Calculus Mindmap - Comprehensive Visual Guide

📋 Introduction

This calculus mindmap provides a visual overview of all major calculus concepts, techniques, and applications essential for JEE Advanced preparation. It covers limits, differentiation, integration, and their applications systematically.

🎯 Calculus Framework

Main Branches:

Calculus
├── Limits and Continuity
├── Differentiation
├── Applications of Derivatives
├── Integration
├── Applications of Integrals
└── Differential Equations

📊 Limits and Continuity

Limits Overview:

Limits and Continuity
├── Basic Concepts of Limits
│   ├── Definition of Limit
│   ├── Left-hand and Right-hand Limits
│   ├── Existence of Limits
│   └── Limit Notation
├── Limit Theorems and Properties
│   ├── Algebra of Limits
│   ├── Sandwich Theorem
│   ├── Limit of Composite Functions
│   └── Standard Limits
├── Evaluation of Limits
│   ├── Direct Substitution
│   ├── Factorization Method
│   ├── Rationalization
│   ├── Standard Form Limits
│   ├── L'Hospital's Rule
│   └── Series Expansion
├── Limits at Infinity
│   ├── Horizontal Asymptotes
│   ├── Limits as x → ∞
│   ├── Limits as x → -∞
│   └── Infinite Limits
├── Continuity
│   ├── Definition of Continuity
│   ├── Types of Discontinuities
│   │   ├── Removable Discontinuity
│   │   ├── Jump Discontinuity
│   │   └── Infinite Discontinuity
│   ├── Continuity of Composite Functions
│   └── Intermediate Value Theorem
└── Special Limits
    ├── Trigonometric Limits
    ├── Exponential and Logarithmic Limits
    ├── Limits Involving Absolute Value
    └── Limits of Piecewise Functions

Limits Key Formulas:

Essential Limits Formulas:
1. Standard Limits:
   - limₓ→a (xⁿ - aⁿ)/(x - a) = n·a^(n-1)
   - limₓ→0 (sin x)/x = 1
   - limₓ→0 (tan x)/x = 1
   - limₓ→0 (1 - cos x)/x = 0
   - limₓ→0 (aˣ - 1)/x = ln a
   - limₓ→0 (eˣ - 1)/x = 1
   - limₓ→∞ (1 + 1/x)ˣ = e
   - limₓ→0 (1 + x)^(1/x) = e

2. L'Hospital's Rule:
   - For 0/0 or ∞/∞ forms
   - limₓ→a f(x)/g(x) = limₓ→a f'(x)/g'(x)
   - Can be applied repeatedly
   - Check conditions before applying

3. Continuity Conditions:
   - f(a) must exist
   - limₓ→a f(x) must exist
   - limₓ→a f(x) = f(a)
   - All three conditions for continuity

4. Special Cases:
   - limₓ→0 sin(ax)/x = a
   - limₓ→0 (tan⁻¹x)/x = 1
   - limₓ→0 (eˣ - 1 - x)/x² = 1/2

📈 Differentiation

Differentiation Overview:

Differentiation
├── Basic Concepts
│   Definition of Derivative
│   Geometric Interpretation
│   Physical Interpretation
│ └── Existence of Derivative
├── Differentiation Rules
│   Power Rule
│   Product Rule
│   Quotient Rule
│   Chain Rule
│ └── Implicit Differentiation
├── Derivatives of Functions
│   Algebraic Functions
│   Trigonometric Functions
│   Inverse Trigonometric Functions
│   Exponential Functions
│   Logarithmic Functions
│   Hyperbolic Functions
│   └── Inverse Hyperbolic Functions
├── Advanced Differentiation
│   Parametric Differentiation
│   Logarithmic Differentiation
│   Differentiation of Inverse Functions
│   Higher Order Derivatives
│   └── Successive Differentiation
├── Special Differentiation Techniques
│   Differentiation Under Integral Sign
│   Leibniz Rule
│   Faa di Bruno's Formula
│   └── Differentiation of Determinants
└── Applications of Differentiation
    Rate of Change
    Tangents and Normals
    Monotonicity
    Extrema
    Concavity
    └── Curve Sketching

Differentiation Key Formulas:

Essential Differentiation Formulas:
1. Basic Rules:
   - Power Rule: d/dx(xⁿ) = nx^(n-1)
   - Product Rule: d/dx(uv) = u'v + uv'
   - Quotient Rule: d/dx(u/v) = (u'v - uv')/v²
   - Chain Rule: d/dx[f(g(x))] = f'(g(x))·g'(x)

2. Trigonometric Functions:
   - d/dx(sin x) = cos x
   - d/dx(cos x) = -sin x
   - d/dx(tan x) = sec² x
   - d/dx(cot x) = -cosec² x
   - d/dx(sec x) = sec x·tan x
   - d/dx(cosec x) = -cosec x·cot x

3. Inverse Trigonometric Functions:
   - d/dx(sin⁻¹x) = 1/√(1-x²)
   - d/dx(cos⁻¹x) = -1/√(1-x²)
   - d/dx(tan⁻¹x) = 1/(1+x²)
   - d/dx(cot⁻¹x) = -1/(1+x²)
   - d/dx(sec⁻¹x) = 1/(|x|√(x²-1))
   - d/dx(cosec⁻¹x) = -1/(|x|√(x²-1))

4. Exponential and Logarithmic:
   - d/dx(eˣ) = eˣ
   - d/dx(aˣ) = aˣ·ln a
   - d/dx(ln x) = 1/x
   - d/dx(logₐx) = 1/(x·ln a)

5. Higher Order Derivatives:
   - d²y/dx² = d/dx(dy/dx)
   - dⁿy/dxⁿ = nth derivative
   - Leibniz formula for nth derivative of product

📊 Applications of Derivatives

Applications of Derivatives Overview:

Applications of Derivatives
├── Rate of Change
│   Average Rate of Change
│   Instantaneous Rate of Change
│   Related Rates
│ └── Motion Problems
├── Tangents and Normals
│   Equation of Tangent
│   Equation of Normal
│   Angle Between Curves
│ └── Length of Tangent/Normal
├── Monotonicity
│   Increasing Functions
│   Decreasing Functions
│   Critical Points
│ └── First Derivative Test
├── Extrema
│   Local Maximum
│   Local Minimum
│   Absolute Maximum
│   Absolute Minimum
│ ├── First Derivative Test
│ └── Second Derivative Test
├── Concavity and Points of Inflection
│   Concave Up
│   Concave Down
│   Points of Inflection
│ └── Second Derivative Test
├── Curve Sketching
│   Domain and Range
│   Intercepts
│   Symmetry
│   Asymptotes
│ ├── Critical Points
│ ├── Concavity
│ └── Complete Analysis
├── Optimization Problems
│   Maximum-Minimum Problems
│   Geometric Optimization
│ ├── Business Applications
│ └── Engineering Applications
├── Mean Value Theorems
│   Rolle's Theorem
│   Mean Value Theorem
│ ├── Cauchy's Mean Value Theorem
│ └── Generalized Mean Value Theorem
└── Approximation
    Linear Approximation
    Differentials
    Newton-Raphson Method
    └── Error Estimation

Key Applications Techniques:

Essential Applications Techniques:
1. Tangent and Normal Equations:
   - Tangent: y - y₁ = f'(x₁)(x - x₁)
   - Normal: y - y₁ = -1/f'(x₁)(x - x₁)
   - Slope of tangent: m = f'(x₁)
   - Slope of normal: m = -1/f'(x₁)

2. First Derivative Test:
   - f'(x) > 0: Function increasing
   - f'(x) < 0: Function decreasing
   - f'(x) = 0: Critical point
   - Sign change indicates extremum

3. Second Derivative Test:
   - f''(x) > 0: Local minimum
   - f''(x) < 0: Local maximum
   - f''(x) = 0: Test inconclusive
   - Check third derivative if needed

4. Mean Value Theorem:
   - Conditions: f continuous on [a,b], differentiable on (a,b)
   - Conclusion: ∃c ∈ (a,b) such that f'(c) = [f(b)-f(a)]/(b-a)
   - Geometric interpretation: Parallel tangent

5. Optimization Steps:
   - Define objective function
   - Identify constraints
   - Find critical points
   - Test endpoints
   - Determine maximum/minimum

∫ Integration

Integration Overview:

Integration
├── Basic Concepts
│   Definition of Integral
│   Indefinite Integral
│   Definite Integral
│ └── Fundamental Theorem of Calculus
├── Integration Techniques
│   Basic Integration Formulas
│   Substitution Method
│   Integration by Parts
│   Partial Fractions
│ ├── Integration of Trigonometric Functions
│ ├── Integration of Rational Functions
│ └── Special Integrals
├── Advanced Integration
│   Integration by Substitution
│   Integration by Parts
│ ├── Integration of Rational Functions
│ ├── Trigonometric Integrals
│ ├── Integration of Irrational Functions
│ └── Special Substitutions
├── Definite Integration
│   Properties of Definite Integrals
│ ├── Evaluation Techniques
│ ├── Reduction Formulas
│ ├── Improper Integrals
│ └── Gamma and Beta Functions
├── Special Integrals
│   Standard Forms
│ ├── Reduction Formulas
│ ├── Integrals of Special Functions
│ └── Definite Integrals with Special Limits
└── Applications of Integration
    Area Under Curves
    Area Between Curves
    Volume of Revolution
    Arc Length
    Surface Area
    └── Physical Applications

Integration Key Formulas:

Essential Integration Formulas:
1. Basic Integrals:
   - ∫xⁿ dx = x^(n+1)/(n+1) + C (n ≠ -1)
   - ∫1/x dx = ln|x| + C
   - ∫eˣ dx = eˣ + C
   - ∫aˣ dx = aˣ/ln a + C
   - ∫sin x dx = -cos x + C
   - ∫cos x dx = sin x + C
   - ∫sec² x dx = tan x + C
   - ∫cosec² x dx = -cot x + C

2. Integration by Parts:
   - ∫u·dv = u·v - ∫v·du
   - LIATE rule for choosing u
   - Tabular integration method
   - Applications to various problems

3. Substitution Method:
   - u = g(x) substitution
   - Trigonometric substitution
   - Weierstrass substitution
   - Special cases and patterns

4. Partial Fractions:
   - Proper fractions decomposition
   - Repeated linear factors
   - Irreducible quadratic factors
   - Integration of each term

5. Definite Integration:
   - ∫ₐᵇ f(x) dx = F(b) - F(a)
   - Properties of definite integrals
   - Even and odd functions
   - Periodic functions

📊 Applications of Integrals

Applications of Integrals Overview:

Applications of Integrals
├── Area Calculations
│   Area Under Curve
│   Area Between Curves
│ ├── Area with Parametric Equations
│ ├── Area with Polar Equations
│ └── Area of Bounded Regions
├── Volume Calculations
│   Volume by Disk Method
│   Volume by Washer Method
│   Volume by Shell Method
│ ├── Volume of Revolution
│ └── Volume with Cross-Sections
├── Arc Length
│   Arc Length Formula
│ ├── Arc Length in Parametric Form
│ ├── Arc Length in Polar Form
│ └── Surface Area of Revolution
├── Surface Area
│   Surface Area of Revolution
│ ├── Surface Area Formula
│ └── Applications
├── Physical Applications
│   Center of Mass
│   Moments of Inertia
│ ├── Work Done
│ ├── Fluid Pressure
│ └── Hydrostatic Force
├── Economic Applications
│   Consumer Surplus
│ ├── Producer Surplus
│ ├── Present Value
│ └── Continuous Income Streams
└── Engineering Applications
    Center of Gravity
    Moment of Inertia
    Centroid
    └── Statics Applications

Key Applications Formulas:

Essential Applications Formulas:
1. Area Calculations:
   - Area under curve: A = ∫ₐᵇ f(x) dx
   - Area between curves: A = ∫ₐᵇ [f(x) - g(x)] dx
   - Parametric area: A = ∫ y·dx
   - Polar area: A = (1/2)∫ r² dθ

2. Volume by Disk Method:
   - V = π∫ₐᵇ [f(x)]² dx
   - Rotation about x-axis
   - V = π∫ₐᵇ [f(y)]² dy
   - Rotation about y-axis

3. Volume by Shell Method:
   - V = 2π∫ₐᵇ x·f(x) dx
   - Rotation about y-axis
   - V = 2π∫ₐᵇ y·f(y) dy
   - Rotation about x-axis

4. Arc Length:
   - L = ∫ₐᵇ √[1 + (dy/dx)²] dx
   - Parametric: L = ∫ √[(dx/dt)² + (dy/dt)²] dt
   - Polar: L = ∫ √[r² + (dr/dθ)²] dθ

