final report applied calculus for it

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FACULTY OF INFORMATION TECHNOLOGY

PHAM DANG ANH NGOC- 523H0162

FINAL REPORT

APPLIED CALCULUS FOR IT

HO CHI MINH CITY, JANUARY 2024

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FACULTY OF INFORMATION TECHNOLOGY

PHAM DANG ANH NGOC- 523H0162

FINAL REPORT

APPLIED CALCULUS FOR IT

M.A. Pham Kim Thuy

HO CHI MINH CITY, JANUARY 2024

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I would like to express my sincere gratitude to Ton Duc Thang University,the Faculty of Information Technology, and the Department of AppliedAnalysis, as well as to MA. Pham Kim Thuy, for their efforts in creating the bestlearning conditions for me during this period. This report represents theculmination of the knowledge I have acquired and am currently learning, thanksto the dedicated and passionate teachings of Ms. Pham Kim Thuy. With thisknowledge, I have gained a deep understanding of the subject, broadened myperspectives, and can now apply it in practical situations, laying the foundationfor promising future developments.

Ho Chi Minh City, January 6, 2024.

Author,(Signature and Full Name)

Ngoc Pham Dang Anh Ngoc

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COMPLETED PROJECT AT TON DUC THANG UNIVERSITY

I hereby affirm that this research project is my own work and wasconduct under the scientific guidance of Dr. Pham Kim Thuy. The researchcontent and results presented in this thesis are truthful and have not beenpublish in any form prior to this. The author collected the data in the tables,used for analysis, comments, and evaluations, from various sources, asexplicitly stated in the reference section.

Furthermore, this project incorporates some comments, evaluations, anddata from other authors and different organizations, all of which areappropriately cite and reference.

In the event of any academic dishonesty, I take full responsibility forthe content of my project. Ton Duc Thang University is not implicate in any

Ho Chi Minh City, January 6, 2024Author,

(Signature and Full Name))Ngoc

Pham Dang Anh Ngoc

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CONTENTS

LIST OF DRAWINGS...

CHAPTER 1 THEORETICAL FOUNDATIONS...

CHAPTER 2 MAIN CONTENT...

2.2 Solutions...

REFERENCES...

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LIST OF DRAWINGS

Figure 1:Task 4... 9Figure2: Task 6... 12

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CHAPTER 1 THEORETICAL FOUNDATIONS

Task 1:

The graphs of even and odd functions have characteristic symmetry properties. A function y=f (x ) is an

f'(a) Is the slope of y=f(x)atx=a.

The tangent line to y=f(x) at (a, f (a)) is the line passing through (a, f (a))with slopef '(a):

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y−f(a)=f '(a)(x−a)

⟺ f(c)≥ f(x) For all x near c (for all x in an open interval containingc)f Has a local (or relative) minimum at c ∈ D

⟺ f(c)≤ f(x) For all x near c (for all x in an open interval containingc)Let f be continuous andca critical number off.

Suppose fis differentiable near c(except possibly atc).If f ' changes from positive to negative at c,Then f has a local maximum atc.

If f ' changes from negative to positive at c,Then f has a local minimum atc.

If f 'does not changes sign at c,Then f has no local max/min atc.

Task 6:

∆ L=

√

(∆ x)2+(∆ y)2∆ L

∆ x=

√

(

∆ y∆ x

)

2

dLdx= lim∆ x→ 0

∆ L∆x=

√

(

If f ' is continuous on[a,b], then the length (arc length) of the curve y=f (x )from thepoint A=(a,f (a)) to the point B=(b,f(b))is the value of the integral:

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(bn) Be series such that0 ≤an≤bnFor all n (Or for all n≥ N¿ then

n =1∞

(an)diverges ⇒

∑

(an) Be series

Suppose lim

an

|

=L where 0 ≤ L≤∞If 0 ≤ L<1,then

∑

(an) is convergent

If 1< L≤∞ ,then

∑

(an) is divergent

If L=1 ,the convergenceof

∑

i=0n

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CHAPTER 2 MAIN CONTENT

2.1 Topics

Task 1:Tell whether the following functions are even, odd, or neither. Give reasonsfor your answer. (1.0 point)

+xf(x)= 4

−4f(x)= x

x→5−¿ ¿

Task 3:Find the derivatives dy

dx of the following functions: (1.0 point)y=

√

x−4

√

x +4y=

(√

On what open intervals is f increasing or decreasing?

