Calculus by Muhammad Umer

By Muhammad Umer

This book is focused towards calculus and some applications of calculus in Biology and Physics. Its topics include but aren't limited to partial derivatives , limits , differential equations , differentiation and integration.

• Language: English
• Publisher: Muhammad Umer
• Release date: Jul 1, 2022
• ISBN: 9781005398637

Book preview

Calculus by Muhammad Umer

The author of this book is Muhammad Umer. Umer is a student of A level 1st year at Roots International school in Lahore , Pakistan. This book is focused towards calculus. It also includes few undergraduate level calculus topics and a touch of calculus in biology and physics.

Note

This book may not be reproduced but with the permission of author.

Should you like to report a mistake or something that you think is incorrect , please don’t hesitate to email me at [e-mail] . Infact , If you do , I’ll really appreciate that.

Many times , the term we has been used. This is just to create a sense of team work progress and does not means that the book has multiple authors.

Context

Chatper 1 : Differentiation

Chapter 2 : Integration

Chapter 3 : Trigonometric , logarithmic and exponential integration

Chapter 4 : Functions

Chapter 5 : Partial derivatives

Chapter 6 : Differential equation

Chapter 7 : Limits

Chapter 8 : Geometric progression

Chapter 9 : Calculus and Biology

You’ll find a few applications of calculus in biology and an idea of how calculus is used in economics.

Chapter 10 : Calculus and Physics

In the end , you’ll find some basic info about optimization.

Introduction to Calculus

Calculus was developed by Newton and leibniz in the 17th century.

It is done by process that are based on summation of infinitesimal differences. Calculus deals with derivatives and integrals of functions , calculus can be be divided in these two parts.

Derivatives are releated to differentiation and integrals refer to integration.

The derivative is the measure of the rate of change of a function whereas integral is the measure of the area under the curve. The derivative explains the function at a specific point while the integral accumulates the discrete values of a function over a range of values.

Differentiation

This marks the beginning of chapter 1 , differentiation. I aplogize to have this note added before the start of every chapter. Keeping in view the format of epub files , this was done to let reader know when a new chapter starts. Please do not consider the following headings as new chapters , they are the sub-topics of the same chapter and continue to be until you read a similar note.

Differentiation

Differentiation is the gradient of tangent to a curve. It can be used to find the derivative of a function. In addition to that , it allows us to find the rate of change e.g. rate of change of velocity with respect to time.

There are different rules that apply to different operators that we need to differentiate. Like if we need to differentiate two variables that multiply , we use the product rule.

Similarly , to differentiate two variable that are divided by each other , we will use the quotient rule. Another rule which we will talk about is the chain rule , which has further three types ; the power rule , the exponential rule and the logarithmic rule.

If we want to differentiate any equation , then , the differentiation of an alone constant is zero. Alone constant can be any number. For example in the equation 3𝑥+7 , the differentiation of 7 ( which is an alone constant) is zero. The

derivative of a variable is equal to its coefficient. The coefficient of variable 𝑥 in 3𝑥+7 is 3. So the complete differentiation of the equation 3𝑥+7 is 3+0 which is equal to 3. Its reason will be explained later.

The differentiation of an expression is represented of dy/dx. If we need to say that 𝑥 is the differentiation of an expression , we can simply say that dy/dx is equal to 𝑥.

The Power Rule

This rule only applies to a variable or an equation that has a power on it , power which shows that something is multiplied by itself the power number of times e.g. 𝑥² , 𝑥⁵ , 3𝑥⁷ , (4𝑥+9)³ etc.

If any expression has a power on it , we can say that the operator is power and when we know that the operator is power , we know that we need to apply the power rule.

There are steps that