Calculus | Differential and Integral Calculus

In mathematics, Calculus deals with continuous change. It is also called infinitesimal calculus or “the calculus of infinitesimals”. The Two major concepts of calculus are Derivatives and Integrals.

• The derivative gives us the rate of change of a function. It describes the function at a particular point while the integral gives us the area under the curve.
• The integral gives us the area under the curve. Integral gathers the different values of a function over a number of values.

In general, classical calculus Calculus is the study of the continuous change of functions. In this article, we have provided everything related to Math Calculus for Beginners. Definition, examples, and practice questions will help you not only learn calculus theory but also practice calculus.

What is Calculus?

Calculus, a branch of mathematics that deals with the study of rate of change. It was founded by Newton and Leibniz.

Calculus math is commonly used in mathematical simulations to find the best solutions. It aids us in understanding the changes between values that are linked by a purpose. Calculus focuses on core ideas like limits, functions, integration, differentiation, and so on.

Calculus mathematics is classified into two parts:

• Differential Calculus
• Integral Calculus

Calculus Definition

Calculus is the part of mathematics that deals with the properties of derivatives and integrals of quantities like area, volume, velocity, acceleration, etc.

Check: Integral Calculus

Basic Calculus

Basic Calculus involves the study of differentiation and integration. Both concepts are based on the concepts of limits and functions. Fundamental concepts such as continuity and exponents serve as the bedrock for advanced calculus.

Basic calculus involves learning two types of calculus:

• Differential Calculus: used to determine the rate of change
• Integral Calculus: used to find quantity based on known rates of change.

Check: Calculus Cheat Sheet

Calculus Topics

Depending upon the variety of topics covered under calculus, we can divide the topics into different categories as listed below,

• Precalculus
• Calculus 1
• Calculus 2

Precalculus

Precalculus is a domain of mathematics that consists of trigonometry and algebra created to get students ready for the preparation of calculus.

In precalculus, advanced mathematics is mainly focused upon which includes functions and quantitative reasoning. Major topics studied under precalculus are,

• Functions
• Inverse Numbers
• Complex Numbers
• Rational Function

Calculus 1

Calculus consists of topics mainly focusing on differential calculus and similar concepts like limits and continuity. Some topics under Calculus 1 are, 

• Limits
• Derivatives
• Applications of Derivatives

Calculus 2

Calculus 2 is all about the mathematical study of change that occurred during the modules of Calculus 1. Some of the topics covered under Calculus 1 are,

• Differential Equations
• Sequence and Series
• Application of Integrals
• Trapezoidal Rule

Check: Differential Calculus

Calculus Functions

Functions in calculus denote the relationship between two variables, which are independent and the dependent variable. 

Let’s examine the following diagram.

We can see that there is an INPUT, a box, and an OUTPUT. For example, assume we want to bake a cake. We would require the following ingredients.

• Refined Flour
• Sugar
• Eggs
• Butter
• Baking powder 
• Baking soda

The above example can be represented as a function as shown below,

Let’s take another example, 

y = 5x

Value of x	Value of y
1	5
2	10
3	15
4	20

From the above example, we can see that the value of y depends on the value of x. We can conclude that.

• INPUT is independent of the OUTPUT
• INPUT is independent of the OUTPUT
• OUTPUT depends on the INPUT
• The box is accountable for the change of the INPUT to the OUTPUT

In calculus,

• INPUT is an independent variable
• OUTPUT is a dependent variable
• The box is a function

Check: Domain Relational Calculus in DBMS

Types of Calculus 

Calculus Mathematics can be divided into two types:

• Differential Calculus
• Integral Calculus.

Both differential and integral calculus consider the effect of a small shift in the independent variable on the equation as it approaches zero.

Both discrete and integral calculus serves as a basis for the higher branch of mathematics known as Analysis.

Check: Tuple Relation Calculus in DBMS

Differential Calculus

• Differential calculus is used to solve the problem of calculating the rate at which a function changes in relation to other variables.
• To obtain the optimal answer, derivatives are utilized to determine a function’s maxima and minima values.
• It primarily handles variables like x and y, functions like f(x), and the variations in x and y that follow.
• dy and dx are used to symbolize differentials.
• The process of differentiating allows us to compute derivatives. The derivative of a function is given by dy/dx or f’ (x).

Let’s go over some of the important subjects covered in basic differential calculus.

Limits

Limit is used to calculate the extent of closeness to any term or upcoming term. A limit is denoted with the help of the limit formula as,

lim_x⇢c f(x) = A

This expression is understood as “the limit of f of x approaches c equals A”.

