Calculus 1 project report

This report will explain the theoretical basis of Derivative, Integral, and Differential equations, and study the application of the three fields by solving 5 practical problems: 2 Deriv

NATIONAL UNIVERSITY OF HO CHI MINH CITY

HO CHI MINH CITY UNIVERSITY OF TECHNOLOGY

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CALCULUS 1PROJECT REPORT

Lecturer: Ms Phan Thị Khánh Vân

Class: CC13

First of all, we are grateful to Phan Thi Khanh Van, M.S for assigning this project and teaching us the exercise lectures on CALCULUS 1 Then, we would like to express our gratitude to the people who assisted us with their help and guidance in this project; it would not have been possible otherwise

We are proud to demonstrate the result of our research and work We have made every effort to produce the best outcome, although there may be some limitations in our report.

ROLE OF THE MEMBERS IN THIS PROJECT

1 Nguyễn Hữu Dũng

2 Nguyễn Quốc Anh

3 Đoàn Quang Đăng

4 Tôn Tuyết Nhi

TABLE OF CONTENTS

1 INTRODUCTION 4

2 APPLICATION OF DERIVATIVE 5

3 APPLICATION OF INTEGRAL 8

4 APPLICATION OF DIFFERENTIAL EQUATION 13

5 CONCLUSION 23

6 REFERENCES 24

1 INTRODUCTION

Mathematics has wide real-life applications in finance, shopping, time management, and health tracking In engineering, it's critical for designing structures, calculating dimensions, and modeling physical phenomena Engineers heavily use math to solve complex equations, analyze data, and optimize systems, ensuring efficient, safe, and functional structures and technologies Overall, mathematics acts as a powerful tool in multiple domains, enabling problem-solving, and data analysis, enhancing our daily lives, and driving technological advancements.

Matlab excels as an invaluable ally for engineers' computations It's a specialized, high-level programming language fine-tuned for numerical computations and data analysis With its diverse range of functions and tools, it empowers engineers to tackle intricate mathematical problems, run simulations, and meticulously analyze data Its adaptable nature and user-friendly interface are indispensable, enabling engineers to streamline calculations, accelerate research, and pioneer innovations across diverse engineering domains This report will explain the theoretical basis of Derivative, Integral, and Differential equations, and study the application of the three fields by solving 5 practical problems: 2 Derivative problems, 2 Integral problems, and 1 Differential equation problem In addition, we will also provide the Euler method solution to calculate approximately the result for the Differential equation problem This report will go through each problem to explain the script and illustrate the way of solving them mathematically and solving them using Matlab.

function f(x) = 2x + 3x - 12x + 5., using the first derivative3 2test.

Solution: The given function is f(x) = 2x + 3x - 12x + 53 2f'(x) = 6x + 6x - 122

f'(x) = 0; 6x - 6x - 12 = 0, 6(x + x - 2) = 0, 6(x - 1)(x + 2) = 02 2Hence the limiting points are x = 1, and x = -2

Let us take the points in the immediate neighborhood of x = 1 The points are {0, 2}

f'(0) = 6(0 + 0 - 2) = 6(-2) = -12, and f'(2) = 6(2 + 2 - 2) = 6(4) 2 2

= +24

The derivative of the function is negative towards the left of x =

1, and is positive towards the right Hence x = 1 is the local minima

Let us now take the points in the immediate neighborhood of x =-2 The points are {-3, -1}

f'(-3) = 6((-3) + (-3) - 2) = 6(4) = +24, and f'(-1) = 6((-1) + (-1)2 2-2) = 6(-2) = -12

The derivative of the function is positive towards the left of x = -2, and is negative towards the right Hence x = -2 is the local maxima.

Therefore, the local maximum is 2, and the local minimum is 1

II/ Problem

There is a shop selling shoes at the price of $150 per pair ofshoes With this price, the store sells about 40 pairs This storehas a discount program It is estimated that if the store reducedeach pair by $5 the number of pairs sold would increase by 15pairs Determine the selling price so that the store can earn thegreatest profit, knowing that the initial import price of each pair

is $95

- Solution:

Call x is the price of each pair in the store (x: $; $95 ≤ x ≤ $150)

We know that:

- If $150 each pair they can sell 40 pairs

- If they discount $5 each pair, they will be able to sell more 10pairs

- If they discount $150 - x each pair they will be able to sellmore:

105(150−x) = 2(150−x)The amount of pairs sells with cost x:

40 + 2(150−x) = −¿2x + 340Call F(x) is the equation of income that the store raises whensell shoes:

F(x) = (−¿2x + 340) × (x−¿95) = −¿2X2 + 530x −¿ 32300

So, we need to find a local max of F(X) when (x: $; $95 ≤ x ≤

$150)

F '(x) = −¿4x + 530 = 0 ⟹x = 132.5

The function F(x) is continuous from 95 to 150:

