GUJARAT TECHNOLOGICAL UNIVERSITY (GTU)

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Applied Mathematics (Course Code: 4320001)

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Competency-focused Outcome-based Green Curriculum-2021 (COGC-2021)
Semester: II

1. RATIONALE

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This course is an extension of the course based on Mathematics of first semester namely Applied Mathematics. The course is designed to inculcate its applications in relevant branch of engineering and technology using the techniques of Differentiation, Integration, Differential equations, Matrix theory and Statistics.

Calculus is a branch of Mathematics that calculates how matter, particles and heavenly bodies actually move. With calculus, we can find how the changing conditions of a system affect us, we can control a system. Derivatives are useful to find maxima and minima of the function, velocity and acceleration and also useful for many engineering optimization problems.

Definite integrals are a powerful tool to help us realize and model the world around us. Differential equations are widely applied to model natural phenomena, engineering systems and many other situations. Matrix analysis is a valuable tool used in nearly all the engineering sciences.

Statistics can be defined as a type of mathematical analysis which involves the method of collecting and analyzing data and then summing up the data into a numerical form for a given set of factual data or real-world observations. This course further develops the skills and understanding of mathematical concepts which underpin the investigative tools used for modeling and analysis in a wide range of applications in engineering.

2. COMPETENCY

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The purpose of this course is to help the student to attain the following industry identified competency through various teaching learning experiences:

•  Solve broad-based technology problems using the principles of Applied mathematics.

3. COURSE OUTCOMES (COs)

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The practical exercises, the underpinning knowledge and the relevant soft skills associated with the identified competency are to be developed in the student for the achievement of the following COs:

• a) Demonstrate the ability to Crack engineering related problems based on Matrices.
• b) Demonstrate the ability to solve engineering related problems based on applications of differentiation.
• c) Demonstrate the ability to solve engineering related problems based on applications of integration.
• d) Develop the ability to apply differential equations to significant applied problems.
• e) Solve applied problems using the concept of mean.

4. TEACHING AND EXAMINATION SCHEME

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Note: Out of 30 marks under the theory CA, 10 marks are for assessment of the micro-project to facilitate integration of COs and the remaining 20 marks is the average of 2 tests to be taken during the semester for the assessing the attainment of the cognitive domain UOs required for the attainment of the COs.

Legends:

• L: Lecture
• T: Tutorial/Teacher Guided Theory Practice
• P: Practical
• C: Credit
• CA: Continuous Assessment
• ESE: End Semester Examination

5. SUGGESTED PRACTICAL/TUTORIALS EXERCISES (During Tutorial Hours)

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The following practical outcomes (PrOs)/Tutorials are the sub-components of the COs. Some of the PrOs/Tutorials marked ‘*’ (in approx. Hrs column) are compulsory, as they are crucial for that particular CO at the ‘Precision Level’ of Dave’s Taxonomy related to ‘Psychomotor Domain’.

Note

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1. More Practical Exercises/Tutorials can be designed and offered by the respective course teacher to develop the industry relevant skills/outcomes to match the COs. The above table is only a suggestive list.
2. The following are some sample ‘Process’ and ‘Product’ related skills (more may be added/deleted depending on the course) that occur in the above listed Practical Exercises/Tutorials of this course required which are embedded in the COs and ultimately the competency.

Geometric Thinking

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Comprehend geometric concepts to interpret solutions by applying apt results to solve well defined Engineering problems.

Algebraic Thinking

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Create, interpret, use, and analyze expressions, equations, and inequalities in a variety of contexts.

6. MAJOR EQUIPMENT/ INSTRUMENTS REQUIRED

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This major equipment with broad specifications for the PrOs is a guide to procure them by the administrators to usher in uniformity of practicals in all institutions across the state.

7. AFFECTIVE DOMAIN OUTCOMES

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The following sample Affective Domain Outcomes (ADOs) are embedded in many of the above-mentioned COs and PrOs. More could be added to fulfill the development of this competency.

• a) Work as a leader/a team member.
• b) Follow ethical practices.
• c) Realize importance of green Mathematics.

