Department of Mathematics, University of Delhi
Semester-I
GE-1: Calculus OR GE-1: Analytic Geometry and Theory of Equations
GE-1: Calculus
Total Marks: 100 (Theory: 75, Internal Assessment: 25)
Workload: 5 Lectures, 1 Tutorial (per week) Credits: 6 (5+1)
Duration: 14 Weeks (70 Hrs.) Examination: 3 Hrs.
Course Objectives: The main aim of this course is to learn about applications of derivatives for
sketching of curves and conics and application of definite integrals for calculating volumes of
solids of revolution, length of plane curves and surface areas of revolution. Various notions related
to vector-valued functions and functions of several variables are also discussed in this course.
Course Learning Outcomes: This course will enable the students to:
i) Sketch the curves in Cartesian and polar coordinates as well as learn techniques of
sketching the conics.
ii) Visualize three dimensional figures and calculate their volumes and surface areas.
iii) Understand limits, continuity and derivatives of functions of several variable and vector-
valued functions.
Unit 1: Applications of Derivatives and Limits
The first derivative test, Concavity and inflection points, Second derivative test, Curve sketching
using first and second derivative test; Limits at infinity, Horizontal asymptotes, Vertical
asymptotes, Graphs with asymptotes; L’Hôpital’s rule.
Unit 2: Applications of Definite Integrals
Volumes by slicing, Volumes of solids of revolution by the disk method, Volumes of solids of
revolution by the washer method, Volume by cylindrical shells, Length of plane curves, Arc length
of parametric curve, Area of surface of revolution.
Unit 3: Conics, Vector-Valued Functions and Partial Derivatives
Techniques of sketching conics, Reflection properties of conics; Polar coordinates, graphing in
polar coordinates; Vector-valued functions: Limits, Continuity, Derivatives, Integrals, Arc length,
Unit tangent vector, Curvature, Unit normal vector; Functions of several variables: Graphs and
level curves, Limits and continuity, Partial derivatives and differentiability, The chain rule,
Directional derivatives and gradient vectors, Tangent plane and normal line, Extreme values and
saddle points.
References:
1. Anton, Howard, Bivens, Irl, & Davis, Stephen (2013). Calculus (10th ed.). John Wiley &
Sons Singapore Pvt. Ltd. Reprint (2016) by Wiley India Pvt. Ltd. Delhi.
2. Strauss, M. J., Bradley, G. L., & Smith, K. J. (2007). Calculus (3rd ed.). Dorling Kindersley
(India) Pvt. Ltd. (Pearson Education). Delhi. Sixth impression 2011.
3
Department of Mathematics, University of Delhi
Additional Reading:
i. Thomas, Jr. George B., Weir, Maurice D., & Hass, Joel (2014). Thomas’ Calculus (13th
ed.). Pearson Education, Delhi. Indian Reprint 2017.
Teaching Plan (GE-1: Calculus):
Weeks 1 and 2: The first derivative test, Concavity and inflection points, Second derivative test, Curve
sketching using first and second derivative test.
[2] Chapter 4 (Section 4.3).
Weeks 3 and 4: Limits at infinity, Horizontal asymptotes, Vertical asymptotes, Graphs with asymptotes;
L’Hôpital’s rule.
[2] Chapter 4 (Sections 4.4, and 4.5).
[1] Chapter 3 (Section 3.3), and Chapter 6 (Section 6.5).
Weeks 5 and 6: Volumes by slicing, Volumes of solids of revolution by the disk method, Volumes of
solids of revolution by the washer method, Volume by cylindrical shells.
[1] Chapter 5 (Sections 5.2, and 5.3).
Week 7: Length of plane curves, Arc length of parametric curves, Area of surface of revolution.
[1] Chapter 5 (Sections 5.4, and 5.5).
Week 8: Techniques of sketching conics, Reflection properties of conics.
[1] Chapter 10 (Section 10.4).
Week 9: Polar coordinates, Graphing in polar coordinates.
[1] Chapter 10 (Section 10.2).
Week 10: Vector-valued functions: Limit, continuity, Derivatives, Integrals, Arc length, Unit tangent
vector, Curvature, Unit normal vector.
[1] Chapter 12 (Sections 12.1 to 12.5).
Weeks 11 and 12: Functions of several variables: Graphs, Level curves, Limits and continuity, Partial
derivatives and differentiability.
[1] Chapter 13 (Section 13.1 to 13.4).
Week 13: Functions of several variables: The chain rule, Directional derivatives and gradient vectors.
[1] Chapter 13 (Sections 13.5, and 13.6).
Week 14: Functions of several variables: Tangent plane and normal line, Extreme values and saddle points.
[1] Chapter 13 (Sections 13.7, and 13.8).
Facilitating the Achievement of Course Learning Outcomes
Unit Course Learning Outcomes Teaching and Learning Assessment Tasks
No. Activity
1. Sketch the curves in Cartesian (i) Each topic to be explained Student
and polar coordinates as well as with examples. presentations.
learn techniques of sketching the (ii) Students to be involved in Participation in
conics. discussions and encouraged discussions.
2. Visualize three dimensional to ask questions. Assignments and
figures and calculate their (iii) Students to be given class tests.
volumes and surface areas. homework/ assignments. Mid-term
3. Understand limits, continuity and (iv) Students to be encouraged examinations.
derivatives of functions of to give short presentations. End-term
several variable and vector- examinations.
valued functions.
Keywords: Concavity, Asymptotes, Curve sketching, L’Hôpital’s rule, Volumes of solids of
revolution, Sketching of conics, Vector-valued functions, Functions of several variables.