At what points, if any, doesf assume local maximum and minimum values?

Task 6:Find all curves through a point where x=1 whose arc length is thefollowing L value: (1.0 point)

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The series a2+a a4+ 8+a16+…+ a2n+… diverges

Determine the convergence or divergence of the following series. Explain indetails.

Task 8:Find all values of x such that the following series is absolutely convergent:(1 point)

n =1∞

n xn

(n+1) (2 x +1)n

Task 9:One thousand earphones sell for $55 each, resulting in a revenue of (1000)($55) = $55,000. For each $5 increase in the price, 20 fewer earphones are sold. Forex., if the price of each earphone is $60, there will be 980 (1000 – 20) earphonessold; if the price of each earphone is $65, there will be 960 (1000 – 20 – 20)earphones sold; so on. Find the revenue in case the price of each earphone is $255(1 point)

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+(− )x¿ 2−x

Since f(−x)≠ f (x )and(−x)≠−f (x) , so f (x) is neither even nor odd. f(x)=x3

+ x

Domain of function: D=R ⟹ ∀ x ∈ D ,−x ∈ Df(−x)=(−x)3

+(− )x¿−x3−x

¿−(x¿¿3+ x)¿¿−f (x)

So f (x) is odd function f(x)= 4

x4−4Domain of function:x4−4 ≠ 0

−2) (x2+2)≠ 0(x2 x+2>0 ∀ )⟹ x2−2≠ 0

⟹ x ≠

√

2, x≠−

√

⟹ D={x ∈ R∨x ≠

√

2, x≠−

√

2}⟹ ∀ x ∈ D ,−x ∈ Df(−x)= 4

(− )x4

−4¿ 4

−4¿f (x)

So f (x) is even function f(x)= x

−4

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Domain of function:x4−4 ≠ 0

−2) (x2+2)≠ 0(x2 x+2>0 ∀ )⟹ x2−2≠ 0

⟹ x ≠

√

2, x≠−

√

⟹ D={x ∈ R∨x ≠

√

2, x≠−

√

2}⟹ ∀ x ∈ D ,−x ∈ Df(−x)= (− )x

(− )x4

−4¿− x

−4¿−f (x)

So f (x) is odd function

Task 2: lim 555x2

x →5−555=0−¿¿¿

¿ , lim

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Task 3:

√

x +4y'

√

x−4)'(

√

x +4)−(

√

x−4) (

√

x+4)'

√

x +4)−(

√

x −4) 12

√

x +4)2

¿(

√

x +4−

√

x +4)2

√

x+4)2

y '=−10

(√

x10−1

)

'

(√

x10 −1

)

−11

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¿ −10

⟹ f'(0)=2e0

⟹ f'(0)=2

Calculate f (x) at x=0:⟹ f(0)=2 e0

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⟹

[

sin

(

x+4π

)

=0sin

(

x −π

)

=0⟹

[

x +4π=kπ(k ∈ Z )

x−π4=kπ(k ∈ Z )

⟹

[

x=kπ−π4(k ∈ Z )x=kπ+π4(k ∈ Z )

Since the domain is restricted to 0≤ x≤ π2

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⟹

[

x=3 π4 ,x=

7 π4x=π

4, x=5 π

So the critical numbers of f :x ∈

{

π4,

3 π4 ,

5 π4 ,

7 π4

}

Domain of function: D=¿ ¿]

3 π4

5 π4

7 π

f (x)f(0)

