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Limits in Calculus

Calculus | Differential and Integral Calculus

Last Updated : 07 Apr, 2025

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Calculus was founded by Newton and Leibniz. Calculus is a branch of mathematics that helps us study change. It is used to understand how things change over time or how quantities grow, shrink, or accumulate. There are two main parts of calculus:

Differential Calculus: It helps us calculate the rate of change of one quantity concerning another. This rate of change is called the derivative.
- Example: Finding how fast a balloon inflates as you pump air into it.
- Calculating the slope of a hill (steepness).

Integral Calculus: helps us calculate the total accumulation of change. This accumulation is called the integral.
- Example: Calculating the area under a curve (e.g., finding the distance traveled by a car when you know its speed at every moment).
- Determining the total rainfall collected in a reservoir.

Note: The process of finding the value of a derivative is called differentiation, and the process of finding the value of an integral is called integration.

Basic of Calculus

This section covers the basics of calculus, including functions, limits, and continuity. You will learn key techniques for finding limits and understanding discontinuities in functions.

Functions
- Domain and Range of a Function
Limits
Formal Definition of Limits
- One-Sided Limits
- Infinite Limits
- Limits at Infinity
Techniques for Finding Limits
Properties of Limits
Continuity of Functions
Discontinuity
- Types of Discontinuity

Differential Calculus

This section explores differential calculus, focusing on derivatives and their applications. You will learn differentiation rules, including the power, product, quotient, and chain rules, along with real-life applications such as rate of change, extrema, and curve sketching.

Differentiability
- Mean Value Theorem: Cauchy’s Mean Value Theorem
- Rolle’s Mean Value Theorem
Derivative
- Derivative by First Principle
- Algebra of Derivative of Functions
Rules for Differentiation
Formulas for Differentiation
Examples of Derivatives
Application of Derivatives
Concavity and Points of Inflection
Curve Sketching
Partial Derivatives
Higher Order Derivatives
Antiderivatives
Real-Life Application of Differentiation

Integral Calculus

This section covers the fundamentals of integral calculus, exploring the concept of integration and its relationship to differentiation. You will learn various methods of integration, such as substitution and integration by parts, and apply these techniques to solve real-world problems involving areas, volumes, and surfaces.

Introduction to Integration
- Antiderivative: Integration as an Inverse Process of Differentiation
- Fundamental Theorem of Calculus
Types of Integrals
Riemann Sum
- Riemann Sums in Summation Notation
Functions defined by Integrals
Integration Formulas
Methods of Integration
Application of Integration
Line Integral
Surface Integral
Double Integration
Triple Integral

Differential Equations

This section introduces differential equations, covering their types, including ordinary and partial differential equations, and methods for solving them. You will explore key concepts such as order, degree, and techniques like exact and separable equations, with a focus on first and second-order differential equations.

Introduction to Differential Equations
- Order and Degree of Differential Equations
- to Differential Equations
Types of Differential Equations

Also, Check Calculus Cheat Sheet.

Practice for Calculus

This section provides a series of practice quizzes and questions to reinforce your understanding of key calculus concepts. You’ll test your knowledge on limits, continuity, maxima and minima, and integration through interactive exercises.

Limits – Quiz
Continuity of Function – Quiz
Maxima and Minima – Quiz
Integration – Quiz
Practice Questions on Calculus

Programs for Calculus

This section offers practical programming solutions for implementing calculus operations. You’ll learn how to write efficient code in Python and MATLAB, enhancing your skills in applying mathematical concepts through programming.

Calculus with Python
Program to Differentiate Given Polynomial
Program to Calculate Double Integration
Calculus in MATLAB
Diffrentiation in MATLAB

Limits in Calculus

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