Calculus — Limits, Derivatives & Integrals - Concept map
Calculus
Related concepts
Limits
Sub concepts
- Definition of limits
- Limit theorems
- Evaluating limits
- Indeterminate forms
- Applications in continuity
Relationship to central
Foundation of calculus - describes behavior of functions as input approaches a value
Derivatives
Sub concepts
- Definition using limits
- Power rule
- Product rule
- Quotient rule
- Chain rule
- Applications in optimization
Relationship to central
Measures instantaneous rate of change using limits
Integrals
Sub concepts
- Definite integrals
- Indefinite integrals
- Fundamental Theorem of Calculus
- Basic integration rules
- Applications in area and volume
Relationship to central
Reverse process of differentiation - finds area under curves
Functions
Sub concepts
- Polynomial functions
- Rational functions
- Trigonometric functions
- Exponential functions
- Logarithmic functions
Relationship to central
Mathematical objects that limits, derivatives, and integrals operate on
Concept connections
- LimitsDerivativesDerivatives are defined using limits to find instantaneous rate of change
- DerivativesIntegralsIntegration is the reverse process of differentiation (antiderivative)
- LimitsIntegralsDefinite integrals are defined as limits of Riemann sums
- Fundamental Theorem of CalculusDerivativesFTC Part 1 shows that derivatives undo integrals
- Fundamental Theorem of CalculusIntegralsFTC Part 2 provides method for evaluating definite integrals
- Limit TheoremsDerivative RulesMany derivative rules are derived from limit theorems
- Power RuleIntegration RulesIntegration of x^n reverses the power rule for derivatives
- Product RuleIntegration by PartsIntegration by parts is derived from the product rule
- Chain RuleSubstitution MethodU-substitution in integration reverses the chain rule
- ContinuityLimitsA function is continuous if its limit equals its function value
Diagrams
Complete Calculus Concept Map
Comprehensive overview showing the three main branches of calculus and their interconnected concepts, rules, and applications.
Problem-Solving Strategy for Limit Evaluation
Step-by-step decision tree for solving limit problems, showing when to use different techniques like factoring, rationalization, or L'Hospital's rule.
Derivative Rules Application Guide
Decision flowchart showing which differentiation rule to apply based on the structure of the given function.
Integration Strategy and Fundamental Theorem Application
Systematic approach to integration problems, distinguishing between definite and indefinite integrals and showing application of the Fundamental Theorem of Calculus.
Distribution of Calculus Topics in UPCAT Math
Shows the typical distribution of calculus topics in UPCAT mathematics exams, helping students prioritize their study time effectively.