# Calculus

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# 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.

## Learning Outcomes

Upon successful completion of this course, students will be able to:

• understand the importance of linear functions in mathematics.
• deals with the major problems of differential and integral calculus.
• recognize other important classes of functions (such as trigonometric and rational functions), and will be able to use calculus with these functions.

## 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

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)

To find radius of curvature (Intrinsic and Parametric form)

## Session 5 (Project 2)

To find radius of curvature (Pedal form)

## Session 9 (Project 3)

To find asymptotes of different curves (implicit form)

## Session 14 (Project 4)

To trace some Cartesian curves (Folium of Descartes, Astroid)

## Session 15 (Project 5)

To trace some polar curves (Cardioid, Lemniscate of Bernaulli)

## Session 16 (Project 6)

To trace few more curves (Cartenary, Cycloid etc) and Loops

## Session 17 (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

## Session 18 (Practice 2)

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)

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)

Reduction formula for integrals of tan^nx, Cot ^nx, sec^nx, cosec^nx

## Session 25 (Project 8)

To find the Reduction formula for integral of sin^pxcos^qx for different values of p and q and different cases

## Session 29

Area of plane regions for parametric curve

## Session 32 (Project 9)

To find the area bounded by closed curves

## Session 36 (Practice 4)

Tracing of conics in Cartesian coordinates/polar coordinates

## Session 37 (Practice 5)

Sketching ellipsoid, hyperboloid of one and two sheets

## Session 38 (Practice 6)

Sketching elliptic, cone, elliptic, paraboloid, hyperbolic paraboloid using Cartesian coordinates

## Session 39

Introduction to vector calculus: Definitions and Characteristics of vectors

https://www.slideshare.net/ManmohanDash/engineering-physics-vector-calculus

## Session 41

Directional derivatives and directional derivatives along any line

https://slideplayer.com/slide/14556957/

## Session 45

Second order differential operator

## Session 46 (Project 10)

To explain about vector, different types of vector and vector algebra with suitable example

## Session 47 (Project 11)

To explain the geometrical significance of Gradient of scalar point function with some examples

## Session 48 (Project 12)

To discuss Laplacian operator and its different properties