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
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
“++