Derivatives - Points of local maxima and local minima

  • Definition of a local maximum and a local minimum
  • Key points on the graph of a function
  • Determining local maxima and local minima algebraically
  • Identifying critical points
  • Using the first derivative test
    • If f’(x) changes sign from positive to negative, there is a local maximum at x.
    • If f’(x) changes sign from negative to positive, there is a local minimum at x.
  • Using the second derivative test
    • If f’’(x) > 0, there is a local minimum at x.
    • If f’’(x) < 0, there is a local maximum at x.
  • Examples and applications
    • Finding local extrema on real-world problems
    • Function optimization
  • Exercise problems to reinforce understanding “++

Derivatives - Points of local maxima and local minima (Contd.)

  • Critical points in terms of derivatives
    • f’(x) = 0 or f’(x) does not exist
  • Global maxima and global minima
    • Definition and difference from local extrema
    • The extreme value theorem
  • Identifying global extrema on a closed interval
    • Checking the function values at the endpoints and critical points
  • Absolute maxima and absolute minima
    • Definition and examples
  • Examples and applications
    • Solving optimization problems
    • Economic analysis using derivatives
  • Exercise problems to further enhance understanding “++

Integration - Basics

  • Introduction to integration
    • Definition and purpose
    • Connection to differentiation
  • Indefinite and definite integrals
    • Notation and meanings
    • Relationship between indefinite and definite integrals
  • Antiderivatives
    • Definition and examples
  • Rules of integration
    • Constant rule
    • Power rule
    • Sum and difference rules
    • Constant multiple rule
  • Examples of integrating polynomials “++

Integration - Basics (Contd.)

  • Examples of integrating exponential functions
  • Examples of integrating trigonometric functions
  • Examples of integrating logarithmic functions
  • Integration by substitution
    • Procedure and steps involved
    • Examples to illustrate the concept
  • Integration by parts
    • Formula and application
    • Examples to demonstrate the method
  • Exercises to practice integration techniques
  • Real-world applications of integration “++

Integration - Techniques

  • Integration by partial fractions
    • Decomposition of rational functions
    • Examples to illustrate the method
  • Integrating using trigonometric substitutions
    • Guidelines and examples
  • Integration of rational functions using long division
  • Integration of rational functions using synthetic division
  • Integration of algebraic functions using logarithmic substitution
  • Trigonometric integrals
    • Formulas and techniques for integrating trigonometric functions
  • Complex integration problems “++

Differential Equations - Introduction

  • Definition of differential equation
  • Types of differential equations
    • Ordinary differential equations (ODE)
    • Partial differential equations (PDE)
  • Order and degree of differential equations
  • Solving differential equations
    • Analytically and numerically
  • Initial value problems (IVP)
    • Definition and significance
    • Solving IVP using integration techniques
  • Examples of solving first-order differential equations “++

Differential Equations - First-Order Equations

  • Separable differential equations
    • Procedure and examples
  • Linear differential equations
    • Solutions using integrating factor
    • Homogeneous and non-homogeneous equations
  • Bernoulli differential equations
    • Substitution method for solving
  • Exact differential equations
    • Conditions and solutions “++

Differential Equations - Second-Order Equations

  • Definition and examples of second-order differential equations
  • Homogeneous linear differential equations with constant coefficients
    • Finding the characteristic equation and roots
    • Solutions based on the nature of roots
      • Real and distinct roots
      • Complex roots
      • Repeated roots
  • Non-homogeneous linear differential equations
    • Solving using the method of undetermined coefficients or variation of parameters
  • Examples illustrating the concepts “++

Differential Equations - Applications

  • Applications of differential equations in physics, engineering, and economics
  • Modeling population growth
    • Logistic growth model
    • Exponential growth model
  • Newton’s Law of Cooling
    • Modeling temperature change
  • Harmonic motion and oscillations
    • Examples of harmonic motion
    • Simple harmonic motion equations
  • Predator-prey models

Derivatives - Points of local maxima and local minima

  • Definition of a local maximum and a local minimum
  • Key points on the graph of a function
  • Determining local maxima and local minima algebraically
  • Identifying critical points
  • Using the first derivative test

Slide 12:

  • If f’(x) changes sign from positive to negative, there is a local maximum at x.
  • If f’(x) changes sign from negative to positive, there is a local minimum at x.
  • Using the second derivative test
  • If f’’(x) > 0, there is a local minimum at x.
  • If f’’(x) < 0, there is a local maximum at x.

