Functions and Their Graphs
- Concept of a Function: A function represents a relationship where each input corresponds to exactly one output.
- Types of Functions: Linear, quadratic, polynomial, exponential, logarithmic, and trigonometric functions.
- Graphical Representation: Functions can be visualized using graphs, helping us understand their behavior.
- Key Features: Domain (set of inputs), range (set of outputs), intercepts, symmetry, and turning points.
- Applications: Used to model real-life situations such as population growth, financial interest, and speed-time relations.
Limits of Functions
- Concept of Limit: Describes the value a function approaches as the input gets close to a specific point.
- Left-hand and Right-hand Limits: Approaching from the negative and positive sides, respectively.
- Continuity and Discontinuity: A function is continuous if its graph has no breaks.
- Real-life Analogy: Like a vehicle approaching a certain speed, even if it never exactly reaches it.
Definition and Properties of the Derivative
- Meaning of Derivative: Represents the rate of change of a function.
- Conceptual Understanding: Interpreted as the slope of the tangent to a curve or instantaneous speed.
- Properties: Linearity, sum/difference, product, and constant multiples make derivatives easier to compute.
- Applications: Widely used in motion analysis, optimization, and scientific modeling.
Integral Calculus Formulas | Indefinite, Definite, Double & Triple Integrals
Table of Derivatives
- Overview: Contains the derivatives of commonly used functions.
- Purpose: Acts as a quick reference in problem-solving.
- Conceptual Use: Helps identify recurring patterns in calculus.
Higher Order Derivatives
- Concept: Taking the derivative of a derivative.
- Interpretation: The second derivative of displacement is acceleration, the third derivative relates to jerk.
- Applications: Used in mechanics, curvature studies, and optimization.
Applications of Derivative
- Rate of Change: Explains how variables evolve over time.
- Maxima and Minima: Helps optimize profit, minimize costs, or maximize efficiency.
- Tangents and Normals: Used in geometry and motion paths.
- Monotonicity: Determines whether a function is increasing or decreasing.
Differential
- Concept: Represents a small change in a variable and the resulting change in the function.
- Relation with Derivative: Differentials approximate small changes using derivatives.
- Applications: Used in error estimation, physics, and predictions.
Multivariable Functions
- Functions of Two or More Variables: Example: cost depending on labor and raw material.
- Partial Derivatives: Rate of change with respect to one variable while holding others constant.
- Applications: Economics (cost functions), thermodynamics, and temperature distribution in physics.
Analytic Geometry Formulas | Lines, Circles, Conic Sections & 3D Geometry
Differential Operators
- Definition: Symbols or tools used for differentiation.
- Common Operators: d/dx (ordinary derivative), ∂/∂x (partial derivative).
- Applications: Solving complex equations in engineering, physics, and applied mathematics.
