# Chapter 4 – Calculus

The following Topics and Sub-Topics are covered in this chapter and are available on MSVgo:

Introduction

You must have come across a situation where you might have needed to solve critical mathematical problems like an object gradually accelerating or decelerating to the slope of a curve. This might be a case not only in an exam but sometimes in real-life situations too. Therefore you must have a thorough understanding of the subject Calculus so that you can solve these questions easily.

You can use calculus to solve numerous problems and equations in mathematics. It has a great role in play, especially for those who opt for a career in science, engineering, designing, and even finance and business sectors. Calculus has a key role to play in today’s Physics, technology, and Science.

#### Classifications of Calculus

Integration:

When you are looking to calculate integral, then you’ll need to use Integration.

Calculus maths can be segregated into two different categories; they are:

• Differential Calculus
• Integral Calculus

Both forms of calculus deal with the impact on a minute change on an independent variable which eventually becomes zero. These two calculus branches act as a build-up towards another elevated branch of mathematics referred to as “analysis”.

#### Basics of Differential Calculus

This mode of Calculus refers to the rate of change in a certain quantity in regards to another one. For instance, velocity is a rate at which distance changes regarding the time in a specific direction. If a function can be termed as f(x), then f’(x) = dy/dx; x isn’t equal to 0 where f(x) is the function’s derivative, x is an independent variable and y is a dependent variable.

#### Derivatives

Whenever we are talking about Differential Calculus, we can’t ignore derivatives. If you want to demonstrate the rate changes, you can do so using the derivative function. Another name for the derivative is the slope. The Derivative is used to measure the steepness of the graph of a function. It shows the change ratio within the value of a function with respect to the change in the independent variable.

If you want to express y’s derivative in regards to x, you can do so like this: dy/dx. This is called the Derivative of a Function.

#### Basics of Integral Calculus

In this topic, you will understand the integrals as well as their properties. Integration is a crucial topic, and its process is just the opposite of differentiation.

Here are a few basic formulas for Integral Calculus:

1. kƒ(x) dx = k ƒ(x) dx
2. [ƒ(x)g(x)] dx = ƒ(x) dxg(x) dx
3. k dx = kx + C
4. xn dx = + C, n-1
5. ex dx = ex + C
6. ax dx = + C, a 0, a

#### Differentiation and Integration

The essence of calculus is Differentiation. The definition of derivative can be explained in a way that it is an instant range of change in functionality that is based on one of the variables. It can be compared to locating a tangent’s slope to the function at a certain point.

Some of the basic formulas for Differentiation include:

• dk/dx = 0 where k is constant
• d(x)/dx = 1
• d(kx)/dx = k where k is constant
• d (xn)/dx = nxn-1

#### Limits

One of the essential aspects of Calculus is Limits. Limits are useful for defining integrals, continuity, and also the derivatives. Let’s discuss the limit of a function:

If we take a function “f” which can be defined as some open interval and consists of numbers such as “a” or might be at “a” itself. In that case, you can write the limit of a function like:

lim x → af(x) = L, if given e > 0, there exists d > 0 such that 0 < |x – a| < d implies that |f(x) – L| < e. This means that the limit f(x) as “x” reaches “a” is “L.”

Limits and Continuity:

They are one of the most important topics as far as Calculus is concerned. Since the limit is already discussed, let’s take a look at what continuity is all about.

The topic of continuity is interesting and important too. A very easy way for testing continuity is to find out whether a pen can follow the graph of a function without you taking it off from a paper. While studying both precalculus and calculus, you need to understand the conceptual definition; however, moving ahead, a technical explanation is also necessary. With limits, the way for you to define continuity will be simple.

Continuity and Discontinuity:

As we already have understood that continuity can be defined as a pen following the graph’s function without lifting it up from a paper, here is what discontinuity is all about.

Discontinuity is of four types: Removal, Infinite, and Jump.

• Removal Discontinuity is expressed as: f(a) = lim x → af(x) f(a) = lim x → af(x)
• Infinite Discontinuity is expressed as:
lim x → 0− f (x) = lim x → 0 − xsin1x = 0 [Since -1 Similarly, lim x → 0 + f(x) = lim x → 0 + xsin1x = 0, [f(0) = 0]. Thus, lim x → 0−f (x) = lim x → 0 + f(x) = f(0).
• Jump Discontinuity is expressed as: lim x → a + f(x) ≠ lim x→a − f(x)

1.  What is calculus in simple terms?
Ans. Calculus is a branch of mathematics that helps in understanding different changes in values, all of which are concerning a function. Calculus mainly has two types; they are differential and integral calculus.
2. What are the 4 concepts of calculus?
Ans. The significant calculus concepts include:

• Limits
• Continuity
• Derivative
• Integration
3. How do you explain limits in calculus?
Ans. Limits are an integral and vital part of calculus as they are used in describing the integrals, continuity and derivatives too.
If we take a function “f” which can be defined as some open interval and consists of numbers such as “a” or might be at “a” itself. In that case, you can write the limit of a function like:
Limx → af (x) = L, if given e > 0, there exists d > 0 such that 0 < |x – a| < d implies that |f (x) – L| < e. This means that the limit f (x) as “x” reaches “a” is “L.”
4. Who is the father of calculus?
Ans. Sir Isaac Newton invented Calculus.
5. What is the most difficult type of math?
Ans. According to different people, calculus is the toughest math to crack.

