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Pailan College of Management and Technology B.B.A Lessons Plan 2nd Semester 2012 – 2013 MATHEMATICS II Faculty Nabanita Maity Course Title B.B.A Course Code BBA-202 Objective of the Students will develop computational skills in calculus needed for more advanced calculus-based courses. They will learn and retain basic knowledge in the core branches of mathematics. subject Teaching Methodology Black board , Class exercises, Assignments. Assessment of the Students will be on the basis of the following weightage :- 1. End Term Exam 70 2. Attendance 5 3. Assignment 5 4. Mid Term. I Exam 20 5. Mid Term. II Exam 20 Total best of two 100 Marks Text Books S.N.Dey , N.K.Nag. Reference books Bhanja Ganguly , Amitava mitra ,Das Mukherjee. Course Duration 40 lectures. Time per 1 hour each Lecture 1 Module Detailed of Course to be covered Topic Determinants of order 2 and 3; minors and cofactors; expansion of determinants; properties of determinants; Cramer’s rule for solving simultaneous equations in two or three variables. 1 Matrices: Different types of matrices; Matrix Algebra – addition, subtraction and multiplication of matrices; Singular and non-singular matrices; adjoint and inverse of a matrix; elementary row / column operations; Solution of a system of linear equations using matrix algebra. Determinants: Vectors: Row and column vectors and their significance Assignment Question 1.For a matrix A 2 -1 1 3 Show that A2 – 5A+7I2 =0 and hence find A-1. 2. Find A-1 when A = 3. If x +y +z =0 1 4 2 16 Show that 1 1 1 X y z X3 y3 z3 =0 4. Solve the following equation by Cramer’s rule i) x + y + z = 3 , 2y + z = 3 , z =1 ii) 2x +3y +z = 8 , 3x – 5y +2x = 7 , x+y=z = 4 5. Determine the matrices A and B where 2B + A = 1 2 0 and 6 3 -3 -5 3 1 6.If A = 7.Show that 1 2 1 1 -4 1 3 0 -3 and 2A –B = 2 1 5 2 -1 6 0 1 2 B= 2 1 1 1 -1 0 2 1 -1 (a2 +b2 )/c c c a (b2 +c2)/a a b b (c2 + a2)/b 8. Show that AB ≠BA where A = 1 4 -1 1 and B = 3 7 -3 -2 2 Compute AB. = 4abc. No. of Lectures. 9 Module 2 Detailed of Course to be covered Topic Idea of conics as sections of a cone Brief ideas of Foci, Directrix, Eccentricity and Latus Rectum; Equations of parabola, ellipse, hyperbola and rectangular hyperbola in standard form. No. of Lectures 7 Assignment Questions 1. Find the points on the curve y = x2 -4x +9 the tangents of which pass through the origin. 2. The equation of a parabola is y2 = 6( x+y) find the vertex, equation of axis, focus , latus rectum. 3. The equation of a hyperbola is x2 - y2 = a2 . find out the eccentricity , focus and latus rectum. 4. If the ellipse x2/a2 + y2/b2 = 1 passes through (-3,2) and its eccentricity be √3/5 then find the latus rectum. 5. If the centre , vertex and eccentricity of a hyperbola be (2,4), (6,4), (√5) then find its equation. 6. The equation of the directrix of a hyperbola is x-y+3 = 0. Its focus is (-1,1) and eccentricity is 3. Find the equation of the hyperbola. 7. If the straight line 3x +4y +1 is a tangent to the hyperbola y2 = 4ax. Then find the value of a. 8. Find the equation of an ellipse for which the principal axes are along the coordinate axis , the length of the latus rectum is 4 and eccentricity is 1/3. 9. In the hyperbola 9x2 – 4y2 + 54x +16y + 29 = 0 find the coordinates of the centre , length of latus rectum, eccentricity, coordinates of foci. 3 Module Detailed of Course to be covered Topic Limits: 3 No. of Lectures Notation and meaning of limits; Fundamental theorems on limits; Evaluation of limits of algebraic, exponential and logarithmic functions. 