Biomedical engineering mathematics i

لكية اهلندسة جامعة املنيا قسم هندسة القوى املياكنيكية 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

𝑓 on the interval [𝑎, 𝑏] is the

following limit

𝑓 𝑥𝑏

𝑎

𝑑𝑥 = lim𝑚𝑎𝑥∆𝑥𝑖⟶0

𝑓 𝑧𝑖 ∆𝑥𝑖 ,𝑛

𝑖=1

Where the numbers 𝑥0, 𝑥1, …….., 𝑥𝑛 satisfy 𝑎 = 𝑥0 ≤ 𝑥1 ≤ …. ≤ 𝑥𝑛 = 𝑏 ,

where ∆𝑥𝑖 = 𝑥𝑖 −𝑥𝑖−1 , for each

𝑖 = 1,2, … . , 𝑛, and 𝑧𝑖 is an arbitrarily

chosen in the interval [𝑥𝑖−1,𝑥𝑖].

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

D 5


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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

(2𝑥 − 𝑦

2+𝑧

3)

𝑥=𝑦2 +1

𝑥=𝑦/2

4

0

3

0

𝑑𝑥𝑑𝑦𝑑𝑧

By applying the transformation

𝑢 =2𝑥 − 𝑦

2, 𝑣 =𝑦

2,𝑤 =𝑧

3

And integrating over an appropriate region in 𝑢𝑣𝑤-space.

Solution:


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Multiple Integrals

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Line Integrals

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Suppose that 𝑓(𝑥, 𝑦, 𝑧) is a real-valued function and we

wish to integrate it over the curve 𝒓 𝑡 = 𝑔 𝑡 𝒊 + 𝑕 𝑡 𝒋+ 𝑘 𝑡 𝒌, 𝑎 ≤ 𝑡 ≤ 𝑏 (shown in the figure)

The line integral of 𝑓 over 𝐶 is

𝑓 𝑥, 𝑦, 𝑧𝐶

𝑑𝑠

If 𝒓 𝑡 is smooth for 𝑎 ≤ 𝑡 ≤ 𝑏 ( 𝑣 = 𝑑𝒓/𝑑𝑡 is

continuous and never 0), then the length of the curve C

can be calculated by

𝑠 𝑡 = 𝐯(𝜏)𝑏

𝑎

𝑑𝜏

Then the integral of 𝑓 over 𝐶 can be written as

𝑓 𝑥, 𝑦, 𝑧𝐶

𝑑𝑠 = 𝑓(𝑔 𝑡 , 𝑕 𝑡 , 𝑘 𝑡 )𝑏

𝑎

𝐯(𝑡) 𝑑𝑡

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 𝑓 𝑥, 𝑦, 𝑧 = 𝑥 − 3𝑦2 + 𝑧 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 2𝑥𝑐𝑑𝑠, where C consists of the arc 𝐶1 of the parabola 𝑦 = 𝑥2 from

(0,0) to (1,1) followed by the vertical line segment 𝐶2 from (1,1) to (2,2).

8

8


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Line Integrals

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Multiple Integrals

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تفكيركم، نمط تغيير إلى ثم بعمق فيها التأمل إلى أدعوكم الفاشل، تفكير عن الناجح تفكير يميز فرقًا عشر أربعة إليكم :الفاشلين تفكير مع وليتنافر الناجحين تفكير مع لينسجم

.املشكلة يف يفكر والفاشل احلل، يف يفكر الناجح1.

.أعذاره تنضب ال والفاشل أفكاره، تنضب ال الناجح2.

.اآلخرين من املساعدة يتوقع والفاشل اآلخرين، يساعد الناجح3.

.حل كل يف مشكلة يرى والفاشل مشكلة، كل يف حلا يرى الناجح4.

."صعب ولكنو ممكن احلل" :يقول والفاشل "ممكن لكنو صعب احلل" :يقول الناجح5.

.يعطيو وعد من أكثر اإلجناز يف يرى ال والفاشل يلبيو، التزاماا اإلجناز يعترب الناجح6.

