Syllabus: Differential Calculus
Function and their graphs (polynomial and rational functions, logarithmic and exponential functions, trigonometric functions and their inverses, hyperbolic functions and their inverses, combination of such functions)
Limits of Functions: Basic limit theorems with proofs, limit at infinity and infinite limits,
Continuous functions. Algebra of continuous functions. Properties Continuous functions on closed and boundary intervals (no proof required).
Differentiation: Tangent lines and rates of change. Definition of derivative, one-sided derivatives. Rules of differentiation (proofs and applications). Successive differentiation. Leibnitz theorem. Related rates. linear approximations and differentials.
Rolle’s theorem: Lagrange’s and Cauchy’s mean value theorems. Extrema of functions. problems involving maxima and minima. Concavity and points of inflection.
Taylor’s theorem with the general form of the remainder ; Lagrange’s and Cauchy’s forms the remainder. Taylor’s series. Differentiation and integration of series. Validity of Taylor expansions and computations and computations with series. indeterminate forms. L-Hospital’s rules.
Integrals: Antiderivatives and indefinite-integrals. Techniques of Integration. Definite Integration using antiderivatives. Definite Integration using Riemann sums. Fundamental theorems of Calculus, Basic properties of Integration. Integration by reduction.
Application of Integration: Plane areas. Solids of revolutions. Volumes by cylindrical shells volumes by cross-sections. Arc length and Surface of revolution. Improper integrals. Gamma and Beta functions. Graphing in polar coordinates. Tangents to polar curves. Area and length in polar coordinates.
Course Objective: This course is designed to achieve the following objectives
Understanding of the basic concepts of differential and integral calculus
The usage of Matlab in order to facilitate understanding and visualization of mathematical problems
Theoretical and practical preparation enabling students to apply the acquired knowledge and skills in professional and specialist courses.
Learning Outcomes: By the end of this course the students must learn the following
define the basic concepts and principles of differential and integral calculus of real functions and sequences and series
interpret the geometric meaning of differential and integral calculus
apply the concept and principles of differential and integral calculus to solve geometric and physical problems
analyze the properties of functions based on graph obtained using Matlab
organize solving of complex problems by combining the acquired mathematical concepts and principles
Calculus - Titas Prokashoni
Differential Calculus & Integral Calculus - Md. Abdul Matin