**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**