Applications of Slope of the Curve

• Physics: Motion of an object
• Economics: Rate of change of cost or revenue
• Engineering: Analysis of circuits
• Biology: Growth rates of populations
• Computer Science: Optimization algorithms

Tangent and Normal Lines

• Definition of tangent line
• Definition of normal line
• Formula for finding the equation of a tangent line at a point
• Formula for finding the equation of a normal line at a point
• Example: Finding tangent and normal lines for a given function

Continuity of a Function

• Definition of continuity
• Types of discontinuities: Removable, Jump, and Infinite
• Theorems for checking continuity
  • Intermediate Value Theorem
  • Extreme Value Theorem
• Example: Determine the points of discontinuity for a given function

Differentiability of a Function

• Definition of differentiability
• Differentiability vs continuity
• Rules for differentiating functions
  • Power Rule
  • Chain Rule
  • Product Rule
  • Quotient Rule
• Example: Find the derivative of a given function using the chain rule

Integration

• Introduction to integration
• Definition of the integral
• Properties of integrals
  • Linearity
  • Constant multiple
  • Sum/Difference
  • Substitution
• Example: Evaluate a definite integral using substitution method

Indefinite Integrals

• Definition of indefinite integral
• Rules for finding indefinite integrals
  • Power Rule
  • Constant Rule
  • Exponential Rule
  • Trigonometric Rule
• Example: Evaluate indefinite integrals using the power rule

Fundamental Theorem of Calculus

• Statement of the Fundamental Theorem of Calculus
• Part 1: Evaluating definite integrals
• Part 2: Finding antiderivatives
• Example: Compute a definite integral using the Fundamental Theorem of Calculus

Area under a Curve

• Concept of area under a curve
• Riemann sums
• Definite integral as a limit of Riemann sums
• Example: Calculate the area under a curve using definite integrals

Differential Equations

• Introduction to differential equations
• Definition of a differential equation
• Types of differential equations
  • Ordinary differential equations
  • Partial differential equations
• Example: Solve a first-order differential equation with initial conditions

Integral Calculus - Slope of the Curve

• Introduction to slope of the curve
• Derivative of a function
• Definition of slope at a point
• Slope formula: (m = \frac{{dy}}{{dx}})
• Finding the slope at a given point

##21. Applications of Integration

• Area calculation
• Volume calculation
• Work and energy calculation
• Probability distribution functions
• Finding arc length

##22. Techniques of Integration

• Substitution method
• By parts method
• Partial fraction decomposition
• Trigonometric substitution
• Integration by trigonometric identities

##23. Applications of Differential Equations

• Growth and decay problems
• Population dynamics
• Electric circuits
• Fluid flow problems
• Heat transfer

##24. Linear Algebra

• Vectors in space
• Matrices and determinants
• Systems of linear equations
• Eigenvalues and eigenvectors
• Diagonalization of matrices

##25. Vector Calculus

• Gradient, divergence, and curl
• Line integrals
• Surface integrals
• Green’s theorem
• Stoke’s theorem

##26. Probability and Statistics

• Probability distributions
• Random variables
• Central limit theorem
• Hypothesis testing
• Regression analysis

##27. Complex Analysis

• Complex numbers and operations
• Complex functions and their properties
• Complex integration
• Residue theorem
• Conformal mapping

##28. Numerical Methods

• Approximation and interpolation
• Numerical integration
• Solving differential equations numerically
• Root finding methods
• Optimization techniques

##29. Discrete Mathematics

• Sets and relations
• Graph theory
• Combinatorics
• Boolean algebra
• Propositional and predicate logic

##30. Mathematical Logic

• Propositional logic
• First-order logic
• Proof techniques
• Logical reasoning
• Applications of mathematical logic