BOARD OF TECHNICAL EXAMINATION – KARNATAKA SUBJECT: APPLIED MATHEMATICS – II For II - SemesterDIPLOMA COURSES OF ALL BRANCHES Contact hour per week: 04 hour per Semester: 64 UNIT NO. CHAPTER TITLE Contact CONTACT HR. 6 16 6 14 4 2 12 4 64 DIFFERENTIATION CALCULUS 1 2 3 4 5 6 7 8 LIMITS DIFFERENTIATION APPLICATIONS OF DIFFERENTIATION INTEGRAL CALCULUS INDEFINITE INTEGRATION DEFINITE INTEGRATION APPLICATIONS OF DEFINITE INTEGRATION DIFFERENTIAL EQUATIONS TESTS AND ASSIGNMENT TOTAL HOUR REFFERENCE BOOKS: 1. Applied Mathematics –II By W.R Neelakanta. Sapna Publications. 2. Applied Mathematics –II By Dr. D S Prakash S Chand Publications 3. Text Books of PUC-2 mathematics. 4. Applied Mathematics –II for Polytechnics- By different Authors. 5. Engineering Mathematics. Applied Mathematics – II Page 1 of 11 HOUR Questions to be set (2 Marks) PART.D - - 6 06 04 TOTA 64 L Questions to be answered 15 10 10 07 08 05 APPLIED – MATHEMATICS – II Marks Part A Part B Part C Part D 2 X 10 = 20 5 X 07 = 35 5 X 05 = 25 5 X 04 = 20 Total = 100 No. of Questions to be Answered 10 07 05 04 Applied Mathematics – II Page 2 of 11 .B 2 6 2 Questions to be set (5 Marks) PART.A 1 2 3 4 5 6 7 6 16 6 14 4 2 12 2 4 2 4 3 Questions to be set (5 Marks) PART.QUESTION PAPER BLUE PRINT FOR APPLIED MATHEMATICS – II UNIT NO.C 6 1 1 Questions to be set (5 Marks) PART. of Questions to be set 15 10 08 06 No. Explicit and implicit function. Problems. UNIT – 2: DIFFERENTIAL CALCULUS. UNIT – 4: INTEGRAL CALCULUS. Rules of integration (only statement) 1. cosx and tanx with respect to ‘x’ from first principle method. List of standard derivatives. Definition of Integration. when x tend to ‘∞’ and x tend to ‘a’. sinx. Parametric functions. product and quotient of functions. Hyperbolic functions and inverse of hyperbolic functions. Integrals of functions involving a2 + x2 . Derivatives of function of a function (Chain rule). Variables and Constants. Geometrical meaning of derivative. ∫ { f(x) ± g(x) }dx = ∫ f(x)dx ± ∫ g(x)dx problems. 16 Hr. Some important integrals of the type dx 1 dx x x 1. List of standard integrals.Diploma Courses of All Engineering Branches II Semester Sub: Applied Mathematics II CONTENTS UNIT – 1: LIMITS. UNIT – 3: APPLICATIONS OF DIFFERENTIATION. 6 Hr. Types of function: Direct and Inverse functions. Maxima and minima of a function. Concept of x tends to ‘a’. Definition of increasing and decreasing function. Derivations of algebraic and trigonometric limits. 2. Problems. lim 1 + = e 4. Derivatives of functions of xn.∫ kf ( x) dx = k ∫ f ( x) dx. Implicit functions. lim = log e a 2. Odd and even functions (Definition with examples). Definition of limit of a function. Problems on limit of a function by factorization. Successive differentiation up to second order. lim =1 3. 6 Hr. Derivatives of inverse Trigonometric functions. Definition of derivative of a function. Derivative as a rate measure. Integration by substitution method. Definition of function. ∫ = sin −1 + c with proof. rationalization when x tend to ‘0’. Logarithmic differentiation. a2 – x2 and their radicals. Equation of tangent and normal to the curve y = f(x) at a given point. Rules of differentiation: Sum. lim(1 + n ) n = e x →0 x →0 n →∞ n →0 x x n Simple problems on standard limits. Problems. Definition of increment and increment ratio. Problems. Problems on rules.∫ 2 = tan −1 + c 2. 2 2 2 a x +a a a a −x Applied Mathematics – II Page 3 of 11 . 