final report applied calculus for information technology

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FACULTY OF INFORMATION TECHNOLOGY

DƯƠNG MINH HIẾU-523H0028NGUYỄN CAO KỲ-523H0049

FINAL REPORTAPPLIED CALCULUS FORINFORMATION TECHNOLOGY

HO CHI MINH CITY, 2024

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DƯƠNG MINH HIẾU-523H028NGUYỄN CAO KỲ-523H0049

FINAL REPORTAPPLIED CALCULUS FORINFORMATION TECHNOLOGY

M.A. Phạm Kim Thủy

HỒ CHÍ MINH CITY, 2024

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First of all, I want to thank Ton Duc Thang University, Department of Information Technology, software engineering industry and I want to thank my teacher, Ms. Pham Kim Thuy. Perhaps because of exposure to a completely new learning environment and creative instructors, my mind felt excited, happy, and surprised. What pleased me most was that, in the first semester, I had the opportunity to interact with Ms. Pham Kim Thuy's Applied Calculus for Information Technology curriculum. I would like to express my sincere gratitude for her passion for teachingand imparting knowledge, which has provided an important support for me to have a source of information about mathematics. applications for the information technology industry.

In addition, I would like to thank the speaker who contacted Ton Duc Thang University, helping me complete this article successfully. Both provided their experiences, such as how to write an article, or how to organize it, so that I could filter the material more clearly.

Finally, I want to wish Ton Duc Thang University, the Department of Information Technology, the Department of Applied Calculus for Information Technology and Ms. Pham Kim Thuy to achieve achievements in life, achieve all goals, especially inthe field of education. As a first-year student, I found that eliminating the limitations in this essay was a challenge, so I looked forward to the advice and comments of the class teacher, Ms. Pham Kim Thuy, to help me complete it. into this essay better in the near future

Ho Chi Minh city, 7th January, 2024 Author

Dương Minh Hiếu Nguyễn Cao Kỳ

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AT TON DUC THANG UNIVERSITY

I hereby declare that this is my own research work and is under the scientific guidance of M.A. Phạm Kim Thủy. The research content and results in this topic are honest and have not been published in any form before. The data in the tables for analysis, comments, and evaluation were collected by the author from different sources and clearly stated in the reference section.

In addition, the Project also uses a number of comments, value assessmentsas well as data from other authors and other organizations, all with citations and source notes.

If any nano species are discovered, I will take full responsibility for the content of my Project. Ton Duc Thang University is not related to these

Hồ Chí Minh city , 7th January, 2024 Author

Dương Minh Hiếu Nguyễn Cao Kỳ

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I applied the mathematical rules, laws, and theorems I learned to my final

report in applied calculus for information technology course, such as

Even-Odd functions, Infinite limits, Derivatives of functions, The equation of tangent line, First derivative test, Arc length formula, and The Ratio test. This enabled me to successfully complete the report, fostering a deeper

comprehension of mathematical operations while solidifying the knowledgeI had acquired.

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CHAPTER 1.SOLUTION

1.1 QUESTION 1

Tell whether the following functions are even, odd, or neither. Give reasons for your answer.

 SOLUTION:

 Even function: for all in the domain of . Odd function: for all in the domain of

Now, let’s analyze the given function:

 Substitute into the function: Now, compare with :

If f(x)=f(−x)⟹ Functioneven .If f(x)=−f(−x)⟹ Function odd .If f(x)≠ f(−x)⟹ Neither .

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f(x)= 4

 Determined set: D ∈ R¿−

√

2;

√

2}¿

 f(−x)= 4(−x)4

−4 ⟹ f(x)=f (− )x

 ⟹ Functionis even .

f(x)= x3

 Determined set: D ∈ R¿−

√

2;

√

2}¿

 f(−x)= (−x)3

x4−4 ⟹ f(x)=− (− )fx

1.2 QUESTION 2

Find the following limit lim 555

x2−25 as:

 SOLUTION:

 x → 5+¿¿

As approaches 5 from the right :

Substitute , where is a small positive number approaching zero:

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Since is approaching zero, the term becomes negligible:

This limit is also infinite, as the denominator approaches 0.Limit as approaches 5 from the right is +∞

 x → 5−¿ ¿

As x approaches 5 from the left :

Substitute , where is a small positive number approaching 0:

Since is approaching 0, the term becomes negligible:

This limit is also infinite, as the denominator approaches 0. Limit as approaches 5 from the left is −∞.

