Model Institute of Engineering & Technology
Course Name –Engineering Mathematics-I
Course Code – BSC-101
Lecture No – 1
Topic –Rolle’s theorem
Date –
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 |
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
Lesson Outcomes
Students will be able to:
.
Understand the conditions under which Rolle's theorem applies (continuity on a closed interval, differentiability on the open interval, and equal function values at the endpoints).
Identify points within an interval where the derivative of a function is zero, given that the function satisfies Rolle's theorem's conditions.
Recognize Rolle's theorem as a special case of the Mean Value Theorem.
Apply Rolle's theorem to solve problems related to finding specific points on a curve where the tangent is horizontal.
Course Outcome 1 - Delivery Plan
Course Outcomes | Topics | Blooms Taxonomy |
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 |
ROLLE’S THEOREM
Let f be a function that satisfies the following three hypotheses:
Then, there is a number c in (a, b) such �that f '(c) = 0.
ROLLE’S THEOREM
In each case, it appears there is at least one point (c, f(c)) on the graph where the tangent is horizontal and
thus f '(c) = 0.
ROLLE’S THEOREM
Let’s apply the theorem to the position function s = f(t) of a moving object.
If the object is in the same place at two different �instants t = a and t = b, then f(a) = f(b).
The theorem states that there is some instant of �time t = c between a and b when f '(c) = 0; that is, �the velocity is 0.
In particular, you can see that this is true when �a ball is thrown directly upward.
Example 1
Reflective Questions
Thank You
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