14 Hr. Standard limit (only statement) n 1 a x −1 ex −1 1 1. Problems. Theorems on definite integrals. Simple problems. ∫ ax 2 + bx + c dx.∫ dx 2 2 ( ) UNIT – 5: DEFINITE INTEGRALS. Solution of D E of first degree and first order by variable separable method. Solutions of differential equation of a type 2 d y dy a 2 + b + cy = 0. Find area. ∫ = sec −1 + c ( 3 to 7 no proof) a x x2 − a2 a Integrals of the forms: dx dx px + q px + q ∫ ax 2 + bx + c . example. Linear equations and its solution. ∫ ax 2 + bx + c . dx dx Applied Mathematics – II Page 4 of 11 . volume and r m s value of a function. Problems. ∫ 2 = log = cosh −1 + c + c if a 〉 x 〉 0. Integration by parts. Homogeneous form and its solution. b. ∫ 2 dx Problems.dx 1 x x−a = sinh −1 + c 4. ∫ 2 2a a −x a− x a x2 − a2 dx 1 x 7. 2 Hr. Solution of differential equations reducible to homogeneous form. Definite integrals of the type ∫0 0 1 + tan x 1 + tan x UNIT – 6: APPLICATIONS OF DEFINITE INTEGRALS. 4 Hr. UNIT – 7: DIFFERENTIAL EQUATIONS. Problems. order and degree of differential equation with examples. Definition. ∫ ax 2 + bx + c dx . where a. ∫ 2 = log + c if x 〉 a 〉 0. Definition of Definite integral. Solution of differential equations reducible to variable separable form. 3. c are constants. Solution of differential equations reducible to linear form-Bernoulli’s form. Rule of integration by parts. Problems. π π 1 1 2 dx. 12 Hr. Exact differential equation and its solution. Formation of differential equation by eliminating arbitrary constants up to second order. 2 2a x −a a x+ a a +x dx 1 dx a + x x 5. x 1 Integration of the forms: ∫ e f(x) + f ( x) dx Problems. Problems. 6. Cosx and Tanx with respect to ‘x’. UNIT – 2: DIFFERENTIAL CALCULUS. 1. x →a x − a 1.2. 1. 2.1 Explain calculus as calculation of Infinitesimal values. 1. 1. 1.2. SPECIFIC OBJECTIVES.2.1.4 Derive differentiation of a functions from first principle method xn . constants and functions.1. 1.2.3 To know the derivatives of higher order up to second order.2 To know different methods of differentiation.1. Direct and inverse functions 2.2 Define limit of a function.5 Define 1.1.2. Write formula for standard limits(statement only). 2.Odd and even function with examples.1. GENERAL OBJECTIVES. 1. 1.8. lim 1 + = e = lim(1 + n ) n x →0 x →0 n →∞ n →0 n x x 1. SPECIFIC OBJECTIVES.2.4 Solve problems on limit of a function by rationalization.2.2 Define Independent.2. Explicit and Implicit function 3.1 To understand variables. lim =1 1.3 Define a function. Sinx. lim 3.1 To understand the differentiation of a function in terms of limit of a function. 2. Applied Mathematics – II Page 5 of 11 .1 Explain the concept of x tends ‘a’. 16 Hr.3 Derive the differential co-efficient (dy/dx). xn − an lim = na n −1 for any rational number. n 1 a x −1 ex −1 1 = log e a 2. 2.5 Solve problems on limit of a function when x tends INFINITY. 2.2. 1.1. 2.2 To know the indeterminant form and evaluation of limit of a given function.1 Define increment and increment ratio.1. 1.7 Solve problems on above results.1.GENERAL AND SPECIFIC OBJECTIVES UNIT – 1: LIMITS.2 Define differentiation. GENERAL OBJECTIVES. 1.6 Deduce sin θ lim = 1 where θ is in radians θ →0 θ 1.4 List types of functions. 2.1.9 Solve simple problems on above results. dependent variable and constants with examples. 1. 6 Hr.3 Solve problems on limit of a function by factorization. 