.يبددىا أحلم وأضغاث أوىام لديو والفاشل حيققها، أحلم لديو الناجح7.

.خيدعوك أن قبل الناس اخدع : يقول والفاشل يعاملوك، أن حتب كما الناس عامل" :يقول الناجح8.

.أمل العمل يف يرى والفاشل أمل، العمل يف يرى الناجح9. .مستحيل ىو ملا ويتطلع للماضي ينظر والفاشل ممكن، ىو ملا ويتطلع للمستقبل ينظر الناجح10. .خيتار أن دون يقول والفاشل يقول، ما خيتار الناجح11. .فظة وبلغة بضعف يناقش والفاشل لطيفة، وبلغة بقوة يناقش الناجح12. .القيم عن ويتنازل بالصغائر يتشبث والفاشل الصغائر، عن ويتنازل بالقيم يتمسك الناجح13. .األحداث تصنعو والفاشل األحداث، يصنع الناجح14. .متارسها ال ولكن تعلمها ..الفاشلني صفات ىي ىذه


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Surface Integrals

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Suppose that a surface S has a vector equation

𝒓 𝑢, 𝑣 = 𝑥 𝑢, 𝑣 𝒊 + 𝑦 𝑢, 𝑣 𝒋 + 𝑧 𝑢, 𝑣 𝒌 𝑢, 𝑣 ∈ 𝐷 Assume that the parameter domain D is a rectangle and is

divided into sub rectangles 𝑅𝑖𝑗 with dimensions ∆𝑢 and ∆𝑣. Then

the surface 𝑆 is divided into corresponding patches 𝑆𝑖𝑗 as in

figure here.

The surface integral of 𝒇 over the surface 𝑺 is

𝑓(𝑥, 𝑦, 𝑧)𝑠

𝑑𝑆 = lim𝑚,𝑛→∞

𝑓(𝑃𝑖𝑗)∆𝑆𝑖𝑗

𝑛

𝑗=1

𝑚

𝑖=1

∆𝑆𝑖𝑗 ≈ 𝒓𝑢 + 𝒓𝑣 ∆𝑢∆𝑣

where

𝒓𝑢 =𝜕𝑥

𝜕𝑢𝒊 +𝜕𝑦

𝜕𝑢𝒋 +𝜕𝑧

𝜕𝑢k and 𝒓𝑢 =

𝜕𝑥

𝜕𝑣𝒊 +𝜕𝑦

𝜕𝑣𝒋 +𝜕𝑧

𝜕𝑣k

Are the tangent vectors at a corner of 𝑺𝒊𝒋

𝑓(𝑥, 𝑦, 𝑧)𝑠

𝑑𝑆 = 𝑓(𝑟 𝑢, 𝑣 ) 𝑟𝑢 × 𝑟𝑣 𝑑𝐴𝐷

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 𝑥2𝑠𝑑𝑆, where 𝑆 is the unit sphere

𝑥2 + 𝑦2 + 𝑧2 = 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 𝑭 𝑥, 𝑦, 𝑧= 𝑧𝒊 + 𝑦𝒋 + 𝑥𝒌 Across the unit sphere

𝑥2 + 𝑦2 + 𝑧2 = 1.


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Surface Integrals

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Partial differentiation and application

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Let 𝑓(𝑥, 𝑦) be a function with two variables. If we keep 𝑦 constant and differentiate 𝑓 (assuming 𝑓 is differentiable) with respect to the variable 𝑥, we obtain what is called the partial derivative of 𝑓 with respect to 𝑥 which is denoted by

𝜕𝑓

𝜕𝑥= 𝑓𝑥 = lim

ℎ→0

𝑓 𝑥 + 𝑕, 𝑦 − 𝑓(𝑥, 𝑦)

𝑕

Similarly if we keep 𝑥 constant and differentiate 𝑓 (assuming 𝑓 is differentiable) with respect to the variable 𝑦, we obtain what is called the partial derivative of 𝑓 with respect to 𝑦 which is denoted by

𝜕𝑓

𝜕𝑦 𝑜𝑟𝑓𝑦 = lim

𝑘→0

𝑓 𝑥, 𝑦 + 𝑘 − 𝑓(𝑥, 𝑦)