Calculus is very easy once you understand the concept. Keep practising, and you will love it in no time. However, for those who are looking to crack this difficult subject, download the MSVgo app or reach https://msvgo.com today, where you can interact with several industry experts who can clear your doubts in no time. Happy Learning!

### High School Physics

• Alternating Current
• Atoms
• Communication Systems
• Current Electricity
• Dual nature of Radiation and Matter
• Electric Charges and Fields
• Electricity
• Electromagnetic Induction
• Electromagnetic Waves
• Electrons and Photons
• Electrostatic Potential and Capacitance
• Fluid Pressure
• Force and Acceleration
• Force And Laws Of Motion
• Gravitation
• Internal Energy
• Kinetic Theory
• Law of motion
• Light – Reflection And Refraction
• Magnetic Effects Of Electric Current
• Magnetism and Matter
• Management Of Natural Resources
• Mechanical properties of Fluids
• Mechanical properties of Solids
• Motion
• Motion in a plane
• Motion in a straight line
• Moving Charges and Magnetism
• Nuclear Energy
• Nuclei
• Oscillations
• Our Environment
• Paths of Heat
• Physical world
• Ray optics and optical instruments
• Semiconductor Devices
• Semiconductor Electronics: Materials, Devices and Simple Circuits
• Simple Machines
• Sound
• Sources Of Energy
• Specific and Latent Heats
• Spherical Mirrors
• Static Electricity
• Systems of Particles and Rotational motion
• Thermal properties of matter
• Thermodynamics
• Units and Measurement
• Vectors, Scalar Quantities and Elementary Calculus
• Wave Optics
• Waves
• Work, Power and Energy

### High School Chemistry

• Acids, Bases and Salts
• Alcohols, Phenols and Ethers
• Aldehydes, Ketones and Carboxylic Acids
• Aliphatic and Aromatic Hydrocarbons
• Alkyl and Aryl Halides
• Amines
• Analytical Chemistry
• Atomic Structure
• Atoms And Molecules
• Basic concepts of Chemistry
• Biomolecules
• Carbon And Its Compounds
• Carboxylic acids and Acid Derivatives
• Chemical Bonding and Molecular Structures
• Chemical Energetics
• Chemical Equilibria
• Chemical Kinetics
• Chemical Reactions And Equations
• Chemical Reactions and Their Mechanisms
• Chemistry in Everyday Life
• Chemistry of p-Block elements
• Chemistry of Transition and Inner Transition
• Classification of Elements
• Coordination Compounds
• Cyanide, Isocyanide, Nitro compounds and Amines
• Electrochemistry
• Electrolysis
• Elements, Compounds and Mixtures
• Environmental Chemistry
• Equilibrium
• Ethers and Carbonyl compounds
• Haloalkanes and Haloarenes
• Hydrocarbons
• Hydrogen
• Ideal solutions
• Introduction to Organic Chemistry
• Ionic equilibria
• Matter
• Matter Around Us
• Matter In Our Surroundings
• Metallurgy
• Metals And Non-Metals
• Mole Concept and Stoichiometry
• Natural Resources
• Organic Chemistry – Basic Principles
• Periodic Classification of Elements
• Physical and Chemical Changes
• Physical and Chemical Properties of Water
• Polymers
• Preparation, Properties and Uses of Compounds
• Principles and Processes of Isolation of Elements
• Redox Reactions
• Relative Molecular Mass and Mole
• States of Matter
• Structure Of The Atom
• Study of Compounds
• Study of Gas Laws
• Study of Representative Elements
• Surface Chemistry
• The d-block and f-block elements
• The Gaseous State
• The p-Block Elements
• The Periodic Table
• The s-Block Elements
• The Solid State
• Thermodynamics