1. 4.4 Solve problems on above types.5 Carry out logarithmic Differentiation 2.2 Solve problems on rules of differentiation.3.6 Solve problem of the type xx . 3. Product and quotient of function. 3.3 Find maximum and minimum values of a function. UNIT -3: APPLICATIONS OF DIFFERENTIATION.1.1 Define integration as anti derivative. GENERAL OBJECTIVES.2 Obtain velocity and acceleration for a moving body whose equation of motion is given. 14 Hr.3.2 To illustrate dy / dx as a rate measure.1 Explain derivative as a rate measure.1 To know integration as converse process of differentiation.5 State derivative of ex and log x.2 List of standard integrals. 3. 4.3 To understand maxima and minima of a function. 3.1 Define increasing and decreasing function.2. 2.3.2 Solve problems on Successive differentiation. 4. x 1/x etc.1 Explain integration by substitution method.1 Obtain the second derivative of a function. 3. 2..3.1. 4.2. volume etc.2. 2.3 Solve problems on tangent and normals. 3. 3.1 To understand dy/dx as slope of a tangent.. 4.3 Solve problems on rate measure including variation of area.1. 3.3 Obtain the derivatives of function of a function (Chain Rule). 2.2. Applied Mathematics – II Page 6 of 11 . 3.3. 4.2.2 To understand indefinite integral. Inverse T-functions. no proof. 4. 6Hr.3 State rules of Integration.4 Solve the problems on rules of integration.2 Solve problems on substitution method. ax .2. 4.1 Explain geometrical meaning dy / dx as a slope of tangent. 3. 2. UNIT – 4: INTEGRAL CALCULUS.2.1. SPECIFIC OBJECTIVES.2.1. SPECIFIC OBJECTIVES.1.2.1.2 Find equation of tangent and normal to a curve y = f(x) at a given point.2.2.1 State rules of differentiation: Derivatives of Sum.2.2 State the condition for maxima and minima of a function. 3. x Sin x. Implicit functions & Parametric functions 2. GENERAL OBJECTIVES. 3.1. 2. ∫ 2 dx 5.6 Explain the rule integration by parts.1. SPECIFIC OBJECTIVES.8 Solve problems of the type ∫ e f(x) + f ( x) dx 1. 5.1. π π 1 1 dx.1 To understand the concept of definite integral to eliminate constant of integration. x log x etc. ∫ 2 = log = cosh −1 + c + c if a 〉 x 〉 0. (ax + b) ex. ex sinx. ∫ x = sin −1 + c . SPECIFIC OBJECTIVES.2. ∫ = sec −1 + c ( 3 to 7 no proof) a x x2 − a2 a 4. 6. log x.. 5. ∫ ax 2 + bx + c dx. 4.2.1 To understand definite integral as a tool to find area under the curve.3 Derive 4. Applied Mathematics – II Page 7 of 11 2 Hr.4 Solve definite integrals of the type ∫02 0 1 + tan x 1 + tan x UNIT – 6: APPLICATIONS OF DEFINITE INTEGRALS.3 Solve the problems of the same type as in indefinite integral using limits of integration.2 State theorems on definite integrals. 5. 4.5 Write px + q px + q dx dx ∫ ax 2 + bx + c . 6.2.5 Solve problems on above results. GENERAL OBJECTIVES. 5. 4.4.2.1 State. x sin2x. ∫ ax 2 + bx + c .2. GENERAL OBJECTIVES.2. ∫ 2 = log + c if x 〉 a 〉 0. volume of solid of revolution and r m s value of a function.4 Solve problems on above results.4 Write dx dx 1 x x−a 3. 4 Hr. x 1 4. ∫ ax 2 + bx + c dx . a∫ b f(x) dx as a definite integral. ∫ 2 2a a −x a− x a x2 − a2 dx 1 x 7.∫ dx 1 x = tan −1 + c 2 a x +a a 2 2. x2 Cos x.1. a a −x dx 2 2 ( ) UNIT – 5: DEFINITE INTEGRALS.1. 4. 2 2 2 2a x −a a x+a a +x dx 1 dx a + x x 5.2.∫ = sinh −1 + c 4. .2.7 Solve problems of the type x sinx. 2 Define Order and Degree of D E with examples.9 Obtain the solution of D E of the type a 2 + b dx dx problems only. volume and r m s value of a function.1.2. 