Differential Calculus

Paper Code: 
CMAT112
Credits: 
4
Contact Hours: 
60.00
Max. Marks: 
100.00
Objective: 

This course will enable the students to -

  1. Acquaint the students with fundamental concepts of single variable calculus.
  2. Explore the solution of problems from a mathematical perspective and help to prepare students to succeed in upper level math, science, engineering and other courses that require calculus. 
  3. Determine whether an infinite sequence is convergent or divergent.

Course Outcomes (COs):

 

 Course

Learning outcomes

(at course level)

Learning and teaching strategies

Assessment

Strategies

Course Code

Course Title

 

 

 

 

 

 

 

 

 

CMAT 112

 

 

Differential Calculus (Theory)

 

 

 

 

 

 

The students will be able to –

 

CO7: Describe the consequences of Mean Value theorems for differentiable functions and be able to compute the expression for the derivative of curve, pedal equation for Cartesian and Polar curves.

CO8: Compute the expression for the derivative of a composite function using the chain rule of differentiation, differentiate a relation implicitly and compute the line tangent to its graph at a point, application of Euler's Theorem in homogeneous functions.

CO9: Analyse the extremas of a function on an interval and classify them as minima, maxima or saddles using the second derivative test.

CO10: Determine the chord of curvature, envelopes and asymptotes for polar and cartesian curves.

CO11: Compute the convergence of infinite non-negative series and alternative series using various convergence tests.

CO12: Interpret the value of the first and second derivative as measures of increase and concavity, convexity of functions, compute the critical points of a function on an interval and be able to trace cartesian and polar curves.

Approach in teaching:

 

Interactive Lectures, Discussion, Power Point Presentations, Informative videos

 

Learning activities for the students:

Self learning assignments, Effective questions, presentations, Giving tasks

 

 

 

Quiz, Poster Presentations,

Power Point Presentations, Individual and group projects,

Open Book Test, Semester End Examination

 

 

 

 

 

 

Unit I: 
I
9.00
Taylor’s and Maclaurins theorems with different remainders, Expansion of sin(x), cos(x), ex, log(1+x), (1+x)m, Derivative of an arc, Pedal equation (Cartesian and Polar Curves).
 
Unit II: 
II
9.00
Infinite series of non-negative terms, Convergences (definition), Test for Convergence (Without Proof): Comparison test, Cauchy’s nth root test, D’Alembert’s ratio test, Raabe’s test,D’Morgan’s test, Cauchy’s condensation test, Logarithm ratio test, Gauss test. Alternating Series – Leibnitz Test, Absolute and conditional convergence.
 
Unit III: 
III
9.00
Partial differentiation, Total derivative, Euler’s theorem for homogeneous functions, Maxima and minima of functions of  two independent variables: necessary and sufficient conditions (without proof), Lagrange’s undetermined multipliers ( without proof ) and related problems.
 
Unit IV: 
IV
9.00
Radius, center and chord of curvature, Envelopes (Cartesian curves), Asymptotes (Cartesian and Polar curves).
 
Unit V: 
V
9.00
Multiple points, Classification of double points: Node, cusp, point of inflexion, Tracing of Cartesian and polar curves.
 
Essential Readings: 
  • Shanti Narayan, Differential Calculus, S. Chand & Co. Pvt. Ltd. New Delhi, 2008.
  • M. Ray and G.C. Sharma, Differential Calculus, Shivalal Agarwal & Co. Agra, 2010.
  • Gorakh Prasad, Text Book on Differential Calculus, Pothishala Pvt. Ltd, Allahabad, 2000.
  • H.S. Dhami, Differential Calculus, New Age International (P) Ltd., New Delhi, 2012.
  • Chaurasia, Goyal, Agarwal, Jain , Differential Calculus, RBD, Jaipur, 2020.
References: 
  • Schaum’s, Theory and Problems of Advanced Calculus, Schaum’s outline series New York, 2011.  
  • Ahsan Akhtar and Sabiha Ahsan, A Text Book of Differential Calculus, PHI Ltd. New Delhi, 2009.
  • G.N. Berman, A Problem Book in Mathematical Analysis, Mir Publishers, Moscow, 2016.
  • G.C. Sharma and Madhu Jain, Calculus, Galgotia Publication, Dariyaganj, New Delhi, 2003.
  • Ulrich L. Rohde, G.C. Jain, Ajay K. Poddar and A.K.Ghosh Introduction to Differential Calculus, Wiley Publications USA, 2012.
 
Academic Year: 
2022-2023
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