Diploma Engineering Tutorial (Applied Mathematics) PDF
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Directorate of Technical Education, Gandhinagar - Gujarat
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This document appears to be a tutorial for applied mathematics. The document includes questions and outlines topics like matrices, differentiation and integration. It's structured as a tutorial, suitable for undergraduate students, within an engineering program.
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Diploma Engineering Tutorial (Applied Mathematics) (4320001)[Semester 2 –Branche_________________] Enrolment No Name Branch Academic Term Institute Directorate Of Technical Education Gandhinagar - Gujarat Applied Mathematics (4320001) DTE’s Vis...
Diploma Engineering Tutorial (Applied Mathematics) (4320001)[Semester 2 –Branche_________________] Enrolment No Name Branch Academic Term Institute Directorate Of Technical Education Gandhinagar - Gujarat Applied Mathematics (4320001) DTE’s Vision: To provide globally competitive technical education; Remove geographical imbalances and inconsistencies; Develop student friendly resources with a special focus on girls’ education and support to weaker sections; Develop programs relevant to industry and create a vibrant pool of technical professionals. DTE’s Mission: Institute’s Vision:(Student should write) Institute’s Mission:(Student should write) Department’s Vision:(Student should write) Department’s Mission:(Student should write) i Applied Mathematics (4320001) Programme Outcomes (POs) : 1. Basic and Discipline specific knowledge: Apply knowledge of basic mathematics, science and engineering fundamentals and engineering specialization to solve the engineering problems. 2. Problem analysis: Identify and analyse well-defined engineering problems using codified standard methods. 3. Design/ development of solutions: Design solutions for engineering well-defined technical problems and assist with the design of systems components or processes to meet specified needs. 4. Engineering Tools, Experimentation and Testing: Apply modern engineering tools and appropriate technique to conduct standard tests and measurements. 5. Engineering practices for society, sustainability and environment: Apply appropriate technology in context of society, sustainability, environment and ethical practices. 6. Project Management: Use engineering management principles individually, as a team member or a leader to manage projects and effectively communicate about well-defined engineering activities. 7. Life-long learning: Ability to analyze individual needs and engage in updating in the context of technological changes in field of engineering. Course Outcomes (COs): 1. Demonstrate the ability to Crack engineering related problems based on Matrices. 2. Demonstrate the ability to solve engineering related problems based on applications of differentiation. 3. Demonstrate the ability to solve engineering related problems based on applications of integration. 4. Develop the ability to apply differential equations to significant applied problems. 