final report applied calculus for it

FACULTY OF INFORMATION TECHNOLOGYPHAM DANG ANH NGOC- 523H0162 FINAL REPORTAPPLIED CALCULUS FOR ITHO CHI MINH CITY, JANUARY 2024... FACULTY OF INFORMATION TECHNOLOGYPHAM DANG ANH NGOC- 52

FACULTY OF INFORMATION TECHNOLOGY

PHAM DANG ANH NGOC- 523H0162

FINAL REPORT

APPLIED CALCULUS FOR IT

HO CHI MINH CITY, JANUARY 2024

FACULTY OF INFORMATION TECHNOLOGY

PHAM DANG ANH NGOC- 523H0162

FINAL REPORT

APPLIED CALCULUS FOR IT

Lecturer

M.A Pham Kim Thuy

HO CHI MINH CITY, JANUARY 2024

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, thanks

to 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

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 for the content of my project Ton Duc Thang University is not implicate in any

copyright violations or infringements that may arise during the course of thisproject

Ho Chi Minh City, January 6, 2024

Author,(Signature and Full Name))

NgocPham Dang Anh Ngoc

CONTENTS

LIST OF DRAWINGS

CHAPTER 1 THEORETICAL FOUNDATIONS

CHAPTER 2 MAIN CONTENT

2.1Topics

2.2 Solutions

REFERENCES

LIST OF DRAWINGS

Figure 1:Task 4 9Figure2: Task 6 12

CHAPTER 1 THEORETICAL FOUNDATIONS

fIs differentiable at a if f'(a)exists

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 slope

f '(a):

y−f(a)=f '(a)(x−a)

Task 5:

A critical number of a function f is a number cin the domain of fsuch thateither f '(c)=0∨f '(c) does not exist

It is continuous on[a,b], differentiable on (a,b) then

f'(x)=0 on(a,b)⟺ f, is constant on (a,b)

f'(x)>0 on(a,b)⇒ f, is increasing on (a,b)

f'(x)<0 on(a,b)⇒ f, is decreasing on (a,b)

Let f be a function with domainD

f Has a local (or relative) maximum at c ∈ D

⟺ 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

(bn) Be series such that

0 ≤an≤bnFor all n (Or for all n≥ N¿ then

Q0is the initial quantity of earphones

P0is the initial price per earphone

bis the rate at which the quantity decreases for each $5 increase in price

CHAPTER 2 MAIN CONTENT

x→5− ¿ ¿

x→−5+ ¿¿

x→−5− ¿¿

Task 3:Find the derivatives dy

dx of the following functions: (1.0 point)y=√x−4

Task 5:Given the derivativef'(x)=¿ (1.0 point)

What are the critical numbers off?

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)

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

Determine the convergence or divergence of the following series Explain indetails

3 π4

5 π4

⟹ f(x) Has a local maximum at x=3 π

4 and x=

7 π4

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

⟹ f(x) Has a local minimum at x=π

4 and x=

5 π4

The curves through a point where x=1 ⟹ y=0

To substitutex=0 into the function y=± ln|x|+c

⟹ y=± ln |1|+c

⟹ c=0

So y=ln x and y=−ln x are 2 curves through a point where x=1 whose arc length

is the following L value

Figure2: Task 6

Task 7:

The series a2+a a a4+ 8+ 16+…+a2n+…=∑

The additional terms ak

3 x +1

I denote ∆ u as the difference in the quantity of products after each price increase, R

as the product’s revenue

⟹ R=(1000−∆ u.Q ).t

⟹ R=(1000 20.40− ).255=51000 $

Therefore, when the price of each earphone is 255 $ the revenue is 51000 $

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