The ADOs are best developed through the laboratory/field-based exercises. Moreover, the level of achievement of the ADOs according to Krathwohl’s ‘Affective Domain Taxonomy’ should gradually increase as planned below:

1. Valuing Level in 1st year
2. Organization Level in 2nd year
3. Characterization Level in 3rd year

8. UNDERPINNING THEORY

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The major underpinning theory is given below based on the higher level UOs of Revised Bloom’s taxonomy that are formulated for development of the COs and competency. If required, more such higher level UOs could be included by the course teacher to focus on attainment of COs and competency.

Unit I: Matrices

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Unit Outcomes (UOs):

1. Solve simple problems using the concept of algebraic operations of matrices.
2. Apply the concept of adjoint of a matrix to find the inverse of a matrix.
3. Investigate the solution of system of linear equations using matrices.

Topics and Sub-topics:

• 1.1 Concept of Matrix
• 1.2 Types of Matrices
• 1.3 Addition, Subtraction and multiplication by scalar of matrices
• 1.4 Product of two matrices
• 1.5 Adjoint and Inverse of a matrix of order 2X2 and 3X3
• 1.6 Solution of Simultaneous linear equations of two variables

Unit II: Differentiation and its Applications

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Unit Outcomes (UOs):

1. Apply the working rules and standard forms of differentiation to find the derivative of simple functions.
2. Invoke the concept of Chain Rule to find the derivative of simple functions.
3. Find the derivative of given Implicit and Parametric functions.
4. Apply the standard forms and rules of derivative to find the second order derivative of simple functions.
5. Apply the concept and rules of derivative to solve the problems related to velocity, acceleration and Maxima-Minima of given simple functions.

Topics and Sub-topics:

• 2.1 Concept and Definition of Differentiation
• 2.2 Working rules: Sum, Product, Division
• 2.3 Chain Rule
• 2.4 Derivative of Implicit functions
• 2.5 Derivative of Parametric functions
• 2.6 Logarithmic Differentiation
• 2.7 Successive Differentiation up to second order
• 2.8 Applications: Velocity, Acceleration, Maxima & Minima of given simple functions

Unit III: Integration and its Applications

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Unit Outcomes (UOs):

1. Apply the working rules and standard forms of integration to find the integral of simple functions.
2. Find the integral of simple functions using the method of substitution and integration by parts.
3. Solve given problems related to definite integral and its applications.

Topics and Sub-topics:

• 3.1 Concept and Definition of Integration
• 3.2 Working rules and Integral of standard functions
• 3.3 Method of substitution
• 3.4 Integration by parts
• 3.5 Definite Integral and its properties
• 3.6 Applications: Area and volume

Unit IV: Differential Equations

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Unit Outcomes (UOs):

1. Find the order and degree of given differential equations.
2. Solve Differential Equations related to Variable Separable method.
3. Solve given linear differential equations.

Topics and Sub-topics:

• 4.1 Concept and Definition, Order and Degree of differential equation
• 4.2 Solution of DE of first degree and first order by Variable Separable method
• 4.3 Solution of linear Differential equation

Unit V: Statistics

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Unit Outcomes (UOs):

1. Find Mean for the given data.
2. Calculate Mean deviation for the given data.
3. Calculate Standard deviation for the given data.

Topics and Sub-topics:

• 5.1 Mean for ungrouped and grouped data
• 5.2 Mean deviation and Standard deviation about Mean for ungrouped and grouped data

9. SUGGESTED SPECIFICATION TABLE FOR QUESTIONPAPER DESIGN

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Legends: R=Remember, U=Understand, A=Apply and above (Revised Bloom’s taxonomy)

Note: This specification table provides general guidelines to assist student for their learning and to teachers to teach and question paper designers/setters to formulate test items/questions assess the attainment of the UOs. The actual distribution of marks at different taxonomy levels (of R, U and A) in the question paper may vary slightly from above table.