5. Surface Area:
   - S = 2π∫ₐᵇ f(x)√[1 + (f'(x))²] dx
   - Rotation about x-axis
   - S = 2π∫ₐᵇ x√[1 + (f'(x))²] dx
   - Rotation about y-axis

📊 Differential Equations

Differential Equations Overview:

Differential Equations
├── Basic Concepts
│   Order and Degree
│   Linear and Non-linear
│   Homogeneous and Non-homogeneous
│ └── General and Particular Solutions
├── First Order Differential Equations
│   Variable Separable
│   Homogeneous Equations
│ ├── Linear Equations
│ ├── Exact Equations
│ ├── Bernoulli Equations
│ └── Clauraut's Equation
├── Second Order Differential Equations
│   Homogeneous Linear Equations
│ ├── Non-homogeneous Linear Equations
│ ├── Constant Coefficient Equations
│ ├── Variable Coefficient Equations
│ └── Special Forms
├── Solution Methods
│   Integrating Factor Method
│ ├── Method of Undetermined Coefficients
│ ├── Variation of Parameters
│ ├── Power Series Method
│ └-- Laplace Transform Method
├── Applications
│   Population Growth
│ ├── Radioactive Decay
│ ├── Newton's Law of Cooling
│ ├── Simple Harmonic Motion
│ └-- Electrical Circuits
└── Special Differential Equations
    Bessel's Equation
    Legendre's Equation
    Hermite's Equation
    └-- Laguerre's Equation

Differential Equations Key Methods:

Essential Solution Methods:
1. Variable Separable:
   - dy/dx = f(x)g(y)
   - ∫dy/g(y) = ∫f(x)dx
   - Solve and add constant

2. Linear First Order:
   - dy/dx + P(x)y = Q(x)
   - Integrating factor: I.F. = e^(∫Pdx)
   - Solution: y·I.F. = ∫Q·I.F.·dx + C

3. Homogeneous Equations:
   - dy/dx = f(y/x)
   - Substitute: y = vx
   - dy/dx = v + x(dv/dx)
   - Solve for v, then substitute back

4. Exact Equations:
   - Mdx + Ndy = 0
   - ∂M/∂y = ∂N/∂x (exactness condition)
   - Solution: ∫Mdx + ∫Ndy = C

5. Second Order Linear:
   - Homogeneous: ay'' + by' + cy = 0
   - Characteristic equation: ar² + br + c = 0
   - Solution depends on roots

🎯 Problem-Solving Strategies

Calculus Problem-Solving Framework:

Systematic Approach:
1. Understand the Problem
   - Identify the type (limit, derivative, integral)
   - Determine given information
   - Identify what needs to be found
   - Note any constraints

2. Choose Appropriate Method
   - For limits: direct substitution, L'Hospital, series
   - For derivatives: basic rules, chain rule, implicit
   - For integrals: substitution, parts, partial fractions
   - For applications: set up the integral properly

3. Execute the Solution
   - Apply formulas correctly
   - Show all steps clearly
   - Handle special cases
   - Simplify results

4. Verify the Answer
   - Check units and dimensions
   - Test special cases
   - Consider reasonableness
   - Cross-check with alternative methods

5. Interpret Results
   - Physical meaning of result
   - Limitations of solution
   - Range of validity
   - Applications

Common Problem Types:

Calculus Problem Categories:
1. Limit Problems:
   - Indeterminate forms (0/0, ∞/∞)
   - Limits at infinity
   - Trigonometric limits
   - Exponential limits

2. Derivative Problems:
   - Finding derivatives
   - Equation of tangent/normal
   - Rate of change problems
   - Optimization problems

3. Integral Problems:
   - Indefinite integration
   - Definite integration
   - Area calculations
   - Volume calculations

4. Application Problems:
   - Maximum-minimum
   - Related rates
   - Approximation
   - Physical applications

📈 Performance Tips

Exam Success Strategies:

• Master basic formulas and their conditions
• Practice pattern recognition for choosing methods
• Understand geometric interpretations of calculus concepts
• Focus on applications and real-world problems
• Practice integration techniques extensively
• Understand the connection between differentiation and integration

Use this comprehensive calculus mindmap to master all JEE Advanced calculus concepts! Systematic practice with these visual aids will significantly enhance your problem-solving abilities and understanding. 🎯