Calculus
Teacher
Dr. Goutam Kumar Mahato
Category
Core Courses
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Course Name : Calculus
Code(Credit) : CUTM1511 (3-1-2)
Course Objectives
- To study how things change. It provides a framework for modeling systems in which there is change, and a way to deduce the predictions of such models.
- To construct a relatively simple quantitative models of change, and to deduce their consequences.
Course Outcomes
| COs | Course Outcomes | Mapping Cos with POs (High-3, Medium-2, Low-1) |
| CO1 | Understand the importance of linear functions in mathematics.
|
PO1(3), PO2(2) |
| CO2 | Deals with the major problems of differential and integral calculus.
|
PO2(2) |
| CO3 | Recognize other important classes of functions (such as trigonometric and rational functions), and will be able to use calculus with these functions.
|
PO4(2), PO9(2) |
Course Syllabus
Module-I: Curvature
Curvature, circle of curvature and radius of curvature, Radius of curvature in Cartesian and polar form.
Project 1: To find radius of curvature (Intrinsic and Parametric form)
Project 2: To find radius of curvature (Pedal form)
Module-II: Asymptotes
Asymptotes and determination of asymptotes, Asymptotes parallel to coordinate axis, Rules to find asymptotes.
Project 3: To find asymptotes of different curves in implicit form
Module-III: Curve Tracing
Curve Tracing, Tracing of Cartesian, parametric and polar curves.
Project 4: To trace some Cartesian curves (Folium of Descartes, Astroid)
Project 5: To trace some polar curves (Cardioid, Lemniscate of Bernaulli)
Project 6: To trace few more curves (Cartenary, Cycloid etc) and Loops
Practice 1: Plotting the graphs of the functions eax+b, log (ax + b), 1/(ax + b), sin (ax + b), cos(ax + b), |ax +b| and to illustrate the effect of a and b on the graph
Practice 2: Plotting the graphs of the polynomial of degree 4 and 5, the derivative graph, the second derivative graph and comparing them
Practice 3: Sketching parametric curves (e.g. Trochoid, cycloid, epicycloids, hypocycloid)
Module-IV: Reduction Formula
Reduction formula, Reduction formula involving sine and cosine function sin ^nx cos ^nx, Reduction formula for integrals with limits from 0 to pi/2, Reduction formula for integral of sin^px, Cos^qx and also with limits from 0 to Pi/2.
Project 7: To find the Reduction formula for integrals of tan^nx, Cot ^nx, sec^nx, cosec^nx
Project 8: To find the Reduction formula for integral of sin^pxcos^qx for different values of p and q and different cases
Module-V: Rectification and Quadrature
Rectification of curves, Finding arc length for polar curves, Quadrature and areas of plane regions for Cartesian, parametric and polar curves, Sectorial area.
Project 9: To find the Area bounded by closed curves
Module-VI: Volume of Solid Body Revolution
Volume of solid of revolution, Volume of solid of revolution about any axis, Pappus theorem for volume of revolution.
Practice 4: Tracing of conics in Cartesian coordinates/polar coordinates
Practice 5: Sketching ellipsoid, hyperboloid of one and two sheets,
Practice 6: Sketching elliptic cone, elliptic paraboloid, hyperbolic paraboloid using Cartesian coordinates
Module-VII: Vector Differential Calculus
Introduction to vector calculus: Definitions and characteristics of vectors, algebraic operations, Directional derivatives and directional derivatives along any line, Gradient of scalar point function, Divergence and curl of vector point functions, Second order differential operator the Laplacian operator and its properties.
Project 10: To explain about vector, different types of vector and vector algebra with suitable example
Project 11: To explain the geometrical significance of Gradient of scalar point function with some examples
Project 12: To discuss Laplacian operator and its different properties
Text Books:
1 A Text book of Calculus Part – II : Shanti Narayan
Chapters: 8 (Art. 24, 25, 26), 10 (Art. 33, 34, 35, 36, 37, 38).
2 A Text book of Calculus Part-III : Shanti Narayan
Chapters: 1 (Art 1, 2, 3), 3(Art 7, 8, 9), 4(Art 10, 11, 12) omitting Simpson’s rule), 5(Arts 13, 14), 6(Arts 15, 16).
3 A Textbook of Vector Calculus by Shanti Narayan & P. K. Mittal, S. Chand & Co., 2003
Chapters: 1, 6, 7 (7.1 to 7.6, 7.8 & 7.11)
Reference Books:
1 J. Strauss, G.L. Bradley and K. J. Smith, Calculus, 3rd Ed., Dorling Kindersley (India) P. Ltd. (Pearson Education). Delhi, 2007.
Chapters: 4(4.3, 4.4, 4.5 & 4.7), 9(9.4), 10(10.1-10.4).