Slide 13:

  • Examples and applications
  • Finding local extrema on real-world problems
  • Function optimization
  • Exercise problems to reinforce understanding
  • Critical points in terms of derivatives

Slide 14:

  • f’(x) = 0 or f’(x) does not exist
  • Global maxima and global minima
  • Definition and difference from local extrema
  • The extreme value theorem
  • Identifying global extrema on a closed interval

Slide 15:

  • Checking the function values at the endpoints and critical points
  • Absolute maxima and absolute minima
  • Definition and examples
  • Examples and applications
  • Solving optimization problems

Slide 16:

  • Economic analysis using derivatives
  • Exercise problems to further enhance understanding
  • Integration - Basics
  • Introduction to integration
  • Definition and purpose
  • Connection to differentiation

Slide 17:

  • Indefinite and definite integrals
  • Notation and meanings
  • Relationship between indefinite and definite integrals
  • Antiderivatives
  • Definition and examples
  • Rules of integration

Slide 18:

  • Constant rule
  • Power rule
  • Sum and difference rules
  • Constant multiple rule
  • Examples of integrating polynomials
  • Examples of integrating exponential functions

Slide 19:

  • Examples of integrating trigonometric functions
  • Examples of integrating logarithmic functions
  • Integration by substitution
  • Procedure and steps involved
  • Examples to illustrate the concept
  • Integration by parts

Slide 20:

  • Formula and application
  • Examples to demonstrate the method
  • Exercises to practice integration techniques
  • Real-world applications of integration
  • Differential Equations - Introduction
  • Definition of differential equation
  • Types of differential equations

Slide 21:

“++

Differential Equations - First-Order Equations (Contd.)

  • Separable differential equations
    • Procedure and examples
  • Linear differential equations
    • Solutions using integrating factor
    • Homogeneous and non-homogeneous equations
  • Bernoulli differential equations
    • Substitution method for solving
  • Exact differential equations
    • Conditions and solutions

Slide 22:

“++

Differential Equations - Second-Order Equations (Contd.)

  • Definition and examples of second-order differential equations
  • Homogeneous linear differential equations with constant coefficients
    • Finding the characteristic equation and roots
    • Solutions based on the nature of roots
      • Real and distinct roots
      • Complex roots
      • Repeated roots
  • Non-homogeneous linear differential equations
    • Solving using the method of undetermined coefficients or variation of parameters
  • Examples illustrating the concepts

Slide 23:

“++

Differential Equations - Applications (Contd.)

  • Applications of differential equations in physics, engineering, and economics
  • Modeling population growth
    • Logistic growth model
    • Exponential growth model
  • Newton’s Law of Cooling
    • Modeling temperature change
  • Harmonic motion and oscillations
    • Examples of harmonic motion
    • Simple harmonic motion equations
  • Predator-prey models

Slide 24:

“++

Trigonometry - Basic Concepts

  • Introduction to trigonometry
  • Definition of trigonometric functions: sine, cosine, tangent
  • Radian measure and degree measure
  • Unit circle and its significance
  • Graphs of trigonometric functions

Slide 25:

  • Properties of trigonometric functions
  • Periodicity and amplitude
  • Even and odd functions
  • Transformations of trigonometric functions
  • Trigonometric identities and equations

Slide 26:

  • Pythagorean identities
  • Reciprocal identities
  • Quotient identities
  • Sum and difference formulas
  • Double angle formulas

Slide 27:

  • Solving trigonometric equations
  • Trigonometric values for special angles
  • Trigonometric functions of complementary angles
  • Applications of trigonometry in real-world problems
  • Graphing trigonometric functions

Slide 28:

  • Amplitude, period, and phase shift
  • Finding the equation of a trigonometric function given its graph
  • Solving trigonometric equations using identities and graphs
  • Exercise problems to reinforce understanding

Slide 29:

“++

Probability - Basic Concepts

  • Introduction to probability
  • Sample space and events
  • Probability of an event
  • Laws of probability
    • Addition rule
    • Multiplication rule
    • Complementary rule

Slide 30:

  • Independent and dependent events
  • Conditional probability
  • Bayes’ theorem
  • Permutations and combinations
  • Factorial notation
  • Probability distributions