𝑘


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Partial differentiation and application

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Example 10: Find the partial derivatives 𝑓𝑥 and 𝑓𝑥 if 𝑓(𝑥 , 𝑦) is given by 1. 𝑓 𝑥, 𝑦 = 𝑥2𝑦 + 2𝑥 + 𝑦 2. 𝑓 𝑥, 𝑦 = 𝑠𝑖𝑛 𝑥𝑦 + 𝑐𝑜𝑠 𝑥 3. 𝑓 𝑥, 𝑦 = 𝑥𝑒𝑥𝑦 4. 𝑓 𝑥, 𝑦 = ln(𝑥2 + 2 𝑦) 5. 𝑓 𝑥, 𝑦 = 𝑦𝑥2 + 2 𝑦 6. 𝑓 𝑥, 𝑦 = 𝑥𝑒𝑥 + 𝑦 7. 𝑓(𝑥, 𝑦) = ln(2𝑥 + 𝑦𝑥) 8. 𝑓(𝑥, 𝑦) = 𝑥sin (𝑥 − 𝑦)

Example 11: Find the second partial derivatives 𝑓𝑥𝑥, 𝑓𝑥𝑦, 𝑓𝑦𝑥, and 𝑓𝑦𝑦 if 𝑓 𝑥 , 𝑦 = 𝑥3 + 𝑥2𝑦3 − 2𝑦2


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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

𝜕2𝑢

𝜕𝑥2+𝜕2𝑢

𝜕𝑥2= 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 𝑢(𝑥, 𝑦) = 𝑒𝑥 sin 𝑦 is the solution of the Laplace’s equation.


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The wave equation 𝜕2𝑢

𝜕𝑡2= 𝑎2

𝜕2𝑢

𝜕𝑥2 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, 𝑢 𝑥, 𝑡 if represents the displacement of a vibrating violin string at time 𝑡 and at a distance 𝑥 from one end of the string as in Figure below, then 𝑢 𝑥, 𝑡 satisfies the wave equation. Here the constant 𝑎 depends on the density of the string and on the tension in the string


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Example 13: verify that the function 𝑢(𝑥, 𝑡) = sin(𝑥 − 𝑎𝑡) satisfies the wave equation.


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Exercises: 1. Use the definition of partial derivatives as limits to find 𝑓𝑥 𝑥, 𝑦 and 𝑓𝑦 𝑥, 𝑦 .

𝑓 𝑥, 𝑦 = 𝑥𝑦2 − 𝑥3𝑦 𝑓 𝑥, 𝑦 = 𝑥 (𝑥 + 𝑦2)

2. Verify that the function 𝑢 = 1

𝑥2+𝑦2+𝑧2 is a solution of the three-

dimensional Laplace equation 𝑢𝑥𝑥 + 𝑢𝑦𝑦 + 𝑢𝑧𝑧 = 0

3. Verify that the function 𝑧 = ln(𝑒𝑥 + 𝑒𝑦) is a solution of the differential

equations 𝜕𝑧𝜕𝑥+𝜕𝑧

𝜕𝑦= 1 and 𝜕

2𝑧

𝜕𝑥2𝜕2𝑧

𝜕𝑦2−𝜕2𝑧

𝜕𝑥𝜕𝑦

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:

𝑎1, 𝑎2, 𝑎3, 𝑎4, … , 𝑎𝑛, … = 𝑎𝑛∞

The number 𝑎1 is called the first term, 𝑎2 is the second term, and in general 𝑎𝑛 is the nth term. We will deal exclusively with infinite sequences and so each term 𝑎𝑛 will have a successor 𝑎𝑛+1. If we try to add the terms of an infinite sequence 𝑎𝑛 ∞ we get as expression of the form 𝑎1 + 𝑎2 + 𝑎3 + 𝑎4 +⋯+ 𝑎𝑛 +⋯ which is called an infinite series or just a series and is donated, for short, by the symbol 𝑎𝑛

∞𝑛=1 or

𝑎𝑛.

