Calculus
Calculus in mathematics deals with continuous change. Derivatives and integrals are the two most important parts of calculus. In other words, we can say that calculus is the study of the continuous change of functions. The integral gives us the area under the curve, while the derivative gives us the rate of change of a function. The integral gathers the different values of a function over a number of values, while the derivative describes the function at a particular point.
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Calculus Definition
Calculus, a branch of mathematics founded by Newton and Leibniz, deals with the pace of transition. 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 Math is mostly concerned with certain critical topics such as separation, convergence, limits, functions, and so on.
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
Functions
In calculus, functions 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
Types of Calculus
Calculus Mathematics can be divided into two types: Differential Calculus and 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.
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”.
 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
 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
 SecondOrder 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⇢a}f(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 viceversa condition is not always true.
 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.
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.
Following are the articles that discuss integral calculus deeply:
 Tangents and Normals
 Equation of Tangents and Normals
 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 Usubstitution
 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 Formula
The 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}(x^{n }– a^{n})(xa)=na^{(n1)}
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_{e}a
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
These formulas can be applied to algebraic expressions, trigonometric ratios, inverse trigonometry, and exponential terms.
Integration Formula
Integrals Formulas can be derived from differentiation formulas, and are complimentary to differentiation formulas.
∫ x^{n}.dx = x^{n + 1}/(n + 1) + C
∫ 1.dx = x + C
∫ e^{x}.dx = e^{x} + C
∫(1/x).dx = logx + C
∫ a^{x}.dx = (a^{x}/log a) + C
∫ cos x.dx = sin x + C
∫ sin x.dx = cos x + C
∫ sec^{2}x.dx = tan x + C
∫ cosec^{2}x.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.
f(x).dx= f(t).dt
= –f(x).dx
cf(x).dx = cf(x).dx
f(x) ± g(x).dx = f(x).dx ± g(x).dx
f(x).dx = f(x).dx + f(x).dx
f(x).dx = f(a+bx).dx
f(x).dx = f(ax).dx
f(x).dx = 2f(x).dx if f(2ax)=f(x)
f(x).dx = 0 if f(2ax) = f(x)
f(x).dx = 2 f(x).dx if f(x) is an even function (i.e., f(x)= f(x)).
f(x).dx = 0 if f(x) is an odd function (i.e., f(x) = f(x)).
Differential Equations formula
Differential equations can be compared to general equations because they are higherorder derivatives. In the general equation, the variable ‘x’ is an unknown, and in this case, the variable is the differentiation of dy/dx.
Homogeneous Differential Equation : f(λx, λy)= λ^{n}f(x,y)
Linear Differential Equation: dy/dx +Py = Q
The general solution of the Linear Differential Equation is y.e^{∫P.dx }= ∫(Q.e^{∫P.dx )}.dx + C
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.
Solved Problems on Calculus
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 = 6x^{2}
dy/dx= 6(2x) = 12x
△y = dy/dx △x
△y = (12x) (x/100)
△y = 0.12 x^{2}
So, △y/y = 0.12 x^{2}/6 x^{2 }= 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 x^{3} + y^{3} = 3axy, find dy/dx.
Solution:
Given, x^{3} + y^{3} = 3axy
Differentiating both sides with respect to x, we get:
3x^{2} + 3y^{2} (dy/dx) = 3ay + 3ax (dy/dx)
⇒ {3y^{2} – 3ax} (dy/dx) = 3ay – 3x^{2}
⇒ dy/dx = (3ax – 3x^{2})/(3y^{2} – 3ax)
⇒ dy/dx = (ax – x^{2.})/(y^{2} – ax).
FAQs on Calculus
Q1: What is Calculus?
Answer:
Calculus in mathematics is used to study the rate of change of a function. It consists of study of differentiation and integration.
Q2: What is Differential Calculus?
Answer:
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
Q3: What is Integral Calculus?
Answer:
The process of calculating the area under a curve or a function is called integral calculus.
Q4: What is the maxima and minima of a function?
Answer:
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.
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