Calculus

Calculus is a subset of mathematics concerned with the study of continuous transition. Calculus is also known as infinitesimal calculus or “infinite calculus.” The analysis of continuous change of functions is known as classical calculus. Derivatives and integrals are the two most important ideas of calculus. The integral is the measure of the region under the curve, while the derivative is the measure of the rate of change of a function. The integral accumulates the discrete values of a function over a number of values, while the derivative describes the function at a given point.

Calculus, a branch of mathematics founded by Newton and Leibniz, study 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 Mathematics is generally 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.

Table of Contents

1. Differential Calculus
2. Integral Calculus

Differential Calculus

Differential Calculus deals with the issues of determining the rate of change of a parameter with respect to other variables. Derivatives are used to find the maxima and minima values of a function in order to find the best solution. The analysis of the boundary of a quotient leads to differential calculus. It is concerned with variables such as x and y, functions f(x), and the resulting variations in x and y. Differentials are represented by the symbols dy and dx. Differentiation refers to the method of determining derivatives. A function’s derivative is defined by dy/dx or f’ (x). It denotes that the equation is the derivative of y with respect to x. Let us go through some main topics discussed in simple differential calculus in the following articles:

1. Introduction to Limits
2. Formal Definition of Limits
3. Strategy in Finding Limits
4. Determining Limits using Algebraic Manipulation
5. Limits of Trigonometric Functions
6. Properties of Limits
7. Limits by Direct Substitution
8. Estimating Limits from Graphs
9. Estimating Limits from Tables
10. Squeeze Theorem
11. Introduction to Derivatives
12. Average and Instantaneous Rate of Change
13. Algebra of Derivative of Functions
14. Product Rule – Derivatives
15. Quotient Rule
16. Derivatives of Polynomial Functions
17. Derivatives of Trigonometric Functions
18. Power Rule in Derivatives
19. Application of Derivatives
20. Applications of Power Rule
21. Continuity and Discontinuity
22. Differentiability of a Function
23. Derivatives of Inverse Functions
24. Derivatives of Implicit Functions
25. Derivatives of Composite Functions
26. Derivatives of Inverse Trigonometric Functions
27. Exponential and Logarithmic Functions
28. Logarithmic Differentiation
29. Proofs for the derivatives of eˣ and ln(x) – Advanced differentiation
30. Derivative of functions in parametric forms
31. Second-Order Derivatives in Continuity and Differentiability
32. Rolle’s and Lagrange’s Mean Value Theorem
33. Mean value theorem – Advanced Differentiation
34. Continuity and Discontinuity in Calculus
35. Algebra of Continuous Functions
36. Critical Points
37. Rate of change of quantities
38. Increasing and Decreasing Functions
39. Increasing and Decreasing Intervals
40. Separable Differential Equations
41. Higher Order Derivatives

Integral Calculus

The analysis of integrals and their properties is known as integral calculus. It is primarily useful for the following two functions: 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. Differentiation is the inverse of integration. As separation can be defined as the division of a part into several small parts, integration can be defined as the selection of small parts to form a whole. It is commonly used to calculate area. 

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. 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 the integral calculus deeply:

1. Tangents and Normals
2. Equation of Tangents and Normals
3. Absolute Minima and Maxima
4. Relative Minima and Maxima
5. Concave Function
6. Inflection Points
7. Curve Sketching
8. Approximations & Maxima and Minima – Application of Derivatives
9. Integrals
10. Integration by Substitution
11. Integration by Partial Fractions
12. Integration by Parts
13. Integration using Trigonometric Identities
14. Functions defined by Integrals
15. Indefinite Integrals
16. Definite integrals
17. Computing Definite Integrals
18. Fundamental Theorem of Calculus
19. Finding Derivative with Fundamental Theorem of Calculus
20. Evaluation of Definite Integrals
21. Properties of Definite Integrals
22. Definite Integrals of Piecewise Functions
23. Improper Integrals
24. Riemann Sum
25. Riemann Sums in Summation Notation
26. Definite Integral as the Limit of a Riemann Sum
27. Trapezoidal Rule
28. Areas under Simple Curves
29. Area Between Two curves
30. Area between Polar Curves
31. Area as Definite Integral
32. Basic Concepts of differential equations
33. Order of differential equation
34. Formation of a Differential Equation whose General Solution is given
35. Homogeneous Differential Equations
36. Separable Differential Equations
37. Linear Differential Equations
38. Exact Equations and Integrating Factors
39. Particular Solutions to Differential Equations
40. Integration by U-substitution
41. Reverse Chain Rule
42. Partial Fraction Expansion
43. Trigonometric Substitution
44. Implicit Differentiation
45. Implicit differentiation – Advanced Examples
46. Disguised Derivatives – Advanced differentiation
47. Differentiation of Inverse Trigonometric Functions
48. Logarithmic Differentiation
49. Antiderivatives