1

2𝑛

∞

𝑛=1

=1

2+1

4+1

8+1

16+⋯+

1

2𝑛+⋯ = 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 22𝑛31−𝑛∞𝑛=1 convergent or divergent?


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Infinite Series

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Example 16: Find the sum of the series 𝑥𝑛∞𝑛=0 , where 𝑥 < 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

𝑛 𝑛+1)∞

𝑛=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

𝑛 𝑛+1+1

2𝑛)∞

𝑛=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 𝑛𝑃 , it is a P-series, which we know to be convergent if 𝑃 > 1 and divergent if 𝑃 ≤ 1 .

2. If the series has the form 𝑎𝑟𝑛−1or 𝑎𝑟𝑛, it is a geometric series, which converges if 𝑟 < 1 and diverges if 𝑟 ≥ 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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STRATEGY FOR TESTING SERIES 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 𝑎𝑛 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 𝑎𝑛has some negative terms, then we can apply the comparison test to

𝑎𝑛 and test for absolute convergence. 4. If you can see at a glance that lim

𝑛→∞𝑎𝑛 ≠ 0, then the Test for Divergence should be

used. 5. If the series is of the form (−1)𝑛−1 𝑏𝑛 or (−1)𝑛 𝑏𝑛, 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 𝑎𝑛+1

𝑎𝑛 → 1 as 𝑛 → ∞ 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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STRATEGY FOR TESTING SERIES 7. If 𝑎𝑛 is of the form (𝑏𝑛)𝑛, then the Root Test may be useful. 8. If 𝑎𝑛 = 𝑓(𝑛) , where 𝑓(𝑥)∞

1 𝑑𝑥 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

2𝑛∞𝑛=1 2.

𝑛+1

2𝑛−3∞𝑛=1 3.

𝑘2

𝑘2−1∞𝑘=2

4. 2𝑛∞

𝑛=1 5. ln𝑛2+1

2𝑛2+1∞𝑛=1 6. (cos 1)𝑘∞

𝑘=1


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Power Series

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A power series is a series of the form 𝑐𝑛𝑥

𝑛∞𝑛=0 = 𝑐0+𝑐1𝑥 +𝑐2𝑥

2+𝑐3𝑥3+𝑐4𝑥

4+… . where 𝑥 is a variable and 𝑐𝑛’s are constants called coefficients of the series. What will happen when 𝑥 = 1? A power series may converge for some values of and diverge for other values of . The sum of the series is a function 𝑓 𝑥 = 𝑐0+𝑐1𝑥 +𝑐2𝑥

2+…+𝑐𝑛𝑥𝑛 + … .

What will happen when 𝑐𝑛 = 1for all n? The power series becomes the geometric series 𝑥𝑛∞𝑛=0 =1+𝑥 +𝑥2+𝑥3+𝑥4+… .

which converges when−1 ≤ 𝑥 ≤ 1 and diverges when 𝑥 ≥ 1. More generally, a series of the form

𝑐𝑛(𝑥 − 𝑎)𝑛∞

𝑛=0 = 𝑐0+𝑐1(𝑥 − 𝑎) + 𝑐2(𝑥 − 𝑎)2+…+𝑐𝑛(𝑥 − 𝑎)

𝑛+… . is called a power series in 𝑥 − 𝑎 or a power series centred at 𝑎 or a power series about 𝑎.


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Power Series

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𝑐𝑛(𝑥 − 𝑎)𝑛∞

𝑛=0 = 𝑐0+𝑐1(𝑥 − 𝑎)+ 𝑐2(𝑥 − 𝑎)2+…+𝑐𝑛(𝑥 − 𝑎)𝑛+… .

we have adopted the convention that 𝑥 − 𝑎 0 = 1 even when 𝑥 = 𝑎. Notice also that when 𝑥 = 𝑎, all of the terms are 0 for 𝑛 ≥ 1 and so this power series always converges when 𝑥 = 𝑎. Example 19: For what values of 𝑥 is the series 𝑛! 𝑥𝑛∞

𝑛=0 convergent?


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Power Series

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Example 20: For what values of 𝑥 does this series (𝑥 − 3)𝑛/𝑛∞𝑛=1 converge?