7.2.2.1 Obtain the solution of D E by variable separable method.2.solve problems. SPECIFIC OBJECTIVES.3 Solve problems on above applications. 7.1. 7.7 Obtain the solution of D E by Reducible to homogeneous form – solve problems 7.6 Obtain the solution of D E by Homogeneous form – solve problems.2. -.1. 7.2. 7.2. 7.1. GENERAL OBJECTIVES.4 Obtain the solution of linear D E of the type dx 7. 7. 7.1 To understand the concept of differential equation. dy + Py = Q --solve problems. UNIT – 7: DIFFERENTIAL EQUATIONS.1 Define differential equation with examples. d2y dy + cy = 0. 6. 7.1.2 Solve problems on variable separable method.1 Explain definite integral as a limit of sum (statement). BOARD OF TECHNICAL EXAMINATION – KARANATAKA Applied Mathematics – II Page 8 of 11 .3 Obtain the solution of D E by Reducible to variable separable method – Solve problems.Solve simple 7.2.2 To solve differential equation for unknown functions.8 Obtain the solution of Exact D E --. ************************************************* 12 Hr.2 Write the formulae for finding area.2.3 Formation of D E by eliminating arbitrary constants.5 Obtain the solution of D E by Reducible to linear form (Bernoulli’s form) –solve problems.1. 6. 7. 7.6. Find 4. Evaluate π 2 ∫ tan 0 2 x dx Applied Mathematics – II Page 9 of 11 .MODEL QUESTION PAPER Code: APPLED MATHEMATICS –II ( FOR ALL COURSES) Time: 3 Hrs Maximum marks:100 NOTE: i) Answer any 10 questions in section A. Evaluate ∫ e tan e dx 2 9. Find the slope of a tangent to the curve y = x2+6x – 7 at point ( 1. Evaluate lim 2. Find dy if y = 1 –cos4x dx dy if x2 + y2 = a2 dx dy if x = a sinθ and y = a cosθ dx 6.Evaluate lim x →2 sin 2 3x x → 0 tan 2 4 x x3 − 8 x 4 − 16 3. Evaluate ∫ 0 4 1 dx x 12. –2) 7. Evaluate ∫ sin x dx 10 Evaluate ∫ cos ec (1 − 4 x)dx 11. & 4 questions in D ii) Each question carries 2 marks in section A iii) Each question carries 5 marks in remaining section SECTION – A 1. Find 5. 7 questions in section B 5 questions in section C. Evaluate sin x dx ∫ 0 π 2 13. If s = 4t2 – 4t + 6 then find velocity when t = 2second x x 8. .r. 9. 7. Evaluate ∫ x. Evaluate lim x→0 sin θ θ →0 θ 3sin 2 x − 5 x dy x + 1 = e 4 x − tan x dx 3. 5. Show that sin x dx cos 2 x d (a x ) = ax dx log e a SECTION – B 1. Find dy/dx if x = a(cosθ + θ ) and y = a ( 1 – sinθ ) 8. If xcoy + ysin(x/y) = k find dy/dx. If y = 2 + 3sinh x . Evaluate 3. Evaluate ∫ 15. Find the derivative of tanx w. Find the maximum and minimum value of the fuction 2x3 . Prove geometrically lim 2. Evaluate cos3 x dx ∫ 0 6. Evaluate ∫ sin x dx 2.log x dx x4 7. find dy/dx 3 + 2 cosh x 6.12 x2 + 18x + 5.3 ) 10. Evaluate ∫ 2 dx x +1 Applied Mathematics – II Page 10 of 11 . SECTION – C 3 1. Find the equations of tangent and normal to the curve y2 = 9x at ( 1. find dy/dx. find dy/dx.to x form the first principle 4. Evaluate 4. Evaluate ∫x ∫x ∫ π 2 2 dx − 6 x + 13 2 sin x dx 2x + 3 dx 8 − 2x − x2 5. If y = ex ( sinx – cosx ).14. If y = sinx cosx. ( 3y – 7x + 7 ) dx + ( 7y – 3x + 3 ) dy = 0 5.8. ( 2x + y + 1 ) dx + ( x + 2y + 1 ) dy = 0 dy x 6. find the area bounded by the curve y = 4x – x2 – 3 and x-axis. SECTION – D Solve the following equations. (y3 – 3x2y ) dx – (x3 – 3xy2 ) dy = 0 4.sec2x tany dx + sec2y tanx dy= 0 2. y (1 + x) dx + x (1 + y) dy = 0 3. ey + 1 = e dx 0-o-0-o-0-o-0-o-0-o-0-o-0-o-0-o-0-o-0-o-0 Applied Mathematics – II Page 11 of 11 . 1.