5. Solve applied problems using the concept of mean. ii Applied Mathematics (4320001) Index Enrolment No: Name: Term: Sr no Practical Outcome/Title of experiment Page CO no Date Sign 1 Solve simple problems using the concept of algebraic 1 operations of matrices. 2 Use the concept of adjoint of a matrix to find the inverse 10 of a matrix. Solve system of linear equations using matrices. Use 3 suitable software to demonstrate the geometric meaning 18 of solution of system of linear equations. 4 Solve examples related to 1st rule of derivative, working 23 rules. 5 Solve examples of derivative related to Chain Rule, Implicit 34 functions. 6 Solve the examples derivative of Parametric functions and 45 second order derivative of simple functions. Use concept of derivative to solve the problems related to 7 velocity, acceleration and Maxima-Minima of given simple 56 functions. Use suitable graphical software to visualize the concept of maxima-minima of function. 8 Solve examples of integration using working rules, 64 standard forms of integration and method of substitution. Use the concept of integration by parts to solve related 9 problems. Solve problems related to definite integral 75 using properties. 10 Apply the concept of definite integration to find area and 86 volume. 11 Solve problems of the order, degree of differential 96 equations and Variable Separable method. Apply the concept of linear differential equations to solve 12 given differential equation. Explain the various 104 applications of differential equations in engineering and real life. 13 Solve examples of Mean for the given data. 112 14 Solve examples of Mean deviation and Standard deviation 120 for the given data. Applied Mathematics (4320001) Date: …………… Tutorial No.1 (Unit No. 1: Matrices) Solve simple problems using the concept of algebraic operations of matrices. COURSE OUTCOME Demonstrate the ability to Crack engineering related problems based on Matrices. List of main formulas/working rules: 1 Order of a matrix with 𝑛 rows and 𝑚 columns is 𝒏 × 𝒎. The transpose of matrix 𝐴 = [𝑎𝑖𝑗 ]𝑛×𝑚 (i.e. matrix A of order 𝑛 × 𝑚 with elements 𝑎𝑖𝑗 where, 𝑖 = 1,2,3, … , 𝑛 and 𝑗 = 1,2,3, … , 𝑚.) is 𝐴𝑇 = [𝑎𝑗𝑖 ]𝑚×𝑛. 2 1 2 1 3 5] e.g. Transpose of 𝐴 = [ is 𝐴𝑇 = [3 0 ] 2 0 −1 2×3 5 −1 3×2 For matrices 𝐴 = [𝑎𝑖𝑗 ]𝑛×𝑚 and = [𝑏𝑖𝑗 ]𝑛×𝑚 , 𝐴 ± 𝐵 = [𝑎𝑖𝑗 ± 𝐵𝑖𝑗 ]𝑛×𝑚 (Note that, addition 3 and subtraction between matrices is possible if and only if they have same orders.) For any two matrices 𝐴 and 𝐵, 𝐴 × 𝐵 is possible if and only if, 4 𝒏𝒐. 𝒐𝒇 𝒄𝒐𝒍𝒖𝒎𝒏𝒔 𝒊𝒏 𝑨 = 𝑵𝒐. 𝒐𝒇 𝒓𝒐𝒘𝒔 𝒊𝒏 𝒃. Also, the order of resulting matrix 𝐴 × 𝐵 is 𝒏𝒐. 𝒐𝒇 𝒓𝒐𝒘𝒔 𝒊𝒏 𝑨 × 𝒏𝒐. 𝒐𝒇 𝒄𝒐𝒍𝒖𝒎𝒏𝒔 𝒊𝒏 𝑩. 5 Number of elements in a matrix 𝐴 = [𝑎𝑖𝑗 ]𝑛×𝑚 is 𝒏𝒎. For matrices 𝐴 and 𝐵, (𝐴𝐵)𝑇 = 𝐵𝑇 𝐴𝑇 6 (𝐴𝑇 )𝑇 = 𝐴 (𝐴 + 𝐵)𝑇 = 𝐴𝑇 + 𝐵𝑇 (𝑘 ∙ 𝐴)𝑇 = 𝑘 ∙ 𝐴𝑇 , 𝑘 ∈ 𝑅 1|P ag e Applied Mathematics (4320001) Q.1 Do as directed (ONE MARK QUESTIONS): 1 2 1 −1 U 1 [ ]+[ ] = _____. 3 4 1 3 (a) [2 1] (b) [−2 1] (c) [2 1] (d) [2 −1] 4 7 4 7 4 0 4 7 3 9 0 U 2 Order of [ ] =_____. 