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Model Institute of Engineering & Technology

Course Name –Engineering Mathematics-I

Course Code – BSC-101

Lecture No – 1

Topic – Successive Differentiation and Leibnitz theorem

Date –

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COURSE OUTCOMES

Course Outcomes	Description	Mapping with Program Outcomes and Program Specific Outcomes
CO1	Apply the knowledge of calculus to plot graphs of functions and solve the problem of maxima and minima..	1, 2, 3, 8, 10, 12
CO2	Determine the convergence/divergence of infinite series, approximation of functions using power and Taylor’s series expansion and error estimation.	1, 2, 5, 8, 10, 12
CO3	Apply the concept of definite integrals to calculate area under the curves.	1, 2, 3, 8, 10, 12
CO4	Apply matrix operations and techniques such as Gauss elimination, matrix inversion, and Cramer’s rule to solve linear systems and engineering problems.	1, 2, 5, 8, 10, 12
CO5	Demonstrate knowledge of vector space by solving associated problems	1, 2, 5, 8, 10, 12

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Detailed Syllabus

Section A

UNIT 1: Differential Calculus– I: Leibnitz theorem (without proof), Partial differentiation, Euler’s theorem on homogeneous functions, Asymptotes, Double points, curvature, Curve tracing in Cartesian, polar and parametric forms. (11 HRS)

UNIT 2: Differential Calculus–II: Rolle’s Theorem, Mean value Theorem, Taylor’s and Maclaurin’s series with remainder , indeterminant forms, Taylor series in two variables, Maxima and Minima of functions of two variables, method of Lagrange’s multipliers. (11 HRS)

UNIT 3: Integral Calculus: Definite integrals with important properties, differentiation under the integral sign, Gamma, Beta and error functions with simple problems, applications of definite integrals to find length, area, volume and surface area of revolutions, transformation of coordinates, double and triple integrals with simple problems. (10 HRS)

Section B

Unit 4: Matrices: Matrices, vectors: addition and scalar multiplication, matrix Multiplication, Linear systems of equations, linear Independence, rank of a matrix, determinants, Cramer’s Rule, inverse of a matrix, Gauss elimination and Gauss Jordan elimination. (15 HRS)

Unit 5: Vector Space, linear dependence of vectors, basis, dimension, Linear Transformations, range and kernel of a linear map, rank and nullity, Inverse of a linear transformation, rank nullity theorem , composition of linear maps, Eigen values, eigenvectors, symmetric, skew-symmetric, and orthogonal matrices, Eigen bases. (10 HRS)

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Assessment and Evaluation Plan

Assessment Tools

Quiz

Evaluation

SNAP Test

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Lesson Outcomes

Students will be able to:

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Understand the concept of successive differentiation.
Learn the techniques for finding higher-order derivatives.
Understand and apply Leibnitz’s Theorem for the nth derivative of product of functions.
Solve problems based on these concepts.

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Course Outcome 1 - Delivery Plan

Course Outcomes	Topics	Blooms Taxonomy
CO1	Apply the knowledge of calculus to plot graphs of functions and solve the problem of maxima and minima..	Understanding

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SUCCESSIVE DIFFERENTIATION

What is Successive Differentiation?

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What is Successive Differentiation?

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Examples of Successive Differentiation

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Leibnitz’s Theorem (Statement)

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Reflective Questions

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Summarize

Leibniz's theorem, also known as the Leibniz rule or successive differentiation rule, provides a formula for finding the nth derivative of the product of two functions. It extends the product rule of differentiation to higher-order derivatives. Essentially, if you have two functions, u(x) and v(x), that are both differentiable n times, Leibniz's theorem allows you to calculate the nth derivative of their product, u(x)v(x).

Applications:

It's useful in various areas of calculus and differential equations, especially when dealing with higher-order derivatives of products.

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Thank You

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