final report applied calculus for it

VIETNAM GENERAL CONFEDERATION OF LABOR TON DUC THANG UNIVERSITY FACULTY OF INFORMATION TECHNOLOGY BAI HOC TON BUC THANG AMI VER FLA R= FIRINIED CTE PHAM DANG ANH NGOC- 523H0162 FINAL

VIETNAM GENERAL CONFEDERATION OF LABOR

TON DUC THANG UNIVERSITY FACULTY OF INFORMATION TECHNOLOGY

BAI HOC TON BUC THANG AMI VER FLA R= FIRINIED CTE PHAM DANG ANH NGOC- 523H0162

FINAL REPORT APPLIED CALCULUS FOR IT

HO CHI MINH CITY, JANUARY 2024

VIETNAM GENERAL CONFEDERATION OF LABOR

TON DUC THANG UNIVERSITY FACULTY OF INFORMATION TECHNOLOGY

BAI HOC TON BUC THANG

TON DUC THANG UNIVERSITY

PHAM DANG ANH NGOC- 523H0162

FINAL REPORT APPLIED CALCULUS FOR IT

Lecturer M.A Pham Kim Thuy

HO CHI MINH CITY, JANUARY 2024

ACKNOWLEDGEMENT

I would like to express my sincere gratitude to Ton Duc Thang University, the Faculty of Information Technology, and the Department of Applied Analysis, as well as to MA Pham Kim Thuy, for their efforts in creating the best learning conditions for me during this period This report represents the culmination of the knowledge I have acquired and am currently learning, thanks

to the dedicated and passionate teachings of Ms Pham Kim Thuy With this knowledge, I have gained a deep understanding of the subject, broadened my perspectives, and can now apply it in practical situations, laying the foundation for 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

1 hereby affirm that this research project is my own work and was conduct under the scientific guidance of Dr Pham Kim Thuy The research content and results presented in this thesis are truthful and have not been publish in any form prior to this The author collected the data in the tables, used for analysis, comments, and evaluations, from various sources, as explicitly stated in the reference section

Furthermore, this project incorporates some comments, evaluations, and data from other authors and different organizations, all of which are appropriately 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 this project

Ho Chi Minh City, January 6, 2024

Author,

(Signature and Full Name))

Ngoc Pham Dang Anh Ngoc

1H

CONTENTS

VN obt-ũa':34ÝÃẼẢÝŸỶ

“Z0 9n cccecceccecccssssccesscsecsscceceeccecsssessessseesesseceetsssesssesetteseesetseenteseessnnteas

Figure 1:Task 4

Figure2: Task 6

LIST OF DRAWINGS

CHAPTER 1 THEORETICAL FOUNDATIONS

We say that f(x) has right-hand limit L atc, and write ee sm Le é

If for every number €>0 there exists a corresponding number 6>0 such that for all

x ¢<x<ct+6=|f [x|—L|<e

We say that f(x) has left-hand limit L atc, and write am hi é

If for every number €>0 there exists a corresponding number 6>0 such that for all

Task 4:

The derivative of the function f at a number a is

1a fla+h|—f(a)

f lal=lim h

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' la):

y—f |aj=f '(a)(x-a)

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

==ƒlc]>ƒ|x, For allx near€ (for all x in an open interval containing c)

f Has a local (or relative) minimum at c € D

<= f\c|<f|x)For allx nearc (for all x in an open interval containingc)

Let f be continuous andca critical number off

Suppose fis differentiable near c(except possibly atc )

e Iff' changes from positive to negative atc, Then f has a local maximum atc

e Iff' changes from negative to positive atc, Then f has a local minimum atc

e Iff'does not changes sign atc, Then f has no local max/min atc

0<a,<b,For all n (Or for all n=Né then

» (b„|converges => (a, |converges

The Comparison Test

Let >" | a,| Be series

i=0

n+1

a Suppose lim | a —L where 0< Lem

The general formula for revenue (R) based on the price per earphone (P) and the

initial quantity (Qo¢can be express as

R=Pé

Qwis the initial quantity of earphones

Pois the initial price per earphone

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

CHAPTER 2 MAIN CONTENT

Task 5: Given the derivativef |x|=é (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 are length is the following L value: (1.0 point)

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

nx"

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 For ex., if the price of each earphone is $60, there will be 980 (1000 — 20) earphones sold; 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)

So x.5” —œ¿

e x —-5*°°

lim Gỗ lim é

555

x¬—B5 sanh X~—Š TT -R1xrB]”

Set y=ƒÍX]= 1+2e*

Slope of ƒ(x)=1+2e* atx=0:

10

® fx)

Figure 1:Task 4 Task 5:

f ix)=é

To find the critical numbers of f

x+=kn(k €Z)

T1 — x——=kn(kEZ) 4

HH

ƒ' Íx] Change from negative to positive at x=1 and X= “T—

=ƒ xì Has a local minimum at x=t and x= ot

=Í dụ=+ J ~

=y=+ln|x|+e

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

To substitutex=0 into the function y=+Ink |rc

=y=+ln |l|+c

=c=0

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

is the following L value

Figure2: Task 6

Task 7:

cóc [đya|[ „| (net)! [nellaxell"

Set L¿ lim |——Elim n+s| đụ | nà Ín+1#1]l2x+]] avi) nx ¬

x

—x—]

=| 2x†1

3x+1 2x+1

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

as the product’s revenue

== R=(1000—Au.Q).t

== R=(|1000—20.40).255=51000$

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

17

REFERENCES

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