More articles on Limits, to get better understanding:

• Introduction to Limits
• Formal Definition of Limits
• Strategy in Finding Limits
• Determining Limits Using Algebraic Manipulation
• Limits of Trigonometric Functions
• Properties of Limits
• Limits by Direct Substitution
• Estimating Limits from Graphs
• Estimating Limits from Tables
• Squeeze Theorem

Derivatives

The instantaneous rate at which one quantity changes in relation to another is represented by derivatives. The representation of a function’s derivative is:

lim_x⇢h [f(x + h) – f(x)]/h = A

More articles on Derivatives, to get better understanding.

• Introduction to Derivatives
• Average and Instantaneous Rate of Change
• Algebra of Derivative of Functions
• Product Rule – Derivatives
• Quotient Rule
• Derivatives of Polynomial Functions
• Derivatives of Trigonometric Functions
• Power Rule in Derivatives
• Application of Derivatives
• Applications of Power Rule
• Continuity and Discontinuity
• Differentiability of a Function
• Derivatives of Inverse Functions
• Derivatives of Implicit Functions
• Derivatives of Composite Functions
• Derivatives of Inverse Trigonometric Functions
• Exponential and Logarithmic Functions
• Logarithmic Differentiation
• Proofs for the derivatives of eˣ and ln(x) – Advanced differentiation
• Derivative of functions in parametric forms
• Second-Order Derivatives in Continuity and Differentiability
• Rolle’s and Lagrange’s Mean Value Theorem
• Mean value theorem – Advanced Differentiation

Continuity

A function f(x) is said to be continuous at a particular point x = a if the following three conditions are satisfied – 

• f(a) is defined
• lim_x⇢af(x) exists
• lim_x⇢a– f(x) = lim_x⇢a+ f(x) =f(a)

Continuity and Differentiability

A function is always continuous if it is differentiable at any point, whereas the vice-versa condition is not always true.

More articles on Continuity and Differentiability, to get better understanding

• Continuity and Discontinuity in Calculus
• Algebra of Continuous Functions
• Critical Points
• Rate of change of quantities
• Increasing and Decreasing Functions
• Increasing and Decreasing Intervals
• Separable Differential Equations
• Higher Order Derivatives

Integral Calculus

The study of integrals and their properties is known as integral calculus. It is primarily useful for:

• To compute f from f’ (i.e. from its derivative). If a function f is differentiable in the range under consideration, then f’ is specified in that range.
• To determine the region under a curve.

Integration

Integration is exactly the opposite of differentiation. Differentiation is the partition of a portion into a number of smaller parts, and integration is gathering tiny parts to create a whole. It is frequently applied to area calculations.

Definite Integral

A definite integral has a specified boundary beyond which the equation must be computed. The lower and upper limits of a function’s independent variable are defined, and its integration is represented using definite integrals.

[Tex]\int ^b_af(x).dx=F(x) [/Tex]

Indefinite Integral

An infinite integral lacks a fixed boundary, i.e. there is no upper and lower limit. As a result, the integration value is always followed by a constant value. 

[Tex]\int f(x).dx=F(x)+C [/Tex]

More articles on Integral Calculus, to get better understanding:

• Tangents and Normal
• Equation of Tangents and Normal
• Absolute Minima and Maxima
• Relative Minima and Maxima
• Concave Function
• Inflection Points
• Curve Sketching
• Approximations & Maxima and Minima – Application of Derivatives
• Integrals
• Integration by Substitution
• Integration by Partial Fractions
• Integration by Parts
• Integration using Trigonometric Identities
• Functions defined by Integrals
• Indefinite Integrals
• Definite integrals
• Computing Definite Integrals
• Fundamental Theorem of Calculus
• Finding Derivative with Fundamental Theorem of Calculus
• Evaluation of Definite Integrals
• Properties of Definite Integrals
• Definite Integrals of Piecewise Functions
• Improper Integrals
• Riemann Sum
• Riemann Sums in Summation Notation
• Definite Integral as the Limit of a Riemann Sum
• Trapezoidal Rule
• Areas under Simple Curves
• Area Between Two curves
• Area between Polar Curves
• Area as Definite Integral
• Basic Concepts of differential equations
• Order of differential equation
• Formation of a Differential Equation whose General Solution is given
• Homogeneous Differential Equations
• Separable Differential Equations
• Linear Differential Equations
• Exact Equations and Integrating Factors
• Particular Solutions to Differential Equations
• Integration by U-substitution
• Reverse Chain Rule
• Partial Fraction Expansion
• Trigonometric Substitution
• Implicit Differentiation
• Implicit Differentiation – Advanced Examples
• Disguised Derivatives – Advanced differentiation
• Differentiation of Inverse Trigonometric Functions
• Logarithmic Differentiation
• Antiderivatives

Calculus Formulas

The Calculus formulas used in calculus can be divided into six major categories. The six major formula categories are limits, differentiation, integration, definite integrals, application of differentiation, and differential equations. 