1 −2 0 (a) 2 × 2 (b) 3 × 2 (c) 2 × 3 (d)3 × 3 1 2 3 [ ] × [5 6]=_____. A 5 6 2 1 5 12] 9 8 6 8 −4 −4 (a) [ (b) [ ] (c) [ ] (d)[ ] 10 6 37 36 7 7 3 4 1 U 4 If 𝐴 = [ ] and B = [3 4] then 𝐴 × 𝐵 = ________. 2 (a) [3 4] (b) [2 3] (c) (d)[−5] 6 8 5 7 1 3 4 9 𝑇 U 5 [2 3 0 4 ] = _____. 5 6 8 2 0 −2 5 1 2 5 1 3 4 9 (a) [3 0 6] (b) [3 3 6] (c) [1 2 5] (d)[2 3 0 4] 4 0 8 4 0 8 3 3 6 9 −4 2 9 4 2 5 6 8 2 𝑥−3 2 ] = [5 2] U 6 If [ then x = ____. 4 0 4 0 (a) 𝑥 = 0 (b) 𝑥 = 2 (c) 𝑥 = −8 (d)𝑥 = 8 7 If A is of order 3 × 2 and B is of order 2× 1 then order of matrix AB is _____. U (a) 2 × 2 (b) 2 × 1 (c) 3 × 1 (d)3 × 2 8 If A is a square matrix then all the diagonal elements if 𝐴 − 𝐴𝑇 are ____. R (a) 2 (b) −1 (c) 0 (d)1 9 If A is of order 3 × 4 and B is of order 4 × 2 then no. of elements in AB is___. U (a) 6 (b) 5 (c) 8 (d)1 10 If for a square matrix 𝐴, 𝐴 = −𝐴𝑇 then A is called ________ Matrix. R (a) Symmetric (b) Skew-Symmetric (c) Singular (d) Non-Singular 11 For any matrix 𝐴, (𝐴𝑇 )𝑇 =_____. R (a) 𝐴𝑇 (b) −𝐴𝑇 (c) – 𝐴 (d)𝐴 2|P ag e Applied Mathematics (4320001) 1 2 𝑥−2 2 U 12 If [ ]=[ ], then 𝑥 = ______ and 𝑦 = ______ 0 −3 0 𝑦+1 (a) -1 and -2 (b) 1 and -4 (c) 3 and -4 (d) 3 and -2 0 −4 1 5 U 13 If X + [3 −2]=[−3 2] then X = _____. 4 3 −5 4 1 9 −1 −9 1 1 −1 9 (a) [−6 4] (b) [ 6 −4] (c) [ 0 0] (d)[ 6 4] −9 1 9 −1 −1 1 9 1 14 A matrix 𝐴 is called a _____ matrix if det(𝐴) = 0. R (a) Symmetric (b) Skew-Symmetric (c) Singular (d) Non-Singular 2𝑥 − 3 𝑥 − 5] U 15 If [ is a symmetric matrix then 𝑥 =_____. −3 5 (a) 𝑥 = 0 (b) 𝑥 = 2 (c) 𝑥 = −8 (d) 𝑥 = 8 Q.2 Do as directed (3 0R 4 MARKS QUESTIONS): 1 3 2 1 −1 −2 0 3 0 5 U Let A = [0 1 0] , B = [ 1 1 −1] and C = [6 9 −1] , find 2𝐴 − 4𝐵 + 𝐶. 7 8 9 2 2 2 7 8 −2 3|P ag e Applied Mathematics (4320001) 2 −2 3 8 15 −6 2 A If 𝑀 = [ 5 −7 9] and 𝑁 = [11 4 7] then prove that (𝑀 + 𝑁)𝑇 = 𝑀𝑇 + 𝑁 𝑇. 1 −4 6 13 5 6 3 1 3 −2 3 A A=[ ] and B = [ ] then prove that, 𝐴𝑇 𝐵𝑇 = (𝐵𝐴)𝑇. 2 4 1 1 4|P ag e Applied Mathematics (4320001) 4 1 −1 A If A = [ ] then find 𝐴 + 𝐴𝑇 + 𝐼. 2 3 5 x+3 −6 2 0 −6 2 A If [y + 1 2 0] = [−3 2 0]then find the values of x , y & z. z−3 −21 0 4x −21 0 5|P ag e Applied Mathematics (4320001) 6 −1 2 A If A = [ ] then find A2 + I. −2 4 7 1 2 1 A 1 −1 1 Let A = [ ] and B = [4 2 1] , find AB. 3 2 1 1 7 5 6|P ag e Applied Mathematics (4320001) 8 1 2 0 1 2 3 1 2 3 A If A = [ 1 1 0] and B =[1 1 −1] and C =[1 1 −1] then , prove that −1 4 0 2 2 2 1 1 1 AB = AC. 9 1 −1 2 0 1 A For 𝐴 = [ ] and 𝐵 = [ ] find 𝐴𝐵 and 𝐵𝐴 whichever is possible. 2 3 5 4 2 7|P ag e Applied Mathematics (4320001) 10 1 −1 3 1 A If 𝐴 + 𝐵 = [ ] and 𝐴 − 𝐵 = [ ] then find 𝐴𝐵. 3 0 1 4 8|P ag e Applied Mathematics (4320001) Answer Key: Q-1: Answers 1) (a) 2) (c) 3) (b) 4) (a) 5) (b) 6) (d) 7) (c) 8) (c) 9) (a) 10) (b) 11) (d) 12) (c) 13) (a) 14) (c) 15) (b) Q-2: Answers 1) 13 12 7 4) 3 1 [ ] [2 7 3] 1 7 13 16 8 5) 𝑥 = −3, 𝑦 = −4, 6) −2 6 𝐴2 + 𝐼 = [ ] −6 13 𝑧 = −9 [−2 7 7) 5] 9) AB is not possible, 12 17 10 2 3 5] 𝐵𝐴 = [ 8 2 18 10) −2 −2 𝐴𝐵 = [ ] 0 −6 Link of BISAG Lectures YouTube Channel name: DTEGUJ (Link: https://www.youtube.com/@dteguj8385 ) (Directorate of Technical Education Department, Government of Gujarat) Basic concepts of matrices, Addition, Subtraction, Multiplication and Transpose. 