कैलकुलस: अवकलन और समाकलन

• Calculus is a branch of mathematics focused on rates of change and accumulation of quantities.
• It has two major branches: differential calculus and integral calculus.

Differential Calculus

• Deals with the instantaneous rate of change of a function.
• Key concepts include derivatives and differentiation.
• The derivative of a function at a point gives the slope of the tangent line to the function's graph at that point.
• Derivatives are used to find maxima and minima of functions, analyze rates of change, and model various phenomena in science and engineering.
• Differentiation is the process of finding the derivative of a function.
• Common rules for differentiation include the power rule, product rule, quotient rule, and chain rule.

Integral Calculus

• Deals with the accumulation of quantities.
• Key concepts include integrals and integration.
• The integral of a function over an interval gives the area under the curve of the function's graph over that interval.
• Integrals are used to find areas, volumes, and solve differential equations.
• Integration is the process of finding the integral of a function.
• The fundamental theorem of calculus relates differentiation and integration, stating that they are inverse processes.
• Techniques of integration include substitution, integration by parts, and partial fractions.

Limits

• The concept of a limit is fundamental to calculus.
• A limit describes the value that a function approaches as the input approaches some value.
• Limits are used to define continuity, derivatives, and integrals.
• Evaluating limits involves algebraic manipulation, L'Hôpital's rule, and understanding indeterminate forms.

Functions

• Calculus deals extensively with functions.
• A function is a relation between a set of inputs and a set of permissible outputs with the property that each input is related to exactly one output.
• Functions can be represented graphically, algebraically, or numerically.
• Types of functions include polynomial, trigonometric, exponential, and logarithmic functions.

Derivatives

• The derivative of a function measures its instantaneous rate of change.
• Formally, the derivative of f(x) at x=a is defined as the limit of the difference quotient: f'(a) = lim (h->0) [f(a+h) - f(a)] / h
• The derivative can also be denoted as dy/dx, f'(x), or Df(x).
• Higher-order derivatives (second derivative, third derivative, etc.) represent rates of change of rates of change.

Applications of Derivatives

• Optimization: Finding maximum and minimum values of functions.
• Related Rates: Analyzing how the rates of change of different variables are related.
• Curve Sketching: Using derivatives to determine the shape of a function's graph (increasing/decreasing intervals, concavity, etc.).
• Linearization: Approximating a function with a linear function near a specific point.

Integrals

• The integral of a function represents the area under its curve.
• The definite integral of f(x) from a to b is denoted as ∫[a,b] f(x) dx.
• The indefinite integral of f(x) is a function F(x) such that F'(x) = f(x).
• The fundamental theorem of calculus connects derivatives and integrals.

Applications of Integrals

• Area between curves: Finding the area between two or more curves.
• Volume: Finding the volume of solids of revolution.
• Average Value: Finding the average value of a function over an interval.
• Work: Calculating the work done by a force.

Sequences and Series

• A sequence is an ordered list of numbers.
• A series is the sum of the terms of a sequence.
• Convergence and divergence of sequences and series are important concepts.
• Tests for convergence include the ratio test, root test, and comparison test.

Multivariable Calculus

• Extends the concepts of calculus to functions of multiple variables.
• Partial derivatives measure the rate of change of a function with respect to one variable, holding others constant.
• Multiple integrals are used to find volumes, surface areas, and other quantities in higher dimensions.
• Gradient, divergence, and curl are important concepts in vector calculus.

Differential Equations

• Equations that relate a function to its derivatives.
• Used to model many phenomena in physics, engineering, and other fields.
• Types of differential equations include ordinary differential equations (ODEs) and partial differential equations (PDEs).
• Techniques for solving differential equations include separation of variables, integrating factors, and Laplace transforms.