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 x →−5+¿¿

As approaches -5 from the right :

Substitute , where is a small positive number approaching 0:

Since is approaching 0, the term becomes negligible:

So, this limit is negative infinity, as the denominator approaches 0 from thenegative side.

Therefore, the limit as approaches from the right is ∞.

 x →−5−¿ ¿

As approaches -5 from the left :

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 Substitute , where is a small positive number approaching 0:

 Since is approaching zero, the term becomes negligible:

This limit is positive infinity, as the denominator approaches 0 from the positive side.

Limit as approaches from the left is +∞.

1.3 QUESTION 3

Find the derivative dydx of the following functions:

 SOLUTION:

 To find the derivative of the given function , we can use the

quotient rule. The quotient rule states that if you have a function , then the derivative is given by:

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√

x(√x+4)−(√x−4) 12

√

x

(

√

x+ 4)2

√

x+ 4−

√

x+42

√

x(

√

x+4)2

10. 1

√

x

)

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10−1

)

−11

(

 y'=2 ex

 y'

(x0)=2 e0=2 y(x0)= y0=1+2 e0

 ⇒ The point of tangency is A(x0, y0)= A(0,3)

 The equation of the tangent line to the graph of a function at a point A(0,3): y= y'(x0)(x−x0)+ y0

 y=2(x−0)+3 y=2 x +3

 So, the tangent equation at the point at A(0,3) is y=2 x +3.

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 Let f be continuous and c a critical number of f.

o If f' changes from positive to negative at c then f has a local maximum at c;

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

o If f' does not change sign at c, then f has no local max / min at c.

f'(x)=0 ⇔(sinx cosx+ ) (sinx cosx− )=0

⟺

[

x=−π4 + kπ

(k ϵ Z )

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Consideringk ϵ Z ,we have:

∫

(1−cos 2 x)dx−1

∫

(1+cos 2 x)dxf(x)=1

(

x−sin 2x2

)

−1

(

x+sin2x2

)

+C

2x−sin 2x4 −1

2x−sin 2x4 +C

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Look at the variable table, we have:

(

0;π

)

:f(x)is decreasing.o

(

π

)

:f(x)isincreasing.o

(

3π

4, x=5π4 ⟹ f

(

π

)

=−12 .

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1.6 QUESTION 6

Find all curves through a point where x=1 whose are length is the following L

⟺ f'(x)=1

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 The series a2+a4+a8+ a16+… a2n+…diverges.

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

 SOLUTION:

Setting

Consider the series :

Since this series diverges (assumption) it does not approach zero as n increases.

Since

Since series diverges:

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Series diverges.

Series diverges too.

an+1=¿ ¿

lim

|

an+1

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n → ∞

(n+1)(2 x+1)

(n+2)x

(

∞∞

)

Applied L’Hopital Rules:

n → ∞

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

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 R as the initial revenue, which is $55,000.

 ∆P as the increase in price for each $5 increase, which is $5.

 ∆E as the decrease in the number of earphones sold for each $5 increase,which is 20.

The relationship between the price increase and the decrease in the numberof earphones sold can be expressed as:

∆ E=∆ P5

Now, let’s find the number of earphones sold when the price is $255. The pricehas increased by (255−55)=200 dollars, and for each $5 increase, 20 fewerearphones are sold. So, the number of earphones sold (E) is given by:

E=1000−

(

2005

)

× 20E=1000−(40×20)

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My essay has played a significant role in solidifying my knowledge of mathematics and expanding my programming mindset. Throughout this journey, I've come to appreciate not only the intrinsic value of each piece of knowledge butalso the synergy and application of these concepts in the real world. In the realm of programming, questioning the significance of each subject has become a natural habit. This has helped me not only comprehend why I'm studying a particular subject but also understand its connections and applications in the field of programming. The process of self-learning and posing challenging questions has allowed me to build a profound understanding of the material. This has not only enhanced problem-solving skills but has also fueled intrinsic motivation to continue learning and exploring. Evaluating the value of each test, thesis, and class is like constructing a roadmap for personal development. It serves not only as a tool for identifying strengths and weaknesses but also as a source of motivation to continuously improve and achieve larger goals in the learning journey.

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