1 https://www.youtube.com/watch?v=iULS7nE4v94&t=1724s 2 https://www.youtube.com/watch?v=TAAo3vEo3d8&t=2411s Suggested Activities and website list for aspiring students https://www.mathsisfun.com/algebra/matrix-multiplying.html https://www.mathsisfun.com/algebra/matrix-calculator.html https://www.programiz.com/python-programming/examples/add-matrix 9|P ag e Applied Mathematics (4320001) Date: …………… Tutorial No.2 (Unit No. 1: Matrices) Use the concept of adjoint of a matrix to find the inverse of a matrix. COURSE OUTCOME Demonstrate the ability to Crack engineering related problems based on Matrices. List of main formulas/working rules: Minor: Every element of a square matrix has a unique minor. The minor of element of matrix is the determinant obtained by eliminating the row and the column containing that particular element. For example, 1 𝑎11 𝑎12 In the matrix 𝐴 = [𝑎 𝑎22 ] the minor of 𝑎11 = 𝑎22. 21 𝒂𝟏𝟏 𝑎12 𝑎13 𝑎 𝑎13 In the matrix 𝐵 = 𝒂𝟐𝟏 [ 𝒂𝟐𝟐 𝒂𝟐𝟑 ], the minor of 𝑎21 = | 12 𝑎32 𝑎33 | 𝒂𝟑𝟏 𝑎32 𝑎33 In a similar way we can find the minors for all the elements of the matrix. Sign of Cofactor: The sign of cofactor of each element of a matrix is obtained using below formula. Sign of cofactor of 𝑎𝑖𝑗 = (−1)𝑖+𝑗. For example, 2 Sign of cofactor of 𝑎21 is (−1)2+1 = (−1)3 = −1 which indicates that the sign is negative. Sign of cofactor of 𝑎33 is (−1)3+3 = (−1)6 = 1 which indicates that the sign is positive. Cofactors: The cofactor of 𝑎𝑖𝑗 is denoted by 𝐴𝑖𝑗. 3 Formula: 𝐴𝑖𝑗 = Sign of cofactor of 𝑎𝑖𝑗 × Minor of 𝑎𝑖𝑗. 10 | P a g e Applied Mathematics (4320001) Adjoint of a matrix: The transpose of the cofactor matrix of the square matrix is called the adjoint of the matrix. The adjoint of matrix 𝐴 = [𝑎𝑖𝑗 ]𝑛×𝑛 is denoted by 𝑎𝑑𝑗(𝐴). 4 𝑇 𝑎𝑑𝑗(𝐴) = [𝐴𝑖𝑗 ] Where, 𝐴𝑖𝑗 are the cofactors of the elements 𝑎𝑖𝑗. 𝑛×𝑛 For any square matrix 𝐴, 𝐴 ∙ 𝑎𝑑𝑗(𝐴) = 𝑎𝑑𝑗(𝐴) ∙ 𝐴 = |𝐴| ∙ 𝐼. Where 𝐼 is identity matrix of order same as order of matrix 𝐴. Inverse of a matrix: The inverse of matrix 𝐴 exists if and only if det(𝐴) ≠ 0. 5 The inverse of matrix 𝐴 is denoted by 𝐴−1. 𝑎𝑑𝑗(𝐴) Formula: 𝐴−1 =. (Remember: |𝐴| = det(𝐴). ) |𝐴| Q.1Do as directed (ONE MARK QUESTIONS): 1 For matrix A , If 𝐴−1 exist then 𝐴 × 𝐴−1 = ________. R (a) 𝐼 (b) 0 (c) – 𝐴 (d) −𝐼 −3 2 U 2 The Adjoint of [ ]=________. 0 1 −3 2 1 −2 3 −2 3 0 (a) [ ] (b) [ ] (c) [ ] (d) [ ] 0 1 0 −3 0 1 2 −1 3 For any square matrix 𝐴 of order 2, 𝑎𝑑𝑗(𝑎𝑑𝑗(𝐴))= ________ U (a) 𝐴 (b) 𝐴𝑇 (c) 𝐴−1 (d) 𝑎𝑑𝑗(𝐴) 4 For any square matrix 𝐴, if 𝐴3 + 2𝐴2 − 2𝐴 + 3𝐼 = 0, then 𝐴−1 =________. A 1 1 (a)𝐴2 + 2𝐴 − 2𝐼 (b) 3 (𝐴2 + 2𝐴 − 2𝐼) (c) − 3 (𝐴2 + 2𝐴 − 2𝐼) (d)−(𝐴2 + 2𝐴 − 2𝐼) 2 1 −1 U 5 In the matrix 𝐴 = [3 2 0 ], the cofactor of 3 (i.e. 𝐴21 ) = _____. 4 2 −2 (a) −1 (b) 1 (c) 2 (d) 0 6 For any square matrix 𝐴, (𝐴−1 )−1 = ________. R (a) 𝐴−1 (b) 𝐴 (c) – 𝐴 (d) 𝑎𝑑𝑗(𝐴) 11 | P a g e Applied Mathematics (4320001) 3 1 A 7 The inverse of matrix [ ] = ________. 