Final report applied calculus for information technology

THEORETICAL BASIS1.1 Question 1: Definition of even and odd functions: An even function satisfies f−x=fx for all x in its domain.. An odd function satisfies f−x=−fx for all x in its doma

FACULTY OF INFORMATION TECHNOLOGY

NGUYỄN TRẦN HOÀNG NHÂN – 523H0164 MAI HOÀNG THÁI – 523H0177

FACULTY OF INFORMATION TECHNOLOGY

NGUYỄN TRẦN HOÀNG NHÂN – 523H0164 MAI HOÀNG THÁI – 523H0177

FINAL REPORT

APPLIED CALCULUS FOR INFORMATION TECHNOLOGY

Instructor

M.A PHẠM KIM THỦY

HO CHI MINH CITY, 2023

We extend our heartfelt gratitude to Ton Duc Thang University, Faculty ofInformation Technology, especially the Applied Calculus for InformationTechnology subject, for granting us the invaluable opportunity to do this project Oursincere appreciation goes to M.A Phạm Kim Thủy for her unwavering support,invaluable guidance, and encouragement that were instrumental in the successfulcompletion of our project

Acknowledging the limitations in our knowledge, we acknowledge thepossibility of errors in the execution and finalization of this report We humbly inviteand greatly value the feedback, insights, and contributions from our esteemedteachers and mentors Your input will undoubtedly enhance the quality and accuracy

THE COMPLETED PROJECT AT

TON DUC THANG UNIVERSITY

I hereby declare that this research project is my own work, conductedunder the scientific guidance of M.A Phạm Kim Thủy The research content andresults in this topic are honest and have not been presented in any form before.The data presented in tables used for analysis, comments, and evaluations werecollected by the author from various explicitly cited sources in the referencesection

Furthermore, this project incorporates some comments, assessments, anddata from other authors and organizations, all of which are properly cited andreferenced

Should any form of academic misconduct be identified, I take full responsibility for the content of this project Ton Duc Thang University is

not liable for any copyright violations or infringements that may arise duringthis work (if any)

Ho Chi Minh City, December 26 , th

2023.

Author (Signed and write with full name)

TABLE OF CONTENTS

CHAPTER 1 THEORETICAL BASIS

1.1 Question 1:

1.2 Question 2:

1.3 Question 3:

1.4 Question 4:

1.5 Question 5:

1.6 Question 6:

1.7 Question 7:

1.8 Question 8:

1.9 Question 9:

CHAPTER 2 SOLUTIONS

2.1 Question 1:

2.1.1 Solution:

2.1.2 Summary:

2.2 Question 2:

2.2.1 Solution:

2.2.2 Summary:

2.3 Question 3:

2.3.1 Solution:

2.3.2 Summary:

2.4 Question 4:

2.4.1 Solution:

2.4.2 Summary:

2.5 Question 5:

2.5.1 Solution:

2.5.2 Summary:

2.6 Question 6:

2.6.1 Solution:

2.6.2 Summary:

2.7 Question 7:

2.7.1 Solution:

2.7.2 Summary:

2.8 Question 8:

2.8.1 Solution:

2.8.2 Summary:

2.9 Question 9:

2.9.1 Solution:

2.9.2 Summary:

REFERENCE MATERIAL

CHAPTER 1 THEORETICAL BASIS

1.1 Question 1:

Definition of even and odd functions:

An even function satisfies f(−x)=f(x) for all x in its domain

An odd function satisfies f(−x)=−f(x) for all x in its domain

We say the limit of f(x), as x approaches a, equals L

Sometimes we may simply write:

f(x)→ L

Note: The limit limf(x)

 depends only on the values of f(x) for "x near a"

 is independent of the value of f(x) "at a"

Step 1: Find the y0 by substituting x0 in the function.

