### ENGINEERING MATHEMATICS – II -- Diploma Karnataka 2020

Pre-requisites:

Engineering Mathematics-I, in First Semester Diploma curriculum.

### Course Objectives:

1.     Apply the concept of straight line and conic section in engineering field.

2.     Determine derivatives of functions involving two variables.

3.     Apply the concepts of differentiation in physics and engineering courses.

4.     Evaluate the integrals of functions of two variables.

5.     Apply the concepts of definite integrals and its application over a region.

6.     Solve the ODE of first degree, first order in engineering field.

### Course Contents:

 Topic and Contents Hours Marks Unit-1: COORDINATE GEOMETRY 08hr 23 a.  Straight lines: Different forms of equations of straight lines: y = mx + c, y − y1 = m(x − x1), y − y1 = (y2–y1) (x − x1). x2–x1 General equation of a lineax + by + c = o (graphical representation and statements) and problems on above equations. Equation of lines through a point and parallel or perpendicular to a given line. Problems. b.  Conic Section: Definition of conic section. Definition of axis, vertex, eccentricity, focus and length of latus rectum. Geometrical representation of parabola, ellipse and hyperbola: Equations of parabolay2 = 4ax, 04 hr                     04hr Equation of ellipse x2 + y2 = 1and a2            b2 Equation of hyperbola x2 − y2 = 1(without proof of above 3 a2            b2 equations). Equations of parabola, ellipse and hyperbola with  respect to x-axis as axis of conic. Finding axes, vertices, eccentricity, foci and length of lattice rectum of conics. Problems on finding the above said equations with direct substitution. UNIT – 2: DIFFERENTIAL CALCULUS 15hr 39 Differentiation. Definition of increment and increment ratio. Definition of derivative of a function. Derivatives of functions ofxn, sin x, cos xand tan xwith respect to ‘x’ from first principle method. List of standard derivatives of cosecx, secx, cotx, loge x, ax, ex         etc. Rules of differentiation: Sum, product, quotient rule and problems on rules. Derivatives of function of a function (Chain rule) and problems. Inverse trigonometric functions and their derivatives. Derivative of Hyperbolic functions, Implicit functions, Parametric functions and problems. Logarithmic differentiation of functions of the type uv,where u and v are functions of x.Problems. Successive differentiation up to second order and problems on all the above types of functions. UNIT – 3: APPLICATIONS OF DIFFERENTIATION. 07hr 17 Geometrical meaning of derivative. Derivative as slope. Equations of tangent and normal to the curve y = f(x) at a given point- (statement only). Derivative as a rate measure i.e.to find the rate of change of displacement, velocity, radius, area, volume using differentiation. Definition of increasing and decreasing function. Maxima and minima of a function. UNIT-4: INTEGRAL CALCULUS. 12hr 30 Definition of Integration. List of standard integrals. Rules of integration (only statement) 1.Ú kf (x)dx = k Ú f (x)dx.        2. Ú{f(x) ± g(x)}dx = Úf(x)dx± Úg(x)dx problems. Integration by substitution method. Problems. Standard integrals of the type     1.Ú    dx     = 1 tan-1Ê x ˆ + c                2. Ú     dx       = sin-1Ê x ˆ + c . x2 + a2    a         Á a ˜                                2             2                        Á a ˜ Ë   ¯                              a   - x     Ë                              ¯   3. Ú  dx            = 1 sec-1Ê x ˆ + c  Á    ˜ x    x2  - a2         a           Ë a ¯                                (1 to 3 with proof) 4. Ú   dx      =  1  logÊ x - a ˆ + c     if x > a > 0. x2 - a2     2a       Á x + a ˜ Ë          ¯ 5. Ú   dx      =   1   logÊ a + x ˆ + c     if  a   > x   >  0.              ( 4 & 5 without proof) a2 - x2     2a       Á a - x ˜ Ë          ¯   and problems on above results Integration by parts of the type∫ xnexdx ,∫ xsinxdx, ∫ xcosxdx, ∫ xlogxdx , ∫ logxdx,∫ tan–1 x dx, ∫ x sin2 x dx , ∫ x cos2 x dxwhere n=1, 2. Rule of integration by parts. Problems UNIT – 5: DEFINITE INTEGRALS AND ITS APPLICATIONS 05 hr 22 Definition of Definite integral. Problems on all types of integration methods. Area, volume, centres of gravity and moment of inertia by integration method. Simple problems. UNIT – 6: DIFFERENTIAL EQUATIONS. 05 hr 14 Definition, example, order and degree of differential equation with examples. Formation of differential equation by eliminating arbitrary constants up to second order. Solution of O. D. E of first degree and first order by variable separable method. Linear differential equations and its solution using integrating factor. Total 52 145     Course Delivery:

The Course will be delivered through lectures, class room interaction, exercises, assignments and self-study cases. On successful completion of the course, the student will be able to:

1.     Formulate the equation of straight lines and conic sections in different forms.

2.     Determine the derivatives of different types of functions.

3.     Evaluate the successive derivative of functions and its application in tangent, normal, rate measure, maxima and minima.