3 2 2 −1 −3 3 2 1 − 3] −1 1 ] (c) [ 3 (d) [ 1 (a)[ (b) [ ] − 3] 2 −3 3 1 −2 −1 1 3 8 The sign of cofactor of the element 𝑎32 =________. U (a) ± (b) + (c) − (d) Undefined Q.2 Do as directed (3 0R 4 MARKS QUESTIONS): 1 1 −1 A If A = [ ] then show that A. A−1 = I. 2 3 12 | P a g e Applied Mathematics (4320001) 2 3 1 A For A = [ ], prove that 𝐴2 − 5𝐴 + 7𝐼 = 0 then find 𝐴−1. −1 2 3 3 0 1 −1 0 2 A If A=[0 1 −1] and B =[ 3 1 2 ] find (A + B)−1 1 2 5 1 −1 −1 13 | P a g e Applied Mathematics (4320001) 4 3 −1 2 A Find the inverse of [4 1 −1]. 5 0 1 5 −1 1 2 0 A For 𝐴 = [ ] and 𝐵 = [ ], Show that (𝐴𝐵)−1 = 𝐵−1 𝐴−1. 2 3 3 4 14 | P a g e Applied Mathematics (4320001) 6 −4 −3 −3 A If A = [ 1 0 1 ] then prove that Adj A = A. 4 4 3 7 cosx −sinx A If A=[ ] then find matrix B, such that AB = BA = I. sinx cosx 15 | P a g e Applied Mathematics (4320001) 8 1 2 2 A If 3𝐴 = [ 2 1 −2], then prove that 3𝐴−1 = 𝐴𝑇. −2 2 −1 16 | P a g e Applied Mathematics (4320001) Answer Key: Q-1: Answers 1) (a) 2) (b) 3) (a) 4) (c) 5) (d) 6) (b) 7) (c) 8) (c) Q-2: Answers 2) Hint: 3) 7 3 6 − Use 7𝐴−1 = 5𝐼 − 𝐴 11 11 11 10 2 7 2 1 (𝐴 + 𝐵)−1 = − − 11 11 11 1 2 4 𝐴−1 = [7 7] − − 1 3 [ 11 11 11 ] 7 7 4) 1 1 1 7) cos 𝑥 sin 𝑥 − 𝐵=[ ] 2 2 2 − sin 𝑥 cos 𝑥 9 7 11 𝐴−1 = − − 2 2 2 5 5 7 [− 2 − 2 2 ] Link of BISAG Lectures YouTube Channel name: DTEGUJ (Link: https://www.youtube.com/@dteguj8385 ) (Directorate of Technical Education Department, Government of Gujarat) Basic concepts of matrices, Addition, Subtraction, Multiplication and Transpose. 1 https://www.youtube.com/watch?v=I2Jdvo8ZAmE&t=1849s 2 https://www.youtube.com/watch?v=geQY5avNew4&t=196s Suggested Activities and website list for aspiring students https://www.mathsisfun.com/algebra/matrix-inverse.html 17 | P a g e Applied Mathematics (4320001) Date: …………… Tutorial No.3 (Unit No. 1: Matrices) Solve system of linear equations using matrices. Use suitable software to demonstrate the geometric meaning of solution of system of linear equations. COURSE OUTCOME Demonstrate the ability to Crack engineering related problems based on Matrices. List of main formulas/working rules: Geometric interpretation and meaning of solution: A linear equation with two variables represents a line in the 𝑥𝑦-plane. The solution of 1 system of linear equations with two variables gives the point of intersection of the lines present in the system. Solution of system of linear equations: 𝑎1 𝑥 + 𝑏1 𝑦 + 𝑐1 = 0 𝑎2 𝑥 + 𝑏2 𝑦 + 𝑐2 = 0 Step-1: Convert the system of linear equations in to a matrix equation. 𝐴𝑋 = 𝐵 𝑎 𝑏1 𝑥 −𝑐1 Where, 𝐴 = [ 1 ] , 𝑋 = [𝑦] and 𝐵 = [−𝑐 ] 𝑎2 𝑏2 2 2 Step-2: Check whether 𝐴 is a non-singular matrix or not. (i.e. |𝐴| ≠ 0) (Note: The system has unique solution if and only if 𝐴 is a non-singular matrix.) If |𝐴| ≠ 0, then go to step-3. Step-3: Find 𝐴−1. Step-4: 𝐴𝑋 = 𝐵 implies, 𝑋 = 𝐴−1 𝐵. Hence, compute 𝐴−1 𝐵 which will give the solution. 18 | P a g e Applied Mathematics (4320001) Q.1 Do as directed (ONE MARK QUESTIONS): Do the system of linear equation 𝑥 + 𝑦 + 1 = 0, 3𝑥 + 3𝑦 − 7 = 0 has a unique U 1 solution? (a) Yes (b) No If the linear equations in a system with two variables represent parallel lines then the U 2 system has a unique solution (a) False (b) True (c)Uncertain The system of linear equations 4𝑥 − 6𝑦 + 1 = 0, 𝑎𝑥 − 3𝑦 = 0 do not have unique U 3 solution if 𝑎= _____. (a)𝑎 = 0 (b)𝑎 = 1 (c)𝑎 = 2 (d) 𝑎 = −2 4 The solution of the system of linear equation 𝑥 = 0, 2𝑦 = 4 is the point ________. U (a)(2,0) (b)(−2,0) (c) (0,2) (d) (0, −2) The matrix equation representing the system of linear equations, U 5 2𝑥 + 4𝑦 − 3 = 0, 5𝑥 + 2𝑦 + 2 = 0, is _____. 