Limits Formulas

Limits Formulas help in estimating the values to a definite number and are defined either to zero or to infinity. 

• Lt_x⇢0(xⁿ – aⁿ)(x-a)=na^(n-1)
• Lt_x⇢0(sin x)/x = 1 
• Lt_x⇢0(tan x)/x = 1
• Lt_x⇢0(e^x – 1)/x = 1
• Lt_x⇢0(a^x – 1)/x = log_ea
• Lt_x⇢0(1 +(1/x))^x = e
• Lt_x⇢0(1 + x)^1/x = e
• Lt_x⇢0(1 + (a/x))^x= e^a

Differentiation Formulas

Differentiation Formulas can be applied to algebraic expressions, trigonometric ratios, inverse trigonometry, and exponential terms.

[Tex]{1.\dfrac{d}{dx}} x^n = nx^{n – 1}\\ 2.\dfrac{d}{dx} Constant = 0\\ 3. \dfrac{d}{dx}  e^x = e^x\\ 4. \dfrac{d}{dx} a^x = ax.loga\\ 5. \dfrac{d}{dx}  log x = 1/x\\ 6. \dfrac{d}{dx}  sin x = cos x\\ 7. \dfrac{d}{dx} cos x = -sin x\\ 8. \dfrac{d}{dx}  tan x = sec^2x\\ 9. \dfrac{d}{dx}  cot x = -cosec^2x\\ 10. \dfrac{d}{dx}  sec x = sec x.tanx\\ 11. \dfrac{d}{dx} cosec x = -cosec x.cot x [/Tex]

Integration Formula

Integration Formulas can be derived from differentiation formulas, and are complimentary to differentiation formulas.

• ∫ xⁿ.dx = x^{n + 1}/(n + 1) + C
• ∫ 1.dx = x + C
• ∫ e^x.dx = e^x + C
• ∫(1/x).dx = log|x| + C
• ∫ a^x.dx = (a^x/log a) + C
• ∫ cos x.dx = sin x + C
• ∫ sin x.dx = -cos x + C
• ∫ sec²x.dx = tan x + C
• ∫ cosec²x.dx = -cot x + C
• ∫ sec x.tan x.dx = sec x + C
• ∫ cosec x.cotx.dx = -cosec x + C

Definite Integrals Formulas

Definite Integrals are the basic integral formulas with limits. There is an upper and lower limit, and definite integrals, that are helpful in finding the area within these limits.

Fundamental theorem of calculus = [Tex]\int ^b_a f'(x)dx = f(b)-f(a) [/Tex]

[Tex]1. \int ^b_a f(x).dx = \int ^b_a f(t).dt    [/Tex]

[Tex]2. \int ^b_a x^n dx = \frac 1 {n+1} [b^{n+1} – a^{n+1}] (for: n\ne 1 [/Tex])

[Tex]3. \int ^b_a1 dx =b-a [/Tex]

[Tex]4. \int^b_a x dx= \frac 1 2[b^2-a^2] [/Tex]

[Tex]5. \int ^b_ax^2 dx=\frac1 3[b^3-a^3] [/Tex]

[Tex]6. \int ^b_a sin(x)dx = -cos(b) + cos(a) [/Tex]

[Tex]7. \int ^b_a cos(x)dx = sin(b) – sin(a) [/Tex]

Differential Equations formula

Differential equations can be compared to general equations because they are higher-order derivatives.

In the general equation, the variable ‘x’ is an unknown, and in this case, the variable is the differentiation of dy/dx.

1. Homogeneous Differential Equation : f(λx, λy)= λⁿf(x,y)
2. Linear Differential Equation: dy/dx +Py = Q
3. The general solution of the Linear Differential Equation is y.e^-∫P.dx = ∫(Q.e^∫P.dx ).dx + C

Advanced Calculus

Advanced Calculus is built upon basic calculus principles such as differentiation and derivatives. It includes other topics like infinite series, power series and so on.