5 Assignment Questions 1. Lt (1 – cos x)/x2 x 0 2. Lt (ex - e-x)/x x 0 3. Evaluate Lt (1m +2m +3m +………nm) / nm+1 n ∞ 4. (m>-1) A function is defined as f(x) = x2 when x > 1 2 when x = 1 x when x < 1 Find Lt f(x) x 1 5. Exercises from S.N.DEY (Vol – 2) Module 4 Detailed of Course to be covered Topic Continuity Continuity of a function at a point x = a and in an interval Assignment Questions 1. Show that the function defined below is continuous at x=1. f(x) = x2 +1 when x > 1 2 when x = 1 2x when x < 1 2. Show that the function defined below is continuous at x=2. f(x) = x2 +4 when x > 2 8 when x = 2 3x2 - 4 when x < 2 3. Exercises from S.N.DEY (Vol – 2) 4 No. of Lectures 2 Module 5 Detailed of Course to be covered Topic Differentiation Meaning and geometrical interpretation of differentiation; Differentiation from first principles; Standard derivatives; Rules for calculating derivatives; Logarithmic differentiation; Derivatives of composite functions, implicit functions and functions defined parametrically. Successive differentiation: Second and higher order derivatives; forming equations with such derivatives. Applications of differentiation: Optimization of functions; Curve sketching; Equations of tangent and normal; Derivative as a rate measurer; Sign of a derivative - increasing and decreasing functions; Partial derivatives: Homogenous functions; Euler’s Th Assignment Questions 1. If y = a cos (logx) + bsin(logx) then show that x2y2 + xy1 + y = 0 2. Verify Euler’s theorem for the function f(x ,y ) = x2+10xy+y2 3. If xy = ex-y then show that dy/dx = log x/ (1+logx)2 4. Verify Euler’s theorem for the function f(x ,y ) = (x2+y2) / (x2-y2) 5. Find dy/dx if y = x2(x4 – 1)5 6. If y = x / √(1+x2) prove that x3 dy/dx = y3. 7. If y = asin(mx) + b cos (mx) then show that y2 = -m2y 8. Verify Euler’s theorem for the function f(x ,y ) = (x2+y2) / (x-y) 9. If y = sin(msin-1x) then show that (1-x2)y2 – xy1 + m2y = 0. 10. If v = sin-1 (x2 +y2)/(x+y), then show that xVx + y Vy = tan V 11. If ax2 + 2hxy +by2 = 1 then show that y2 = (h2 – ab) / (hx +by)2 5 No. of Lectures 10 Module 6 Detailed of Course to be covered Topic Optimization No. of Lectures Optimization of functions of more than one variable: unconstrained and constrained optimization; cases of two variables involving not more than one constraint. 3 Assignment Questions 1. Examples and Exercises from the text book. Module 7 Detailed of Course to be covered Topic Integration Indefinite Integrals: Integration as the inverse of differentiation; Standard integrals; Integration by substitution, by parts and by the method of partial fractions. Definite Integrals: Definite integral as the limit of a sum; Properties of definite integrals; Application of definite integrals in calculating the areas under curves. Assignment Questions п/2 1. ∫ √sinx / (√sinx + √cosx) dx 0 2. Evaluate ∫ x / (x+1)(x+2) dx. 3. 4. 5. 6. 7. 8. No. of Lectures 1 From the first principle find the value of ∫ (3x +5) dx 0 -1 2 Evaluate ∫ x sin x / √(1-x ) dx 1 From the first principle find the value of ∫ (x2 +5) dx 0 Evaluate ∫ x5 / √( 1+x3) dx 1 Evaluate ∫ sin-1 (2x/1+x2) dx 0 Evaluate ∫ ex (1/x - 1/x2) dx 9. Evaluate ∫ x2 / √(1+x3) dx 10. Evaluate ∫ ex(x logx +1) /x dx 11. Examples and Exercises from the text book. 6 10 7