4.     Evaluate the integrations of algebraic, trigonometric and exponential function.

5.     Calculate the area under the curve, volume by revolution, centre of gravity and radius of gyration using definite integration.

6.     Form and solve ordinary differential equations by variable separable method and linear differential equations. CO Course Outcome PO Mapped Cognitive Level Theory Sessions Allotted marks on cognitive levels TOTAL R U A CO1 Formulate the equation of straight lines and conic sections in different forms. 1,2,3,10 R/U/A 08 6 5 12 23 CO2 Determine the derivatives of different types of functions. 1,2,3,10 R/U/A 15 12 15 12 39 CO3 Evaluate the successive derivative of functions and its application in tangent, normal, rate measure, maxima and minima. 1,2,3,10 R/U/A 07 6 5 6 17 CO4 Evaluate the integrations of algebraic, trigonometric                           and                           exponential function 1,2,3,10 R/U/A 12 9 15 6 30 CO5 Calculate the area under the curve, volume by revolution, centre of gravity and radius of gyration using definite integration 1,2,3,10 R/U/A 05 6 10 6 22 CO6 Form and solve ordinary differential equations    by     variable                    separable method     and     linear                 differential equations. 1,2,3,10 R/U/A 05 3 5 6 14 Total Hours of instruction 52 Total marks 145
##### R-Remember; U-Understanding; A-Application

Course outcomes –Program outcomes mapping strength

 Course Programme Outcomes 1 2 3 4 5 6 7 8 9 10 Engineering Maths-II 3 3 3 - - - - - - 3 Method is to relate the level of PO with the number of hours devoted to the COs which address the given PO. If >40% of classroom sessions addressing a particular PO, it is considered that PO is addressed at Level 3

If 25 to 40% of classroom sessions addressing a particular PO, it is considered that PO is addressed at Level 2 If 5 to 25% of classroom sessions addressing a particular PO, it is considered that PO is addressed at Level 1

If < 5% of classroom sessions addressing a particular PO, it is considered that PO is considered not-addressed.

### Reference Books:

1.     NCERT Mathematics Text books of class XI and XII.

2.     Higher Engineering Mathematics by B.S Grewal, Khanna publishers, New Delhi.

3.     Karnataka State PUC mathematics Text Books of I & II PUC by H.K. Dass and Dr. Ramaverma published by S.Chand & Co.Pvt. ltd.

4.     CBSE Class Xi & XII by Khattar & Khattar published PHI Learning Pvt. ltd.,

5.     First and Second PUC mathematics Text Books of different authors.

6.     E-books:www.mathebook.net

7.     www.freebookcentre.net/mathematics/ introductory-mathematics -books.html

### Course Assessment and Evaluation:

 Method What To whom When/where (Frequency in the course) Max Marks Evidence collected Contributing to course outcomes DIRECT ASSMENT *CIE Internal Assessment Tests Student Three tests (Average of Three tests to be computed). 20 Blue books 1 to 6 Assignment s Two    Assignments based on CO’s (Average marks of Two Assignments shall be rounded off    to    the         next higher digit.) 5 Log of record 1 to 6 Total 25 *SEE Semester End Examinatio n End of the course 100 Answer scripts at BTE 1 to 6 INDIRECT ASSESSMENT Student feedback Student Middle of the course -NA- Feedback forms 1 to 3, delivery of the course End of Course survey End of course Questionnaire 1 to 6, Effectiveness of delivery of instructions and   assessment methods

*CIE – Continuous Internal Evaluation    *SEE – Semester End Examination

Note: I.A. test shall be conducted for 20 marks. Average marks of three tests shall be rounded off to the next higher digit.

### Composition of Educational Components:

Questions for CIE and SEE will be designed to evaluate the various educational components (Bloom’s taxonomy) such as:

 Sl. No. Educational Component Weightage (%) 1 Remembering 31 2 Understanding 41 3 Applying the knowledge acquired from the course 25 Analysis Evaluation 3