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Power Series

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The number 𝑅 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 − 𝑥= 1 + 𝑥 + 𝑥2 + 𝑥3 +⋯ = 𝑥2

∞

𝑛=0

𝑓𝑜𝑟 𝑥 < 1……………(1)

Here we were able to represent the function 11−𝑥

by the power series 𝑥2∞𝑛=0 𝑓𝑜𝑟 𝑥

< 1


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Power Series

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Example 21 : Express 1/(1 − 𝑥2) 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/(𝑥 + 2)

Example 23 : Find a power series representation for 𝑥3/(𝑥 + 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 − 𝑥 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 − 𝑥) and its radius of convergence?


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Taylor & Maclaurin Series

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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 𝑓 which is any function that can be represent with a power series; 𝑓 𝑥 = 𝑐0 + 𝑐1 𝑥 − 𝑎 + 𝑐2(𝑥 − 𝑎)

2+𝑐3(𝑥 − 𝑎)3+𝑐4(𝑥 − 𝑎)

4+⋯ 𝑥 − 𝑎 < 𝑅 (1)

The task here is to find the values of 𝑐𝑛 that must be in terms of 𝑓. To begin, as mentioned previously when 𝑥 = 𝑎 in Equation (1) then 𝑓(𝑎) = 𝑐0. Differentiating Equation (1) gives 𝑓′(𝑥) = 𝑐1 + 2𝑐2 𝑥 − 𝑎 + 3𝑐3 𝑥 − 𝑎

2 + 4𝑐4 𝑥 − 𝑎3 +⋯ 𝑥 − 𝑎 < 𝑅 (2)

and substituting of 𝑥 = 𝑎 in Equation (2) gives 𝑓′(𝑎) = 𝑐1

Differentiating Equation (2) gives 𝑓′′ 𝑥 = 2𝑐2 + 2 . 3𝑐3 𝑥 − 𝑎 + 3 . 4𝑐4 𝑥 − 𝑎

2 +⋯ 𝑥 − 𝑎 < 𝑅 (3)

Again we put x=a in Equation (3). The result is 𝑓′′ 𝑎 = 2𝑐2

Repeating the process we get 𝑓′′′ 𝑎 = 3! 𝑐3

and 𝑓(𝑛) 𝑎 = 2 . 3 . 4 . 5 . … . 𝑛𝑐𝑛 = 𝑛! 𝑐𝑛


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Solving this equation for the nth coefficient 𝑐𝑛, We get 𝑐𝑛 =𝑓(𝑛)(𝑎)

𝑛!

THEOREM T&M 1: if 𝑓 has a power series representation (expansion) at a, that is, if

𝑓 𝑥 = 𝑓 𝑛 (𝑎)

𝑛!

∞

𝑛=0

(𝑥 − 𝑎)𝑛 𝑥 − 𝑎 < 𝑅

Then its coefficients are given by the formula 𝑐𝑛 =𝑓(𝑛)(𝑎)

𝑛!

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.

𝑓 𝑥 = 𝑓 𝑛 (𝑎)

𝑛!

∞

𝑛=0

(𝑥 − 𝑎)𝑛= 𝑓 𝑎 +𝑓′ 𝑎

1!(𝑥 − 𝑎) +

𝑓′′(𝑎)

2!(𝑥 − 𝑎)2+

𝑓′′′(𝑎)

3!(𝑥 − 𝑎)3+⋯

The series is called the Taylor series of the function 𝑓 at 𝑎 or about 𝑎 or centred at 𝑎 for the special case 𝑎=0 the Taylor series becomes Maclaurin series

𝑓 𝑥 = 𝑓 𝑛 (0)

𝑛!

∞

𝑛=0

𝑥𝑛 = 𝑓 0 +𝑓′(0)

1!𝑥 +𝑓′′(0)

2!𝑥2 +𝑓′′′(0)

3!𝑥3 +⋯


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Example 26 : Find the Maclaurin series of the function 𝑓 𝑥 = 𝑒𝑥 and its radius of convergence?


