 24 : Express1 1 − 푥 2 as a power series by differentiation Equation 1. What is
the radius of convergence?

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Power Series
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Example 25 : Find a power series representation for ln(1 − 푥) and its radius of
convergence?

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Taylor & Maclaurin Series
12/3/2014 67
Previously we found power series representations for a certain restricted class of functions.
Here we investigate more general problems: Which functions have power series
representations? How can we find such representations?
Let’s start with 푓 which is any function that can be represent with a power series;
푓 푥 = 푐0 + 푐1 푥 − 푎 + 푐2(푥 − 푎)2+푐3(푥 − 푎)3+푐4(푥 − 푎)4+ ⋯ 푥 − 푎 < 푅 (1)
The task here is to find the values of 푐푛 that must be in terms of 푓. To begin, as
mentioned previously when 푥 = 푎 in Equation (1) then 푓(푎) = 푐0.
Differentiating Equation (1) gives
푓′(푥) = 푐1 + 2푐2 푥 − 푎 + 3푐3 푥 − 푎 2 + 4푐4 푥 − 푎 3 + ⋯ 푥 − 푎 < 푅 (2)
and substituting of 푥 = 푎 in Equation (2) gives 푓′(푎) = 푐1
Differentiating Equation (2) gives
푓′′ 푥 = 2푐2 + 2 . 3푐3 푥 − 푎 + 3 . 4푐4 푥 − 푎 2 + ⋯ 푥 − 푎 < 푅 (3)
Again we put x=a in Equation (3). The result is 푓′′ 푎 = 2푐2
Repeating the process we get 푓′′′ 푎 = 3! 푐3
and 푓(푛) 푎 = 2 . 3 . 4 . 5 . … . 푛푐푛 = 푛! 푐푛

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12/3/2014 68
Solving this equation for the nth coefficient 푐푛, We get 푐푛 =
푓(푛)(푎)
푛!
THEOREM T&M 1: if 푓 has a power series representation (expansion) at a, that is, if
푓 푥 =
푓 푛 (푎)
푛!
∞
푛=0
(푥 − 푎)푛 푥 − 푎 < 푅
Then its coefficients are given by the formula 푐푛 =
푓(푛)(푎)
푛!
Substituting this formula for back into the series, we see that if has a power series expansion at,
then it must be of the following form.
푓 푥 =
푓 푛 (푎)
푛!
∞
푛=0
(푥 − 푎)푛= 푓 푎 +
푓′ 푎
1!
(푥 − 푎) +
푓′′(푎)
2!
(푥 − 푎)2+
푓′′′(푎)
3!
(푥 − 푎)3+ ⋯
The series is called the Taylor series of the function 푓 at 푎 or about 푎 or centred at 푎 for the
special case 푎=0 the Taylor series becomes Maclaurin series
푓 푥 =
푓 푛 (0)
푛!
∞
푛=0
푥푛 = 푓 0 +
푓′(0)
1!
푥 +
푓′′(0)
2!
푥2 +
푓′′′(0)
3!
푥3 + ⋯

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Example 26 : Find the Maclaurin series of the function 푓 푥 = 푒푥 and its radius of
convergence?

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12/3/2014 70
From Theorem T&M 1 and example 26 we conclude that if 푒푥 has a power series expansion at 0,
then 푒푥 = 푥푛
푛!
∞푛
=0 .
So how can we determine whether 푒푥 does have a power series representation?
Let’ s investigate the more general question: Under what circumstances is a function equal to
the sum of its Taylor series? In other words, if 푓 has derivatives of all orders, when is it true that
푓 푥 = 푓 푛 (푎)
푛!
∞푛
=0 (푥 − 푎)푛
As with any convergent series, this means that 푓 푥 is the limit of the sequence of partial sums.
In the case of the Taylor series, the partial sums are
푇푛 푥 =
푓 푖 (푎)
푖!
푛
푖=0
(푥 − 푎)푖
= 푓 푎 +
푓′ 푎
1!
푥 − 푎 +
푓′′ 푎
2!
푥 − 푎 2 +
푓′′′ 푎
3!
푥 − 푎 3 + ⋯
+
푓 푛 (푎)
푛!
(푥 − 푎)푛
Notice that 푇푛 is a polynomial of degree 푛 called the nth-degree Taylor polynomial of 푓 at a.

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The first the second, and the third sums are
푇1 푥 = 1 + 푥.
푇2 푥 = 1 + 푥 +
푥2
2!
.
푇3 푥 = 1 + 푥 +
푥2
2!
+
푥3
3!
.
In general, f(x) is the sum of its Taylor series if 푓 푥 = lim
푛→∞
푇푛(푥)
If we let 푅푛 푥 = 푓 푥 − 푇푛 푥 so that 푓 푥 = 푇푛 푥 + 푅푛 푥
푅푛 푥 is called the reminder of the Taylor series. If we can somehow show that lim
푛→∞
푅푛 푥 = 0,
then it follows that lim
푛→∞
푇푛 푥 = lim
푛→∞
[푓 푥 − 푅푛 푥 ] = 푓 푥 − lim
푛→∞
푅푛 푥 = 푓(푥)

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Example 27: find a. 푒−푥2
푑푥. 푏. 푒−푥2 1
0 푑푥.

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Example 28: find lim
푥→0
푒푥−1−푥
푥2

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Example 29: find the first three nonzero terms in the Maclaurin series for
a. 푒푥 sin 푥
b. tan 푥

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Example 29: find the first three nonzero terms in the Maclaurin series for
a. 푒푥 sin 푥
b. tan 푥

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Fourier Series
12/3/2014 77
Fourier series are infinite series that represent periodic functions in terms of cosines and
sines. As such, Fourier series are of greatest importance to the engineer and applied
mathematician. To define Fourier series, we first need some background material. A
function 푓(푥) is called a periodic function if 푓( 푥) is defined for all real 푥, except possibly at
some points, and if there is some positive number p, called a period of 푓( 푥), such that
푓 푥 + 푝 = 푓(푥)
Periodic function of period p

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Fourier Series
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the fundamental period 푓 푥 + 푛푝 = 푓(푥)
Cosine and sine functions having the period 2

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Fourier Series
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The series to be obtained will be a trigonometric series, that is, a series of the form
푎0 + 푎1 cos 푥 + 푏1 sin 푥 + 푎2 cos 2푥 + 푏2 sin 2푥 + ⋯ = 푎0 + 푎푛 cos 푛푥 + 푏푛 sin 푛푥
∞
푛→1
.
푎0, 푎1, 푏1, 푎2, 푏2, … are constants, called the coefficients of the series. We see that each term
has the period 2휋Hence if the coefficients are such that the series converges, its sum will be a
function of period 2휋.
푓(푥) = 푎0 + 푎푛 cos 푛푥 + 푏푛 sin 푛푥
∞
푛→1
Is the Fourier Series of 푓(푥) with Fourier Coefficients

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Fourier Series
To be continued

Biomedical engineering mathematics i

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Biomedical engineering mathematics i