10. SUGGESTED STUDENT ACTIVITIES

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Other than the classroom and laboratory learning, following are the suggested student-related co-curricular activities which can be undertaken to accelerate the attainment of the various outcomes in this course:

Students should conduct following activities in group and prepare reports of about 5 pages for each activity, also collect/record physical evidences for their (student’s) portfolio which will be useful for their placement interviews:

1. Identify engineering problems based on real world problems relevant to content of the unit and solve these problems in the light of free tutorials available on the internet.
2. Explore the opportunity to visit Science city, ISRO or nearby Science centers.
3. Explore the opportunity to visit Mathematics Lab Virtually.
4. Prepare charts showing formulas of differentiation.
5. Prepare charts showing formulas of integrations.
6. Use Graphing calculator to plot the graph of solutions explaining Engineering applications.
7. Communicate mathematical thinking coherently and clearly to other students, peers, and others.

11. SUGGESTED SPECIAL INSTRUCTIONAL STRATEGIES (if any)

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These are sample strategies, which the teacher can use to accelerate the attainment of the various outcomes in this course:

1. Massive open online courses (MOOCs) may be used to teach various topics/sub topics.
2. Guide student(s) in undertaking micro-projects.
3. ‘L’ in section No. 4 means different types of teaching methods that are to be employed by teachers to develop the outcomes.
4. About 20% of the topics/sub-topics which are relatively simpler or descriptive in nature is to be given to the students for self-learning, but to be assessed using different assessment methods.
5. With respect to section No.10, teachers need to ensure to create opportunities and provisions for co-curricular activities.
6. Explore the possibility for understanding the Biosphere through Mathematics.

12. SUGGESTED MICRO-PROJECTS

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Only one micro-project is planned to be undertaken by a student that needs to be assigned to him/her in the beginning of the semester. In the first four semesters, the micro-project are group-based (group of 3 to 5). However, in the fifth and sixth semesters , the number of students in the group should not exceed three.

The micro-project could be industry application based, internet-based, workshop-based, laboratory-based or field-based. Each micro-project should encompass two or more COs which are in fact, an integration of PrOs, UOs and ADOs. Each student will have to maintain dated work diary consisting of individual contribution in the project work and give a seminar presentation of it before submission. The duration of the micro project should be about 14 -16 (fourteen to sixteen) student engagement hours during the course. The students ought to submit micro-project by the end of the semester to develop the industry-oriented COs.

A suggestive list of micro-projects is given here. This has to match the competency and the COs. Similar micro-projects could be added by the concerned course teacher:

• a) Charts: Prepare the Charts of formulae for Matrix, Differentiation, Integration.
• b) Charts: Compare last weather conditions with current weather to predict future weather through chart.
• c) Models: Prepare the cardboard models based on real world applications of derivatives.
• d) Presentation/Seminar: Prepare a presentation/seminar on any relevant topic of interdisciplinary nature.
• e) History of Mathematics : Prepare a write up on the Historical path of Calculus.
• f) Solution of system of linear equations: Form the system of linear equations up to three variables for the given electrical circuit using matrices solve it.
• g) Maxima and Minima: Find a real-world problem related to finding area/volume, form the corresponding function and find maxima/minima. For example, maximize the volume of a box made of a rectangle tin sheet by cutting off squares of same size from each corner and folding up.
• h) Slope of tangent: Find the slope of tangent for the given curves at a given point using derivative and visualize the location using suitable software.
• i) Area/Volume: Find the area of a given closed region or volume of revolution for a given function using integration and visualize using suitable software.
• j) Solution of Differential equation: Form differential equations for real-world problems and plot the graph using suitable software with geometrical interpretation.
• k) Statistics: Collect the data of world of work and find mean, mean deviation and standard deviation for that data.

13. SUGGESTED LEARNING RESOURCES

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14. SOFTWARE/LEARNING WEBSITES

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1. YouTube Channel of DTEGUJ
2. GeoGebra
3. NIOS Online Course Material
4. DPlot
5. Wolfram Mathematica
6. Easy Calculation
7. SciLab
8. NCERT Textbooks (NCERT Textbooks of Mathematics 11th and 12th Science)
9. Desmos Graphing Calculator

15. PO-COMPETENCY-CO MAPPING

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Legend: ’ 3’ for high, ’ 2 ’ for medium, ‘1’ for low and ‘-’ for no correlation of each CO with PO.

16. COURSE CURRICULUM DEVELOPMENT COMMITTEE

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GTU Resource Persons

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NITTTR Resource Persons

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