### High School Biology

• Absorption and Movement of Water in Plants
• Anatomy of Flowering Plants
• Animal Kingdom
• Bacteria and Fungi-Friends and Foe
• Biodiversity and Conservation
• Biofertilizers
• Biological Classification
• Biomedical Engineering
• Biomolecules
• Biotechnology and its Applications
• Biotic Community
• Body Fluids and Circulation
• Breathing and Exchange of Gases
• Cell – Unit of Life
• Cell Cycle and Cell Division
• Cell Division and Structure of Chromosomes
• Cell Reproduction
• Cellular Respiration
• Chemical Coordination and Integration
• Circulation
• Control And Coordination
• Crop Improvement
• Digestion and Absorption
• Diversity In Living Organisms
• Ecosystem
• Environmental Issues
• Excretory Products and their Elimination
• Flowering Plants
• Genes and Chromosomes
• Health and Diseases
• Health and Its Significance
• Heredity And Evolution
• Heredity and Variation
• How Do Organisms Reproduce?
• Human Diseases
• Human Eye And Colourful World
• Human Health and Disease
• Human Population
• Human Reproduction
• Hygiene
• Improvement In Food Resources
• Integumentary System- Skin
• Kingdom Fungi
• Kingdom Monera
• Kingdom Protista
• Life Processes
• Locomotion and Movement
• Microbes in Human Welfare
• Mineral Nutrition
• Molecular Basis of Inheritance
• Morphology of Flowering Plants
• Neural Control And Coordination
• Nutrition in Human Beings
• Organism and Population
• Photosynthesis
• Photosynthesis in Higher Plants
• Plant Growth and Development
• Plant Kingdom
• Pollination and Fertilization
• Pollution; Sources and its effects
• Principles of Inheritance and Variation
• Reproduction and Development in Angiosperms
• Reproduction in Organisms
• Reproductive Health
• Respiration in Human Beings
• Respiration in Plants
• Respiratory System
• Sexual Reproduction in Flowering Plants
• Strategies for Enhancement in Food Production
• Structural Organisation in Animals
• Structural Organisation of the Cell
• The Endocrine System
• The Fundamental Unit Of Life
• The Living World
• The Nervous System and Sense Organs
• Tissues
• Transpiration
• Transport in Plants

### High School Math

• Algebra – Arithmatic Progressions
• Algebra – Complex Numbers and Quadratic Equations
• Algebra – Linear Inequalities
• Algebra – Pair of Linear Equations in Two Variables
• Algebra – Polynomials
• Algebra – Principle of Mathematical Induction
• Binomial Theorem
• Calculus – Applications of Derivatives
• Calculus – Applications of the Integrals
• Calculus – Continuity and Differentiability
• Calculus – Differential Equations
• Calculus – Integrals
• Geometry – Area
• Geometry – Circles
• Geometry – Conic Sections
• Geometry – Constructions
• Geometry – Introduction to Euclid’s Geometry
• Geometry – Three-dimensional Geometry
• Geometry – Lines and Angles
• Geometry – Straight Lines
• Geometry – Triangles
• Linear Programming
• Matrices and Determinants
• Mensuration – Areas
• Mensuration – Surface Areas and Volumes
• Number Systems
• Number Systems – Real Numbers
• Permutations and Combinations
• Probability
• Sequence and Series
• Sets and Functions
• Statistics
• Trignometry – Height and Distance
• Trignometry – Identities
• Trignometry – Introduction

### Middle School Science

• Acids, Bases And Salts
• Air and Its Constituents
• Basic Biology
• Body Movements
• Carbon and Its Compounds
• Cell – Structure And Functions
• Changes Around Us
• Chemical Effects Of Electric Current
• Coal And Petroleum
• Combustion And Flame
• Components Of Food
• Conservation Of Plants And Animals
• Crop Production And Management
• Electric Current And Its Effects
• Electricity And Circuits
• Elements and Compounds
• Fibre To Fabric
• Food production and management
• Force And Pressure
• Forests: Our Lifeline
• Friction
• Fun With Magnets
• Garbage In, Garbage Out
• Getting To Know Plants
• Health and Hygiene
• Heat
• Hydrogen
• Life Processes: Nutrition in Animals and Plants
• Materials: Metals And Non-Metals
• Matter and Its States
• Metals and Non-metals
• Micro Organisms: Friend And Foe
• Motion And Measurement Of Distances
• Motion And Time
• Nutrition In Animals
• Nutrition In Plants
• Organization in Living Things
• Our Environment
• Physical And Chemical Changes
• Pollution and conservation
• Pollution Of Air And Water
• Reaching The Age Of Adolescence
• Reproduction In Animals
• Reproduction In Plants
• Respiration In Organisms
• Rocks and Minerals
• Separation Of Substances
• Simple Machines
• Soil
• Some Natural Phenomena
• Sorting Materials Into Groups
• Sound
• Stars And The Solar System
• Structure of Atom
• Synthetic Fibers And Plastics
• The Living Organisms And Their Surroundings
• Transfer of Heat
• Transformation of Substances
• Transportation In Animals And Plants
• Universe
• Waste-water Story
• Water: A Precious Resource
• Weather, Climate And Adaptations Of Animals To Climate
• Winds, Storms And Cyclones

### Middle School Math

• Area and Its Boundary
• Boxes and Sketches
• Data Handling
• Fun With Numbers
• Heavy and Light
• How Many
• Long And Short
• Mapping
• Measurement
• Money
• Multiplication and Factors
• Multiply and Divide
• Numbers
• Parts and Wholes
• Pattern Recognition
• Patterns
• Play With Patterns
• Rupees And Paise
• Shapes And Angles
• Shapes And Designs
• Shapes and Space
• Similarity
• Smart Charts
• Squares
• Subtraction
• Tables And Shares
• Tenths and Hundredths
• Time