2 Anton, I. Bivens and S. Davis, Calculus, 7th Ed., John Wiley and Sons (Asia) P. Ltd., Singapore, 2002.
Chapters: 6. (6.2-6.5).7(7.8). 8(8.2-8.3, Pages: 532-538), 11(11.1), 13(13.5)
3 B. Thomas and R. L. Finney. Calculus, 9th Ed., Pearson Education, Delhi, 2005
4 Courant and F. John Introduction to Calculus and Analysis (Volumes I & II). Springer-Verlag. New York. Inc., 1989
Session 4 (Project 1) (2 hrs)
To find radius of curvature (Intrinsic and Parametric form)
Session 5 (Project 2) (2 hrs)
To find radius of curvature (Pedal form)
Session 6
Asymptotes and determination of asymptotes
https://www.youtube.com/watch?v=oBmnt5tXRws
Session 7
Asymptotes parallel to coordinate axis
https://www.youtube.com/watch?v=LCfxTH1LEd4
https://slideplayer.com/slide/5279710/
Session 9 (Project 3) (2 hrs)
To find asymptotes of different curves (implicit form)
Session 10
Session 12
Session 13
Session 14 (Project 4) (2 hrs)
To trace some Cartesian curves (Folium of Descartes, Astroid)
Session 15 (Project 5) (2 hrs)
To trace some polar curves (Cardioid, Lemniscate of Bernaulli)
Session 16 (Project 6) (2 hrs)
To trace few more curves (Cartenary, Cycloid etc) and Loops
Session 17 (Practice 1) (2 hrs)
Plotting the graphs of the functions eax+b, log (ax + b), 1/(ax + b), sin (ax + b), cos(ax + b), |ax +b| and to illustrate the effect of a and b on the graph
Session 18 (Practice 2) (2 hrs)
Plotting the graphs of the polynomial of degree 4 and 5, the derivative graph, the second derivative graph and comparing them
Session 19 (Practice 3) (2 hrs)
Sketching parametric curves (e.g. Trochoid, cycloid, epicycloids, hypocycloid)
Session 21
Reduction formula involving sine and cosine function sin ^nx cos ^nx
Session 22
Reduction formula for integrals with limits from 0 to pi/2
Session 23
Reduction formula for integral of sin^px, Cos^qx and also with limits from 0 to Pi/2
Session 24 (Project 7) (2 hrs)
Reduction formula for integrals of tan^nx, Cot ^nx, sec^nx, cosec^nx
Session 25 (Project 8) (2 hrs)
To find the Reduction formula for integral of sin^pxcos^qx for different values of p and q and different cases
Session 27
Session 28
Quadrature and areas of plane regions for Cartesian curve
https://www.youtube.com/watch?v=xec6HTcn2M8
https://www.youtube.com/watch?v=cHk_BKIoIXM&feature=youtu.be
https://www.slideshare.net/vaibhavtandel4/application-of-definite-integrals
Session 30
Area of plane regions for polar curve
Session 32 (Project 9) (2 hrs)
To find the area bounded by closed curves
Session 33
Session 35
Session 36 (Practice 4) (2 hrs)
Tracing of conics in Cartesian coordinates/polar coordinates
Session 37 (Practice 5) (2 hrs)
Sketching ellipsoid, hyperboloid of one and two sheets
Session 38 (Practice 6) (2 hrs)
Sketching elliptic, cone, elliptic, paraboloid, hyperbolic paraboloid using Cartesian coordinates
Session 39
Introduction to vector calculus: Definitions and Characteristics of vectors
https://www.youtube.com/watch?v=KBSCMTYaH1s
https://www.slideshare.net/ManmohanDash/engineering-physics-vector-calculus
Session 40
Algebraic operations on vectors
https://www.youtube.com/watch?v=fNk_zzaMoSs
https://www.slideshare.net/UrmilaBhardwaj/vector-algebra-58263932
Session 41
Directional derivatives and directional derivatives along any line
Session 42
Session 45
Second order differential operator
Session 46 (Project 10) (2 hrs)
To explain about vector, different types of vector and vector algebra with suitable example
Session 47 (Project 11) (2 hrs)
To explain the geometrical significance of Gradient of scalar point function with some examples
Session 48 (Project 12) (2 hrs)
To discuss Laplacian operator and its different properties
Case Studies
Case Studies
Our Main Teachers
Received M.Sc. (Mathematics) from Vinoba Bhave University, Hazaribag and B.Ed. from Ranchi University, Ranchi in the year of 2006 and 2008 respectively. Awarded PhD, in the year of 2013, from Indian School of Mines, Dhanbad. Published 22 research papers in different journals of international repute. Participated and presented 16 research papers in various National and […]