final report applied calculus for it course

Using several mathematics theories like functions, derivates, applications ofderivatives, simple integration and simple multivariable calculus, etc… to solve the questions and explain it

FACULTY OF INFORMATION TECHNOLOGY

NGUYỄN VIỆT GIA TRỰC – 523H0109 TRẦN VĂN HUY – 523H0035

TON DUC THANG UNIVERSITY

FACULTY OF INFORMATION TECHNOLOGY

NGUYỄN VIỆT GIA TRỰC – 523H0109 TRẦN VĂN HUY – 523H0035

FINAL REPORT

APPLIED CALCULUS FOR IT

COURSE

Instructor M.A Phạm Kim Thủy

Ho Chi Minh City, 2024

Finally, we would like to thank our instructor Pham Kim Thuy, who has

continuously shared her professional knowledge and passion with us The enthusiastic support of M.A Pham Kim Thuy has had a significant impact on our learning journey

- 7 January, 2024

Using several mathematics theories like functions, derivates, applications ofderivatives, simple integration and simple multivariable calculus, etc… to solve the questions and explain it in an intuitive way in order to help the readers tounderstand both the questions and how we solve them

CHAPTER 1 INTRODUCTION 1

1.1 Objective of the report 1

CHAPTER 2 THEORETICAL BASIS 1

2.1 Functions, Limits and Continuity 1

2.1.1 Functions 1

2.1.2 Definition of limit 1

2.1.3 One-Side Limits 2

2.2 Derivative and application 2

2.2.1 Definiton of derivative 2

2.2.2 Derivative as a function 2

2.3 Critical Point 3

2.3.1 Definition 3

2.4 Sequence and Series 3

2.4.1 Definition 3

CHAPTER 3 SOLUTIONS 3

3.1 Question 1 3

3.1.1 Function 1 4

3.1.2 Function 2 4

3.1.3 Function 3 4

3.1.4 Function 4 5

3.2 Question 2 5

3.2.1 Function 1 5

3.2.2 Function 2 5

3.2.3 Function 3 6

3.2.4 Function 4 6

3.3 Question 3 6

3.3.1y=x −4 x +4 6

3.3.2y=x 10−1−10 7

3.4 Question 4 7

3.5 Question 5 8

3.6 Question 6 10

3.7 Question 7 10

3.8 Question 8 12

3.9 Question 9 13

CHAPTER 4 Conclusion 14 References 15

CHAPTER 1 INTRODUCTION

1.1 Objective of the report

Using theories in applied calculus to solve the problems Explain it in a way that help the readers to understand every single steps in solving the problems Increase the capability in solving problem, which occurrs in coding and

programming

CHAPTER 2 THEORETICAL BASIS

2.1 Functions, Limits and Continuity

2.1.1 Functions

Let X and Y be two sets:

Let f : X→Ybe a function

o Xis domain of f;

o Y is codomain of f

Unless otherwise stated, X and Y are always taken to be subsets of the set of real numbers ℝ

We make the following convetion:

o If X is not stated, the domain of f is taken to be the largest possible set(∈ ) on which ℝ f is defined

o If Y is not stated, take Y =R

The range is the set of images:

(but not equal) to a, then we write

x→a=¿f (x)→L2.1.3 One-Side Limits

The right – hand limit:

If as x is close to a from the right f (x) is close to L, or simply

The left – hand limit:

If as xclose to afrom the left f (x) is close to L, or

o fis differentiable at a if f'(a) exists

o f '(a) is the slope of y=f (x ) at x=a

Let x=a+ h then h=x−a⇔ x→a We may use an equivalent definition:

f'(a)≔ lim

x→a

f(x)−f (a)x−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)

2.2.2 Derivative as a function

The derivative of f at point x=a :

h→0

f(a+h)−f (a)h

The derivative of f as a function:

f'(x)=lim

h→0

f(x+h)−f (x)h

o At least one of fx(a,b) and fy(a,b) does not exists

Therefore if z=f (x , y) at a local extreme value at (a,b) then (a,b) is acritical point of f

2.4 Sequence and Series

2.4.1 Definition

A sequence is a list of numbers written in a definite order:

a1,a ,a2 3,… an,……

a1: the first term, a2: the second term,… an: the n termth

The sequence is denoted by {an}

CHAPTER 3 SOLUTIONS

3.1 Question 1

The question requires us to tell which function is odd, even or neither

A function is an even function when f(x)=f(−x), ∀ xєD

A function is an odd function when f(−x)=−f (x), ∀ xє D

So to define a function is even or odd, we need to find f (− )x and compare it with f (x) If f(−x)≠f (x ), we continue to find −f (x) and comparing it If

−f(x)=f (− )x We say that the function is neither even nor odd

Comparing (1) to (2) we can see that f(−x)≠f (x ) but −f(x)=f (− )x

We say that function 2 is an odd function

Comparing (1) to (2) we see that f(x)≠f (−x ) and (2) comparint to (3) is

f(−x)≠−f (x ) so the function 4 is neither even nor odd

Because 555 is a positive numerator and x approches 5 from the postive side,the limit is +∞.