Step 2: Find the derivative of the function y =f(x) and represent it by f '(x).Step 3: Substitute the point (x0, y0) in the derivative f(x) which gives theslope of the tangent m

Step 4: Find the equation of the tangent using the point-slope form

y−y0=m(x −x0)

1.5 Question 5:

Definition of critical number:

Let f be a function with domain D Then c ∈ D called a critical number of f

if:

 f ’(c) does not exist, or f ’(c) exists and equal to 0

Definition of local maximum and local minimum:

Let f be a function with domain D

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

⇔ f(c)≥ f(x) for all x near c

 f has a local (or relative) minimum at c ∈ D

⇔ f(c)≤ f(x) for all x near c

Theorem of increasing or decreasing on the interval:

 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]

1.6 Question 6:

Let f be a continuous function on [a , b]

 Divide [a , b] into n equal subintervals, say

 a1: the 1 term; st a2: the 2 term; , nd a n: the n term.th

Theorem The P-Series:

Theorem of Ratio test:

Step 2: Find the number of price increases

Step 3: Substitute the number of price increases into functions to find therevenue

Conclusion of even function test: The function is even.

Odd function test:

¿555 lim 1

x−5lim

1

x+52.2.1.1 As x approaches 5+¿:¿

Substitute x=5+ ¿ ¿ into the expression:

This behavior aligns with the concept that when the denominator approacheszero from the positive side, the fraction tends toward positive infinity

This behavior aligns with the concept that when the denominator approacheszero from the negative side, the fraction tends toward negative infinity.

2.2.1.3 As x approaches −5+ ¿ ¿

:Substitute x=−5+ ¿¿

into the expression:

This behavior aligns with the concept that when the denominator approacheszero from the positive side, the fraction tends toward positive infinity

2.2.1.4 As x approaches −5− ¿¿

:Substitute x=−5− ¿¿

into the expression:

This behavior aligns with the concept that when the denominator approacheszero from the negative side, the fraction tends toward negative infinity.

2√x(√x+4−√x+4)(√x+4)2

Find an equation of the tangent line to the graph of y=1+2e x the point where

Find the derivative of the function y=1+2e x with respect to x Thederivative, denoted as ⅆⅆy x, provides the slope of the tangent line at any given point.Apply sum rule:

f '(x)=(1)'

+(2e x)'

¿2e x

Evaluate the Derivative at x=0:

Substitute x=0 into the derivative to determine the slope of the tangent line

at the point of interest

Given the derivative f '(x)=(sinx +cos x) (sinx −cos x), 0≤ x ≤ π2 (1 point)

What are the critical numbers of f?

On what open intervals is f increasing or decreasing?

At what points, if any, does f assume local maximum and minimum values?2.5.1 Solution:

 Critical numbers:

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

sinx−cosx=0⇔[ √2 sin(x+π4)=0

 Local maximum and local minimum:

Base on variation table:

Local minimum at x=π, x=5π

Local maximum at x=3π

4 , x=7π

4.2.5.2 Summary:

Critical numbers: x=π

4, x=3π

4 , x=5π

4 , x=7π4

Intervals f increase are: [π

+1

du= x

√x2

+1ⅆ xWhen x=1, u=√2 and when x=5, u=√26

Then:

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

Determine the convergence or divergence of the following series Explain indetails

Given a divergent sequence a2n, it suggests that this sequence can either be auniform sequence with all terms being equal, or it may follow a certain pattern,possibly remaining constant or decreasing within a range, but it will never decrease

to zero due to its divergence according to the given condition, therefore a2n willapproach a specific nonzero value when n approaches to infinity.

Considering the first case:

Considering the second case:

a2n may follow a certain pattern, but when n approaches to infinity, a2n willapproaches to a specific nonzero value

As n approaches to infinity, we can consider a2n, a n are equivalent

So: we can consider a2n =v (v is a specific nonzero value) and a2n =a n=vwhen

Because when calculating the limit, x

2x+1 does not depend on n as n approaches infinity, so we can consider it as a constant or a coefficient, and the limit

of a constant or coefficient times a function is equal to the constant times the limit

For ex., 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 – ) earphonessold; so on Find the revenue in case the price of each earphone is $ 255 (1 point).2.9.1 Solution:

For each $ 5 increase in the price, 20 fewer earphones are sold, so:

n is the number of price increases

S is the quality of earphones are sold

C is the cost of each earphone

[3] R W Hamming, [1986], Numerical methods for scientists andengineers, Dover, New York.

[4] Steven C Chapra, [2012], Applied numerical methods with MATLABfor engineers and scientists, McGraw-Hill Education, New York

[5] Timothy A Davis, [2011], MATLAB primer, CRC Press, Boca Raton.Online materials:

[1] www.wikipedia.org