F (95) = 0

F (132.5) = 2812.3

F (150) = 2200

=> when x=132.5 that F (x) is local maximum

=> The highest income when the price of 1 pair is $135

III/ Code Matlab

% Call x is the price of one pair in store ($95 ≤ x ≤ $150)

x = linspace(95,150,56);

% We know that:

% If $150 for each pair they can sell 40 pairs

% If they discount $5 for each pair, they will be able to sell more than 10 pairs

% If they discount $150-x each pair they will be able to sell more pairs:

y= -2*x.^2 + 530*x - 32300;% This function is from b*(x-95)

% To find the highest income we need to find a local maxima offunction y:

3 APPLICATION OF INTEGRAL

I/ Introduction

Integration is a foundational concept in calculus that plays a crucial role in understanding and solving problems related to accumulation, area under curves, and various applications in mathematics and science This report aims to provide a

comprehensive overview of the theory of integration, covering its definition, properties, and applications

Definition and Basics of Integration

Integration involves finding the integral of a function,

representing the accumulation of quantities or the area under a curve The integral is denoted by the symbol ∫ and is expressed

as ∫f(x)dx, where f(x) is the integrand and dx indicates the differential variable

a

b

f (x )dx, where a and b are the limits of

integration

Fundamental Theorem of Calculus

The fundamental theorem of calculus establishes a fundamental connection between differentiation and integration

f F(x) is an antiderivative of f(x) then as∫

a

b

f (x )dx= F(b) - F(a)

Basic examples of integration

-Example of indefinite integration:

Find the antiderivative of f(x)=3x2

F(x) = ∫3x2dx =x3

+ C

Here, C is the constant of integration

-Example of definite integration:

Application of integral in economics

Problem: After t hours of work, a certain worker can produce at a rate of 100 + e −0.5 t units per hour Assuming the individual starts working at 8:00 AM, the question is, how manyunits will the person produce between 9:00 AM and 11:00 AM?Solution:

Let Q(t) represent the quantity produced by a person after t hours, measured from 8 AM

The given derivative is Q'(t) = 100 + e −0.5 t

To find the quantity produced by the person between 9 AM ( t =

1 ) and 11 AM ( t = 3 ), we calculate the integral:

Therefore, the quantity produced by the person from 9 AM to 11

AM is 200.76 units

% Define the function for the production rate

production_rate = @(t) 100 + exp(-0.5*t);

% Prompt the user for the start and end times

start_time = input('Enter the start time (e.g., 9 for 9:00 AM): ');

end_time = input('Enter the end time (e.g., 11 for 11:00 AM): ');

% Convert the time to hours since 8:00 AM

% Display the result

fprintf('Total units produced between %d:00 AM and %d:00 AM: %.2f units\n', start_time, end_time, total_units_produced);

Application of integration in calculating work done Problem: A crane is lifting a container with mass

m=20.103 kg from the ground to a height h=5m at a constant rate Find work done by the crane

W = integral(F, 0, H, 'ArrayValued', true);

% Display the result

disp(['The work done is ', num2str(W), ' (J)'])

4 APPLICATION OF DIFFERENTIAL EQUATION

1 What differential equations are?

In short, a differential equation is an equation that contain an unknow function and one or more functions with its derivatives, that involve one variable (dependent variable) with respect to the other variable (in dependent variable).

y ’ =f ( x)= dy

dx

Here “x” is an independent variable and “y” is a dependent variable

2 What does it use for?

The derivatives of the function define the rate of change of a function at a point It is mainly used in fields such as physics, engineering, biology, … The primary purpose of the differentialequation is the study of solutions that satisfy the equations and the properties of the solutions, so we can do calculations make graphs, predict the future, and so on

3 How many types of them?

First we have to know that there are 2 main orders of the Differential equation, these created all the types of differential equation that we learn Calculus 1:

 First-order Differential equation

It has only the first derivative such as dy/dx, where x and y are the two variables and is represented as:

dy

dx =f (x , y )= y’

 Second-order Differential equation

The equation which includes the second order derivative is the second order Differential equation, it is represented as:

The mixing problem has 2 types: if the liquid has rate in is equal

to rate water out, we can use separable differential equation to solve the problem But if the liquid has a rate different than the rate out, we must use Linear differential equation to solve the problem

1 Separable differential equations.

 The problem:

A tank contains 20 kg of salt dissolved in 5000 L of water Solution that contains 0.03 kg of salt per liter of water enters thetank at a rate of 25 L/min The solution is kept thoroughly mixed and drains from the tank at the same rate How much salt remains in the tank after half an hour?