2 4 𝑥 −3 −2 4 𝑥 −3 −2] [𝑥 ] [−2] 2 4 𝑥 3 (a)[ ][ ] = [ ] (b)[ ] [ ] = [ ] (c)[5 = (d) [ ][ ] = [ ] 5 2 𝑦 2 −5 2 𝑦 2 2 −4 𝑦 3 5 2 𝑦 −2 Q.2 Do as directed (3 0R 4 MARKS QUESTIONS): 1 Solve the system of linear equations 3𝑥 + 𝑦 = 9 , 2𝑥 − 3𝑦 = −5 using matrices. A 19 | P a g e Applied Mathematics (4320001) 2 Solve the system of linear equations 2𝑥 + 3𝑦 = 1, 𝑦 − 4𝑥 = 2 using matrices. A 3 2 1 4 3 Solve the system of linear equations 𝑥 + 𝑦 + 2 = 0, 𝑥 − 𝑦 + 6 = 0 using matrices. A 20 | P a g e Applied Mathematics (4320001) 4 Solve the system of linear equations 2𝑥 + 3𝑦 = 6𝑥𝑦 and 𝑥 − 𝑦 = 𝑥𝑦 using matrices. A 5 Solve the system of linear equations 𝑥 + 𝑦 = 0, 𝑥 − 𝑦 = 0 using matrices and show it’s geometrical interpretation using GeoGebra 21 | P a g e Applied Mathematics (4320001) Answer Key: Q-1: Answers 1) (b) 2) (a) 3) (c) 4) (c) 5) (d) Q-2: Answers 1) 𝑥 = 2, 𝑦 = 3 2) 5 4 𝑥=− ,𝑦 = 14 7 3) 7 7 4) 5 5 𝑥=− ,𝑦 = 𝑥 = ,𝑦 = 10 8 4 9 5) 𝒙 = 𝟎, 𝒚 = 𝟎 Link of GeoGebra Graphing Calculator: https://www.geogebra.org/graphing?lang=en Link of BISAG Lectures YouTube Channel name: DTEGUJ (Link: https://www.youtube.com/@dteguj8385 ) (Directorate of Technical Education Department, Government of Gujarat) Basic concepts of matrices, Addition, Subtraction, Multiplication and Transpose. 1 https://www.youtube.com/watch?v=I2Jdvo8ZAmE&t=1849s 2 https://www.youtube.com/watch?v=geQY5avNew4&t=196s Suggested Activities and website list for aspiring students https://www.mathsisfun.com/algebra/systems-linear-equations-matrices.html 22 | P a g e Applied Mathematics (4320001) Date: …………… Tutorial No. 4 (Unit No. 2: Differentiation and its Applications) Solve examples related to 1st rule of derivative, working rules. COURSE OUTCOME Demonstrate the ability to solve engineering related problems based on applications of differentiation List of main formulas/working rules: Definition of Derivative d 1 Derivative of function f ( x) denoted as f ( x ) or f ( x ) and defined as below dx f ( x h) f ( x ) f (t ) f ( x) f ( x) lim or f ( x) lim h 0 h tx tx 2 Some standard formulas: d d n 1) k 0 ,where k is a constant. 2) x n x n 1 dx dx d x d x 3) a a x log e a 4) e ex dx dx d d 5) sin x cos x 6) cos x sin x dx dx d d 7) tan x sec 2 x 8) cot x cos ec 2 x dx dx d d 9) sec x sec x tan x 10) cos ec x cos ec x cot x dx dx d 1 11) log e x dx x 3 Working Rules of Derivative d d d 1) f ( x) g ( x) f ( x) g ( x) dx dx dx d d 2) kf ( x) k f ( x), k is a constant dx dx d d d 3) f ( x) g ( x) f ( x) g ( x) g ( x) f ( x), (Product rule) dx dx dx d d g ( x) f ( x) f ( x) g ( x) 4) d f ( x) dx dx , g ( x) 0 (Quotient rule) g ( x) 2 dx g ( x) 23 | P a g e Applied Mathematics (4320001) Q.1 Do as directed (ONE MARK QUESTIONS): d U 1 (sin 2 x cos 2 x) _____ dx (a) 1 (b) 0 (c) sin x (d) cos x d R 2 (34) _____ dx (a) 1 (b) 33 (c) 0 (d) 