Calculus and Linear Algebra

Functions, Limit, Continuity and Differentiability. Definition of derivative, physical and geometrical significance of derivative, derivative by first principal. Derivative of sum, difference, product and quotient of functions. Derivatives of polynomial and trigonometric functions, derivative of composite functions, chain rule, derivatives of inverse trigonometric functions, derivative of implicit function. Concept of exponential and logarithmic functions and their derivative. Logarithmic differentiation. Derivative of functions expressed in parametric forms. Second order derivatives. Applications of Derivatives: rate of change, maxima and minima (second derivative test only). Simple problems (that illustrate basic principles and understanding of the subject as well as real life situations). Applications: Drug delivery and Pharmacokinetics, diffusion and mass transfer.

Integration as inverse process of differentiation. Integration of a variety of functions by substitution, by partial fractions and by parts, only simple integrals of the type to be evaluated. Basic properties of definite integrals and evaluation of definite integrals. Applications of the Integrals: Applications in finding the area under simple curves, especially lines, areas of circles/parabolas/ellipses (in standard form only). Applications: Bioreactor engineering, cell growth and biomass estimation, calculation of total drug exposure (Area Under the Curve, AUC), estimation of cumulative drug release.

Definition, order and degree, general and particular solutions of a differential equation. Formation of differential equation whose general solution is given. Solution of differential equations of first order and first degree by method of separation of variables, homogeneous differential equations. Solutions of linear differential equation of the type: 𝑑𝑦 𝑑𝑥 + 𝑝𝑦= 𝑞, where p and q are functions of x. Applications: Population growth, Pharmacokinetics, diffusion and bioreactor modelling.

Definition, Types of Matrices, Addition, Subtraction, Scalar Multiplication and Multiplication of Matrices, Determinants: Definition, Minors, Cofactors, Properties of Determinants. Adjoint, Inverse and solution of system of linear equations. Applications: Solve system of linear equations in bioreactor and metabolic models, diffusion and transport problems.

Introduction to Vector Algebra: Scalar (dot) product, vector (cross) product, scalar triple product, vector triple product, direction cosines and direction ratios, Scalar and vector fields, Gradient Divergence and Curl, Directional Derivatives. Applications: Application of vector in biological systems, Biomechanics, and biomedical image and data analysis. Integration flow.