3 π4

)

(

5 π

7 π4

)

f(2)

⟹ f(x)Is increasing on

(

π4,

3 π4

)

∪

(

5 π

4 ,7 π

⟹ f(x)Is decreasing on

[

0 ,π4

)

∪

(

3 π

4 ,5 π

)

∪

(

7 π4 ,2

]

f '(x) Change from positive to negative at x=3 π4 and x=

7 π4⟹ f(x) Has a local maximum at x=3 π

4 and x=7 π

4f '(x) Change from negative to positive at x=π4 and x=5 π4

⟹ f(x) Has a local minimum at x=π4 and x=

5 π4

Task 6:

The arc length of the curve y=f (x ) from x = a to x = b is L=

∫

b

√

(

f'(x)

)

2

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⟹ L=

∫

x2dx Is the arc length from x = 1 to x = 5 of the curve y=f(x)with

As y=f(x)⟹ f'(x )=dydx⟹

(

x2=±1x⟹ dy=±1

xdx⟹

∫

dy=±

∫

1 xdx⟹ y=± ln|x|+c

The curves through a point where x=1 ⟹ y=0To substitutex=0 into the function y=± ln|x|+c⟹ y=± ln |1|+c

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The series a2+a a a4+ 8+ 16+…+a2n+…=

∑

The series a1+a2

(

a2n2n

)

a2n

2n≤ a2n (a2n>0Andn>1)The series

∑

n =1∞

ak is non-decreasing ⟹ 2n<k <2n +1

According to the topic:a2n≥ a ≥ ak 2n+ 1⟹ak

k≥a2n +12n +1

Sum of digits between 2nand 2n+ 1

2n+ 1ak

2n+ 1a2n+ 12n+1

2n+1ak .

k .(2¿¿n)≥

∑

2n+ 1a2n+ 12n+1.(2n

2n+1ak .

k .(2¿¿n)≥

∑

2n+ 1a2n+ 1

2 .¿a2n +1Is an element of the.

∑

a2n (divergent series)

2n+1a2n +1

2n+1. Isdivergent series

Apply the Comparison Test to the series:ak .

k >a2n+ 12n+1

2n+ 1a2n+ 1

2n+1. Is divergent series⟹

∑

2

n+ 1ak .

k .Is divergent series

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The additional terms ak .

n+… Is divergent series

Task 8:

n =1∞

|

n → ∞

|

n2

+2 n+1n2

+2 nx(2 x+1)

|

+2 n

)

x(2 x+1)

|

⟹ L¿

|

(1+0) x(2 x +1)

|

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⟹

{

x(2 x+1)−1<0

x(2 x+1)+1>0

⟹

{

x−2 x−12 x+1 <0x +2 x+1

2 x+1 >0

⟹

{

−x−12 x +1<03 x +1

2 x +1>0When−x−1

Domain of function: D=

{

x ∈ R∨x ≠−12

}

3 x +1

2 x +1=0 ⇔ 3 x +1=0⇔ x=−1

3 x +1

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⟹ x ∈

(

−∞,−12

)

∪

(

−1

Since ⟨1⟩and ⟨2⟩

⟹

{ [

x ←1x>−12

⟹

[

x←1x >−13

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[1]. Weir, M. D., Hass, J., & Thomas, G. B. [2010]. Thomas' Calculus: EarlyTranscendentals (13th ed.). Pearson Education, Boston..

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final report applied calculus for it

<b> FINAL REPORT</b>

<b>APPLIED CALCULUS FOR IT</b>

<b>FINAL REPORT</b>

<b>APPLIED CALCULUS FOR IT </b>

<b> Ho Chi Minh City, January 6, 2024. </b>

<b>COMPLETED PROJECT AT TON DUC THANG UNIVERSITY</b>

<b> CONTENTS</b>

<b>LIST OF DRAWINGS</b>

<b>CHAPTER 1 THEORETICAL FOUNDATIONS</b>

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