The main aim of this course is to give students some basic ideas of calculus, which is the mathematics of motion. The purpose is not just making the students learn these ideas but to enable them to apply these ideas to solve problems of practical nature. The course will provide the students with the necessary tools to understand and formulate advanced mathematical concepts and an awareness of their relationship to a variety of problems arising in engineering and sciences. Students wishing to major in the sciences, engineering, or medicine are required to have a working knowledge of the calculus and its applications.
COURSE LEARNING OUTCOMES (CLO)
CLO: 1. Define and explain the ideas of differential and integral calculus. (Level: C2)
CLO: 2. Apply the derivatives and integrals for solving different problems arising in Engineering sciences. (Level: C3)
CLO: 3. Use the vector calculus and analytical geometry in multiple dimensions. (Level: C3)
- Limits and Continuity – Three Lectures
- Introduction to Limits
- Rates of Change and Limits
- Definition and Examples
- Relation Between Differentiability and Continuity
- Equations of Tangents and Normals
- Derivative as slope, as rate of change (graphical representation).
- Differentiation & Successive Differentiation and its applications to rate speed and acceleration.
- Maxima and minima of function of one variable and its applications.
- Convexity and Concavity
- Points of inflexion
- Indefinite Integrals
- Definite Integrals
- Integration by Substitution by partial fractions and by parts
- Integration of Trigonometric Functions
- Riemann Sum, Fundamental Theorem of Calculus
- Area Under the Graph of a Nonnegative Function
- Area between curves of improper Integrals.
- Inverse functions
- Hyperbolic and Trigonometric Identities and their relationship.
- Logarithmic and Exponential Functions
- Three-Dimensional Geometry
- Vectors in Spaces
- Rectangular and polar co-ordinate systems in three dimensions.
- Direction Cosines
- Plane (Straight lines) and sphere
- Partial Derivatives
- Partial differentiation with chain rule
- Total Derivative
- Divergence, curl of a vector field.
- Arc-Length and Tangent Vector
- Lengths of curves
- Radius of gyration
- Fubini’s Theorem for calculating double Integrals
- Center of a gravity of a solid of revolution
- Moment of inertia
- Second moment of area
- Centers of pressure and depth of center of pressure
- Triple integrals, volume of a region in space.
- Volume of solids of revolution
- Areas Moments and Centers of Mass
- Curvature, radius and center of curvature