Course objective
The main emphasis of this course is to equip the student with necessary analytic and technical skills to handle problems of mathematical nature as well as practical problems. More precisely, the main target of this course is to explore the different tools for higher order derivatives to plot the various curves and to solve the problems associated with differentiation and integration of vector functions.
Unit I
Hyperbolic functions, higher order derivatives, Leibnitz rule and its applications to problems of the type e^(ax+b) sin x, e^(ax+b) cos x, (ax+b)^n sin x, (ax+b)^n cos x, concavity and inflection points, asymptotes, curve tracing in Cartesian coordinates, tracing in polar coordinates of standard curves, L'Hospital rule, application in business, economics and life sciences.
Unit II
Riemann integration as a limit of sum, integration by parts, reduction formulae, derivations and illustrations of reduction formulae of the type ∫sin^n x dx, ∫cos^n x dx, ∫tan^n x dx, ∫sec^n x dx, ∫(log x)^n dx, ∫sin^n x cos^n x dx, definite integral, integration by substitution.
Unit III
Volumes by slicing, disks and washers methods, volumes by cylindrical shells, parametric equations, parameterizing a curve, arc length, arc length of parametric curves, area of surface of revolution, techniques of sketching conics, reflection properties of conics, rotation of axes and second degree equations, classification into conics using the discriminant, polar equations of conics.
Unit IV
Triple product, introduction to vector functions, operations with vector-valued functions, limits and continuity of vector functions, differentiation, partial differentiation, div, curl and integration of vector functions, tangent and normal components of acceleration.
Learning outcomes
- Trace a curve and find asymptotes.
- Calculate integrals of typical type using reduction formulae, etc.
- Calculate arc length, surface of revolution and know about conics.
- Calculate triple products, gradient, divergence, curl, etc.
Books recommended
- H. Anton, I. Bivens and S. Davis: Calculus, 10th Ed., John Wiley and Sons (Asia) Pvt. Ltd., Singapore, 2002.
- Shanti Narayan, P. K. Mittal: Differential Calculus, S. Chand, 2014.
- R. J. T. Bell: An Elementary Treatise on Coordinate Geometry, MacMillan and Company Limited, 2005.
Books for reference
- James Stewart: Single Variable Calculus, Early Transcendentals, 8th Ed., Cengage Learning, 2016.
- G. B. Thomas and R. L. Finney: Calculus, 9th Ed., Pearson Education, Delhi, 2005.
- M. J. Strauss, G. L. Bradley and K. J. Smith: Calculus, 3rd Ed., Dorling Kindersley (India) Pvt. Ltd. (Pearson Education), Delhi, 2007.
Recorded