(v')2

In this function u is √x+4 and v is √x−4 Using derivative of square root wewill have:

√x(√x+4)2= 4

Using the Chain Rule in the function 2, we’ll have:

In question 4, it requires us to find the equation of tangent line to the graph

y=1+2 ⅇx at the point where x0=0

To find the equation of tangent line we follow these steps:

1 Find the derivative of y

2 Evaluate the derivative at x

3 Find the y-coordinate at x

4 Write the equation of the tangent line with the general form of the equation is y− y1= y'(x0)(x −x1)

Following the steps, we’ll have:

Step 1: Derivative of y=1+2 ⅇx is y'

=2 ⅇx

Step 2: Replace x=0 in the y' we’ll have y'(x0)=2 e0=2

Step 3: The y-coordinate of the function is the point where the tangent line must pass through on the curve when x=0 So to find the y-coordinate, replace x in the function y=1+2 ex, y1=3

Step 4: As we have the general form of the equation, replace all the numbers

The question requires us to:

To find critical numbers of f we need to find derivative of f, which wealready have, then we solve f'(x)=0 After solving f '(x) = 0, we will criticalnumbers of f

We need to antiderivative f '(x) in order to find the intervals which f isincreasing or decreasing

After having all the necessary data, we write a variation table includes f '(x)

f (x) and x we already find above, then calculated the local minimum and maximum

of f based on variation table

Solving f'(x)=0:

{sin x+ cosx=0

sin x−cosx=0

¿{x=3 π4 +kπ

x=π4+kπ

Now we need to antiderivative f '(x):

∫ (sin x +cos x )(sin x−cos x )d

12

0

As we can see, after passing through π

4 and

3 π

4 the sign of equation

changes, it means that π

4 and

3 π

4 is the critical numbers of f (x)

We also see that the function decreases in the interval from 0to π

function f (x) In the variation table we see that local maximum is −1

Therefore, the value of (L) is approximately 4,828

3.7 Question 7

We’re given a1,a2,a3,…an,… are real numbers with these followingconditions:

an>0 ,n∈ Z+ ¿¿

a1≥a2≥a3≥…≥an≥…

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

We’re required to determine the convergence or divergence of the followingseries Explain in details:

bn be series such that:

0 ≤a ≤bn n for all n (Or for all n≥ N)

Beside that we’ll use Stolz – Cèsaro criterion:

Suppose we have bn is a sequence of real numbers which strictly increasing

or decreasing and unbounded, and (an) is sequence of real numbers, then the limit ofsequence an

Based on the condition that an>0:

So ∀ x ∈ R and |x|≤ the series is absolutely convergent.

3.9 Question 9

This question requires us to find revenue of earphones with given data:One thousand earphones sell for $55 each, resulting in revenue of $55000.For each $5 increase in the price, 20 fewer earphones are sold

The question requires us to find the revenue in case the price of eachearphone is $225

Here’s how to find the revenue when earphones are sold $255 each:Initial price: $ 55

Price increase per step: $ 5

Decrease in sales per price increase: 20 earphones

Number of price increasees from $55 to $255: ((255−55 5)/ : 40 timesTotal decreases in sales due to price increases: 40 20=800 earphones

Initial earphones sold: 1000 earphones

Number of earphones sold at $255: 1000 800 200− = earphones

of critical thinking and solving problems, which is very vital in our road to become

an developer

One of the key findings of this report is applying theories into practice Thisinsight not only enriches our understanding of calculus but also helps us reinforcedour knowledge, increase our ability of solving complex probems, underscoring thepower and versatility of calculus in solving real-world challenges

In conclusion, our journey through the world of Applied Calculus has beenimmensely enriching It has not only enchanted our analytical and problem-solvingskills but has also increase our ability of teamworking As we continue to advance

in our academic road, the knowledge and expericenes accquired through this studywill be invaluabe

[3] James Stewart, [2012], Calculus, Brooks/Cole, Belmont.

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

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

[6] Timothy A Davis, [2011], MATLAB primer, CRC Press, Boca Raton