Module 1: Introduction to Calculus
Overview of Calculus
Definition and significance of calculus
Historical development of calculus
Applications of calculus in various fields
Functions and Graphs
Understanding functions and their types
Domain and range of functions
Plotting graphs of basic functions
Limits and Continuity
Concept of limits
Evaluating limits analytically
Continuity and its implications
Lab Activities
Plotting functions using graphing software
Evaluating limits graphically and numerically
Identifying discontinuities in functions
Module 2: Differentiation
Introduction to Derivatives
Definition and geometric interpretation of the derivative
Basic differentiation rules (power, product, quotient, chain rules)
Higher-order derivatives
Applications of Derivatives
Finding tangent and normal lines
Motion along a line: velocity and acceleration
Related rates and optimization problems
Techniques of Differentiation
Implicit differentiation
Differentiation of exponential, logarithmic, and trigonometric functions
Logarithmic differentiation
Lab Activities
Differentiating functions using software tools
Solving real-world problems using derivatives
Graphing derivatives of functions
Module 3: Applications of Derivatives
Curve Sketching
Analyzing critical points and inflection points
Determining concavity and convexity
Sketching graphs of functions based on derivative information
Optimization Problems
Setting up and solving optimization problems
Applications in economics, physics, and engineering
Interpreting and validating results
Mean Value Theorem and L’Hôpital’s Rule
Understanding the Mean Value Theorem and its implications
Applying L’Hôpital’s Rule to evaluate limits
Solving indeterminate forms
Lab Activities
Sketching curves using derivative information
Solving optimization problems from various fields
Applying L’Hôpital’s Rule to complex limits
Module 4: Integration
Introduction to Integrals
Concept of antiderivatives and indefinite integrals
Basic integration rules (power, substitution, integration by parts)
Definite integrals and the Fundamental Theorem of Calculus
Applications of Integrals
Area under a curve
Volume of solids of revolution
Work and fluid pressure
Techniques of Integration
Integration by substitution
Integration by parts
Partial fraction decomposition
Lab Activities
Evaluating integrals using software tools
Calculating areas and volumes using integration
Applying integration techniques to solve real-world problems
Module 5: Applications of Integrals
Arc Length and Surface Area
Finding the length of curves
Calculating the surface area of solids of revolution
Applications in engineering and physics
Differential Equations
Introduction to differential equations
Solving simple first-order differential equations
Applications in modeling real-world phenomena
Improper Integrals
Understanding and evaluating improper integrals
Convergence and divergence of integrals
Applications in probability and physics
Lab Activities
Calculating arc length and surface area using integrals
Solving differential equations with numerical methods
Evaluating improper integrals in applied contexts
Module 6: Sequences and Series
Sequences
Definition and types of sequences
Convergence and divergence of sequences
Limits of sequences
Series
Definition and types of series
Convergence tests (comparison, ratio, root, integral tests)
Power series and Taylor series
Applications of Series
Approximating functions using Taylor series
Fourier series and their applications
Solving differential equations using series
Lab Activities
Analyzing convergence of sequences and series
Approximating functions using Taylor series
Applying series to solve practical problems
Module 7: Analytical Geometry
Introduction to Analytical Geometry
Cartesian coordinates and the distance formula
Slope, intercept, and equation of a line
Conic sections: circles, ellipses, parabolas, hyperbolas
Transformations and Rotations
Translating and rotating axes
Graphing conic sections with transformed coordinates
Applications in physics and engineering
Polar Coordinates
Introduction to polar coordinates
Converting between Cartesian and polar coordinates
Graphing polar equations
Lab Activities
Graphing conic sections using software tools
Solving problems involving transformations and rotations
Plotting and analyzing polar equations
Module 8: Multivariable Calculus
Functions of Several Variables
Definition and examples of multivariable functions
Partial derivatives and gradient vectors
Tangent planes and linear approximations
Multiple Integrals
Double and triple integrals
Applications in calculating area, volume, and mass
Change of variables: polar, cylindrical, and spherical coordinates
Vector Calculus
Vector fields and line integrals
Green’s Theorem, Stokes’ Theorem, and Divergence Theorem
Applications in physics and engineering
Lab Activities
Calculating partial derivatives and gradients
Evaluating multiple integrals using software tools
Applying vector calculus to solve real-world problems
Module 9: Capstone Project and Review
Project Planning and Proposal
Choosing a topic related to calculus and analytical geometry
Defining project objectives and scope
Developing a project plan and timeline
Implementation and Development
Conducting research and analysis
Applying calculus and analytical geometry techniques
Testing and refining the project
Final Presentation and Evaluation
Preparing a project presentation
Demonstrating findings and applications
Receiving and incorporating feedback
Review and Reflection
Reviewing key concepts from the course
Identifying areas for further study and application
Reflecting on the learning experience
Lab Activities
Developing a project proposal and plan
Working on the capstone project
Preparing and delivering the final presentation
Admission Open for this course
Contact Number: 03307615544