Essential areas for advanced study include vector spaces, matrices, and linear transformations. It also delves into vector fields as derivatives, continuous differentiability, tangent space, normal space via gradients, and the dual space with its dual basis, providing profound insights into complex mathematical relationships.

Advanced Calculus helps us understand mathematical concepts like:

• Critical point analysis for multivariate functions
• Vector fields as derivatives
• Tangent space and normal space via gradients
• Fourier series and transforms
• Curvature and torsion
• Multilinear algebra
• Integration of forms
• Quadratic forms
• Generalized Stokes’ theorem
• Dual space and dual basis

Applications of Calculus

Calculus plays a very important role and helps us in:

• Examining a system to discover the best approach to forecast any given circumstance for a function.
• Calculus concepts are widely used in everyday life, whether it is to solve problems with complex shapes, assess survey results, determine the safety of automobiles, design a business, track credit card payments, or determine how a system is developing and how it affects us, etc.
• Economists, biologists, architects, doctors, and statisticians all speak calculus. For instance, engineers and architects employ several calculus ideas to determine the size and design of construction structures.
• Modeling ideas like occurrence and mortality rates, radioactive decay, reaction rates, heat and light, motion, and electricity all employ calculus.

Sample Calculus Problems with Solutions

Problem 1: Find the percentage error in calculating the surface area of a cubical box if an error of 1% is made in measuring the lengths of edges of the cube.

Solution:

According to the given condition,

Let △x be the error in the length and △y be the error in the surface area

Let’s take length as x

△x/x × 100 = 1

△x = x/100

x+△x = x+(x/100)

As, surface area of the cube = 6x²

dy/dx= 6(2x) = 12x

△y = dy/dx △x

△y = (12x) (x/100)

△y = 0.12 x²

So, △y/y = 0.12 x²/6 x² = 0.02

Percentage change in y = △y/y × 100 = 0.02 × 100 = 2

Hence, the percentage error in calculating the surface area of a cubical box is 2%

Problem 2: If x³ + y³ = 3axy, find dy/dx.

Solution:

Given, x³ + y³ = 3axy

Differentiating both sides with respect to x, we get:

3x² + 3y² (dy/dx) = 3ay + 3ax (dy/dx)

⇒ {3y² – 3ax} (dy/dx) = 3ay – 3x²

⇒ dy/dx = (3ax – 3x²)/(3y² – 3ax)

⇒ dy/dx = (ax – x^2.)/(y² – ax).

Practice Questions on Calculus

Q1: Find the indefinite integral: ∫ (3x² + cos(x) – 1/x) dx

Q2: Evaluate the definite integral: ∫ (from 1 to 2) (e^x + x²) dx

Q3: Find the derivative of the function: f(x) = ln(x² + 1)

Q4: Determine the limit (if it exists): lim (x² – 4x + 3) as x approaches 2

Q5: Find the area enclosed by the curve y = x³, the x-axis, and the lines x = -1 and x = 1.

Conclusion of Calculas

Calculus is a fundamental branch of mathematics that deals with change and motion. It provides tools like differentiation and integration, which are essential for analyzing functions, finding rates of change, and solving various real-world problems. With broad applications across science, engineering, economics, and technology, calculus is a cornerstone of modern knowledge and innovation, enabling us to understand and manipulate the world around us with precision and insight.

Calculus – FAQs

What is Calculus?

Calculus is a branch of mathematics that studies the rate of change. It’s used in mathematical models to find optimal solutions. Calculus is concerned with two basic operations, differentiation and integration.

What is Differential Calculus?

Differential calculus is used to study the problems of calculating the rate at which a function changes in relation to other variables. It is represented in the form of f'(x) = dy/dx

What is Integral Calculus?

The process of calculating the area under a curve or a function is called integral calculus. 

What is the maxima and minima of a function?

Maxima is the highest value of a function while minima is the lowest value of a function. Both can be obtained by finding the derivative of a function.

What are the 4 concepts of calculus?

The four main concepts covered under calculus are given below,

• Limits
• Differential Calculus
• Integral Calculus
• Multivariable Calculus

Where is calculus used in real life?

Calculus is fundamental to understanding and describing the behavior of physical systems. Engineers use calculus to analyze and design structures, and electrical circuits. Calculus is used to develop financial models and analyze complex financial derivatives.

What is the Meaning of Calculus?

Calculus, a vital branch of mathematics, studies derivatives and integrals to understand dynamic quantities like area, volume, velocity, and acceleration. It reveals changes within functions and provides essential tools for analyzing complex relationships between variables.