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From Theorem T&M 1 and example 26 we conclude that if 𝑒𝑥 has a power series expansion at 0, then 𝑒𝑥 = 𝑥𝑛

𝑛!∞𝑛=0 .

So how can we determine whether 𝑒𝑥 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 𝑓 has derivatives of all orders, when is it true that

𝑓 𝑥 = 𝑓 𝑛 (𝑎)

𝑛!∞𝑛=0 (𝑥 − 𝑎)𝑛

As with any convergent series, this means that 𝑓 𝑥 is the limit of the sequence of partial sums. In the case of the Taylor series, the partial sums are

𝑇𝑛 𝑥 = 𝑓 𝑖 (𝑎)

𝑖!

𝑛

𝑖=0(𝑥 − 𝑎)𝑖

= 𝑓 𝑎 +𝑓′ 𝑎

1!𝑥 − 𝑎 +

𝑓′′𝑎

2!𝑥 − 𝑎 2 +

𝑓′′′ 𝑎

3!𝑥 − 𝑎 3 +⋯

+𝑓 𝑛 (𝑎)

𝑛!(𝑥 − 𝑎)𝑛

Notice that 𝑇𝑛 is a polynomial of degree 𝑛 called the nth-degree Taylor polynomial of 𝑓 at a.


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The first the second, and the third sums are 𝑇1 𝑥 = 1 + 𝑥.

𝑇2 𝑥 = 1 + 𝑥 +𝑥2

2!.

𝑇3 𝑥 = 1 + 𝑥 +𝑥2

2!+𝑥3

3!.

In general, f(x) is the sum of its Taylor series if 𝑓 𝑥 = lim𝑛→∞𝑇𝑛(𝑥)

If we let 𝑅𝑛 𝑥 = 𝑓 𝑥 − 𝑇𝑛 𝑥 so that 𝑓 𝑥 = 𝑇𝑛 𝑥 + 𝑅𝑛 𝑥

𝑅𝑛 𝑥 is called the reminder of the Taylor series. If we can somehow show that lim𝑛→∞𝑅𝑛 𝑥 = 0,

then it follows that lim𝑛→∞𝑇𝑛 𝑥 = lim

𝑛→∞[𝑓 𝑥 − 𝑅𝑛 𝑥 ] = 𝑓 𝑥 − lim

𝑛→∞𝑅𝑛 𝑥 = 𝑓(𝑥)


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Example 27: find a. 𝑒−𝑥2𝑑𝑥. 𝑏. 𝑒−𝑥

21

0𝑑𝑥.


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Example 28: find lim𝑥→0

𝑒𝑥−1−𝑥

𝑥2


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


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


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Fourier Series

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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 𝑓(𝑥) is called a periodic function if 𝑓( 𝑥) is defined for all real 𝑥, except possibly at some points, and if there is some positive number p, called a period of 𝑓( 𝑥), such that

𝑓 𝑥 + 𝑝 = 𝑓(𝑥)

Periodic function of period p


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Fourier Series

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the fundamental period 𝑓 𝑥 + 𝑛𝑝 = 𝑓(𝑥)

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

𝑎0 + 𝑎1 cos 𝑥 + 𝑏1 sin 𝑥 + 𝑎2 cos 2𝑥 + 𝑏2 sin 2𝑥 + ⋯ = 𝑎0 + 𝑎𝑛 cos 𝑛𝑥 + 𝑏𝑛 sin 𝑛𝑥

∞

𝑛→1

.

𝑎0, 𝑎1, 𝑏1, 𝑎2, 𝑏2, … are constants, called the coefficients of the series. We see that each term has the period 2𝜋Hence if the coefficients are such that the series converges, its sum will be a function of period 2𝜋.

𝑓(𝑥) = 𝑎0 + 𝑎𝑛 cos 𝑛𝑥 + 𝑏𝑛 sin 𝑛𝑥

∞

𝑛→1

Is the Fourier Series of 𝑓(𝑥) with Fourier Coefficients


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Fourier Series

To be continued

Biomedical engineering mathematics i

Engineering

Transcript of Biomedical engineering mathematics i