 Solution:

Let y(t) be the amount of salt (in kilograms) after t minutes We are given that y(0) = 20 and we want to find y(30) We do this

by finding a differential equation satisfied by y(t) Note that dy dt

is the rate of change of the amount of salt, so:

dy=¿

Where (rate in) is the rate at which salt enters the tank and (rate out) is the rate at which salt leaves the tank We have:

rate in ¿(0.03kg

L)(25 L min)=0.75 kg

min

The tank will always contain 5000 L of liquid, so the

concentration at time is 5000y (t) (measured in kilograms per liter) Since the brine flows out at a rate of 25 L/min, we have:

rate out ¿(y(t)

5000

kg

L) (25 L min)=y (t )200

kg min

Thus, from the rate of change of the amount of salt equation, weget:

dy

dt=0.75−y(t)

200 =150− y (t)200

Solving this separable differential equation, we get:

Since y(t) is continuous and y(0) = 20 and the right side is never

0, we deduce that 150 – y(t) is always positive Thus |150-y| =

y( 30 )=150−130 e

−30 200

3L/min If y(t) is the amount of salt (in kilograms) after t minutes, show that satisfies the differential equation:y

Let y(t) be the amount of salt (in kilograms) after minutes We t

are given that y(0) = 0 and we want to find y(20) We do this by finding a differential equation satisfied by y(t) Note that dy dt is the rate of change of the amount of salt, so:

min

Since solution is drained from the tank at a rate of 3 L/min, but salt solution is added at a rate of 5 L/min, the tank, which starts out with 100 L of water, contains (100 + (5-3)t) L of liquid after

t min Thus, the salt concentration at time t is:

y (t)

100+2t

kg L

Therefore the salt leaves the tank at a rate of:

Rate out ¿( y(t)

100+2 t

kg

L) (3 L min)= 3y(t)

100+2t

kg min

Thus, from the rate of change of the amount of salt equation, weget:

3 100+2 tdx

III MATLAB’S CODES:

1 Application of the Separable differential equation

% Input value

initial_salt = input('Please input the value for the initial amount

of salt in the tank (kg): ');

initial_volume = input('Please input the value for the initial volume of water in the tank (L): ');

rate_water_in = input('Please input the value for the inflow rate

of salt solution (L/min): ');

rate_water_out = input('Please input the value for the outflow rate of salt solution (L/min): ');

C_in = input('Please input the value for the concentration of salt

in incoming brine (kg/L): ');

time_to_simulate = input('Please input the value for the time to simulate (minutes): ');

% Define the differential equation

dydt = @(t, y) (rate_water_in * C_in) - ((rate_water_out * y) / (initial_volume + (rate_water_in - rate_water_out) * t));

% Solve the differential equation using ode45

[t, y] = ode45(dydt, [0, time_to_simulate], initial_salt);

% Plot the solution

plot(t, y);

xlabel('Time (minutes)');

ylabel('Amount of salt in the tank (kg)');

title('Salt Concentration in the Tank Over Time');

% Display the concentration after time_to_simulate

concentration_after = y(end);

fprintf('Concentration of salt after %.2f minutes: %.2f kg/L\n', time_to_simulate, concentration_after);

 The result:

2 Application of the Linear differential equation.

% Input value

initial_salt = input('Please input the value for the initial amount

of salt in the tank (kg): ');

initial_volume = input('Please input the value for the initial volume of water in the tank (L): ');

rate_water_in = input('Please input the value for the inflow rate

of salt solution (L/min): ');

rate_water_out = input('Please input the value for the outflow rate of salt solution (L/min): ');

C_in = input('Please input the value for the concentration of salt

in incoming brine (kg/L): ');

time_to_simulate = input('Please input the value for the time to simulate (minutes): ');

% Define the differential equation

dydt = @(t, y) (rate_water_in * C_in) - ((rate_water_out * y) / (initial_volume + (rate_water_in - rate_water_out) * t));

% Solve the differential equation using ode45

[t, y] = ode45(dydt, [0, time_to_simulate], initial_salt);

% Plot the solution

plot(t, y);

xlabel('Time (minutes)');

ylabel('Amount of salt in the tank (kg)');

title('Salt Concentration in the Tank Over Time');

% Display the concentration after time_to_simulate

concentration_after = y(end);

fprintf('Concentration of salt after %.2f minutes: %.2f kg/L\n', time_to_simulate, concentration_after);

 The result:

5 CONCLUSION

Through this Calculus 1 major assignment, our group has learned how to work together and also about Matlab symbolism We have learnt how to solve Derivatives, Integrals, and Differential equations problems, and perform the Euler method using Matlab symbolic calculation along with the knowledge to choose an appropriate stepsize in order to obtain a certain accuracy.

In addition, our group would like to send our most sincere gratitude to Phan Thi Khanh Van, M.S, the lecturer in class CALCULUS 1 (EXERCISE) CC13, is also the instructor for this assignment Thanks to your wholehearted guidance, our group completed the report on schedule and resolved the problems encountered Your guidance has been the guideline for all of the group actions and maximized the supportive relationship between the lecturers and the students in the educational environment.

Additionally, we would also like to sincerely thank Dr Le Xuan Dai, the lecturer in class CALCULUS 1 CC07 for teaching the theory very carefully so that we have a solid foundation to carry out this major assignment

This is also the first time our group has done a major assignment and written a report for it And, we might have made some mistakes Therefore, we hope to receive your sympathy for our shortcomings in this report.