35 3 d dx tan x 2 x _____ U (a) sec 2 x (b) sec2 x 2 x log e 2 (c) sin x (d) cos x dy U 4 y 5e x 4 then ____. dx (a) 5e x 4 (b) 5e x (c) e x 4 (d) 5e x 3 5 f ( x) x 2 2 x 11 then f (0) _____. U (a) 1 (b) 11 (c) 2 (d) 0 6 y x then y ____. U 1 (a) (b) x (c) x (d) 1 2 x d R 7 log x _____ dx 1 (a) x (b) (c) sin x (d) cos x x d U 8 log 4 _____ dx 1 (a) 4 (b) (c) 0 (d) log x 4 dy U 9 If y sec2 x tan 2 x than ____. dx (a) 1 (b) 0 (c) tan x (d) sec x 10 f ( x) 3x then f (1) _____ A 1 (a) 3 (b) (c) 3loge 3 (d) loge 3 3 24 | P a g e Applied Mathematics (4320001) Q.2 Do as directed (3 0R 4 MARKS QUESTIONS): 1 Find the derivative of following functions using definition of derivative. A (1) f ( x) x 2 (2) f ( x) e x (3) f ( x) sin x (4) f ( x) log x 25 | P a g e Applied Mathematics (4320001) 2 Find the derivative of following functions using 1st rule of derivative. A (1) f ( x) 2 x3 4 (2) f ( x) 5x (3) f ( x) cos x 3 Find d dx 4 x3 2 x 2 3 A 26 | P a g e Applied Mathematics (4320001) 4 Find d 3 x dx x 3 3 A 5 d 2 x3 3x 2 3 A Find dx x 27 | P a g e Applied Mathematics (4320001) 6 Find d x dx e sin x A 7 Find d 2 dx x log x A 28 | P a g e Applied Mathematics (4320001) 8 Find d dx 2 x3cox A 9 Find d x dx 3 tan x 2 A 29 | P a g e Applied Mathematics (4320001) 10 d x2 1 A Find dx x 2 1 11 d log x A Find dx x 30 | P a g e Applied Mathematics (4320001) 12 1 sin x A For y , find y. 1 sin x 13 tan x A For f ( x) , find f ( x ). x 31 | P a g e Applied Mathematics (4320001) 14 d a b sin x A Find dx a sin x b 15 Find the derivative of f ( x) 2 x sin x x3 cos x A 32 | P a g e Applied Mathematics (4320001) Answer Key: Q-1: Answers 1) (b) 2) (c) 3) (b) 4) (b) 5) (c) 6) (a) 7) (b) 8) (c) 9) (b) 10) (c) Q-2: Answers 1) (1) 2x (2) e x (3) cos x 1 (4) x 2) (1) 6x 2 (2) 5 x log e 5 (3) sin x 3) 12 x 2 4 x 4) 3x 2 3x log e 3 5) 4x 3 2 3 6) e x (sin x cos x) x 7) x 2 x log x 8) 2 x3 sin x 6 x 2 cos x 9) 3x (sec2 x log e 3 tan x) 10) 4x x 2 2 1 11) 1 log x 12) 2cos x x2 1 sin x 2 13) x sec2 x tan x 14) b 2 a 2 cos x x2 a sin x b 2 15 2 x cos x 2sin x x3 sin x 3x 2 cos x Link of BISAG Lectures YouTube Channel name: DTEGUJ (Link: https://www.youtube.com/@dteguj8385 ) (Directorate of Technical Education Department, Government of Gujarat) Differentiation and Its applications 1 https://bit.ly/2IYAMip 2 https://bit.ly/2IYAS9L Suggested Activities and website list for aspiring students GEOMETRICAL MEANING OF DERIVATIVE : https://www.geogebra.org/m/DbZh24kJ https://archive.nptel.ac.in/courses/111/106/111106146/ https://www.whitman.edu/mathematics/calculus_online/chapter03.html 33 | P a g e Applied Mathematics (4320001) Date: …………… Tutorial No. 5 (Unit No. 2: Differentiation and its Applications) Solve examples of derivative related to Chain Rule, Implicit functions. COURSE OUTCOME Demonstrate the ability to solve engineering related problems based on applications of differentiation List of main formulas/working rules: Chain Rule (Differentiation of composite function) Let y gof be a real valued function which is a composite of two functions g and f. y gof x g ( f ( x)) g (u ) 1 dy Suppose u= f(x). g (u ) f ( x) dx dy dy du dx du dx 2 Some standard formulas: d 1 d 1 1) (sin 1 x) 2) (cos1 x) dx 1 x2 dx 1 x2 d 1 d 1 3) (tan 1 x) 4) (cot 1 x) dx 1 x2 dx 1 x2 d 1 d 1 5) (sec1 x) 6) (co sec1 x) dx x x2 1 dx x x2 1 Q.1 Do as directed (ONE MARK QUESTIONS): dy U 1 y e2 x then ____. dx e2 x (a) 1 (b) (c) e 2 x (d) 2e 2 x 2 34 | P a g e Applied Mathematics (4320001) dy A 2 y sin 3 x then ____. dx cos 3 x (a) (b) 3cos3x (c) sin3x (d) cos3x 3 dy A 3 y log(2 x 1) then ____. dx 1 2 1 (a) (b) (c) (d) cos x 2x 1 2x 1 x 1 dy A 4 xy 1 then ____. dx 1 1 (a) (b) (c) 1 (d) 0 x2 x2 d R 5 (cos 1 x) ___ dx 1 1 (a) 1 (b) 0 (c) (d) 1 x2 1 x2 d R 6 (tan 1 x) ___ dx 1 1 1 1 (a) (b) (c) (d) 1 x2 1 x 2 1 x2 1 x2 d U 7 (sin 1 x cos 1 x) ___ dx 1 1 (a) 1 (b) 0 (c) (d) 1 x2 1 x2 d U 8 (tan 1 x cot 1 x) ___ dx 1 1 (a) 1 (b) 0 (c) (d) 1 x2 1 x2 dy A 9 If y e tan x then ____. dx tan x tan x 2 tan x 2 tan x (a) e tan x (b) e tan x (c) e sec x (d) e sec x dy A 10 y log(5 x) then ____. dx 1 5 1 1 (a) (b) (c) (d) 5 x 5x x 35 | P a g e Applied Mathematics (4320001) Q.2 Do as directed (3 0R 4 MARKS QUESTIONS): 1 Find d sin x dx e A 2 Find d dx sin( x 2 ) A 36 | P a g e Applied Mathematics (4320001) 3 Find d dx sin 2 x A 4 Find d dx log x 2 1 A 37 | P a g e Applied Mathematics (4320001) 5 Find d dx log x2 a2 A 6 Find d 4x dx e cos 3 x A 38 | P a g e Applied Mathematics (4320001) 7 d sin(log x) 1 A Prove that 2 cos(log x) sin(log x) dx x x 8 d A Prove that log sec x tan x sec x dx 39 | P a g e Applied Mathematics (4320001) 9 d sin x A Prove that log cos ec x dx 1 cos x 10 dy If x3 y3 3axy then find. dx 40 | P a g e Applied Mathematics (4320001) 11 dy A If x cos y y cos x 3 0 then find. dx 12 dy A If x y a then find. dx 41 | P a g e Applied Mathematics (4320001) 13 dy cos( x y ) A If y sin( x y ) then prove that . dx 1 cos( x y ) 14 dy y cos( xy ) 1 A If x y sin( xy ) then prove that . dx 1 x cos( xy ) 42 | P a g e Applied Mathematics (4320001) 15 dy A If e x e y e x y then find. dx Answer Key: Q-1: Answers 1) (d) 2) (b) 3) (b) 4) (a) 5) (c) 6) (c) 7) (b) 8) (b) 9) (c) 10) (d) Q-2: Answers 1) cos x esin x 2) 2 x cos( x2 ) 3) 2cos x sin x 4) 2x x 1 2 5) x 6) e3 x (3cos 2 x 2sin 2 x) x a2 2 10) dy ay x 2 11) dy y sin x cos y . dx y 2 ax dx cos x x sin y 12) dy y 15) dy . e y x dx x dx 43 | P a g e Applied Mathematics (4320001) Link of BISAG Lectures YouTube Channel name: DTEGUJ (Link: https://www.youtube.com/@dteguj8385 ) (Directorate of Technical Education Department, Government of Gujarat) Differentiation and Its applications 1 https://bit.ly/2x5QlSM 2 https://bit.ly/2QvIULE Suggested Activities and website list for aspiring students https://www.whitman.edu/mathematics/calculus_online/chapter03.html 44 | P a g e Applied Mathematics (4320001) Date: …………… Tutorial No. 6 (Unit No. 2: Differentiation and its Applications) Solve the examples derivative of Parametric functions and second order derivative of simple functions. COURSE OUTCOME Demonstrate the ability to solve engineering related problems based on applications of differentiation List of main formulas/working rules: Differentiation of Parametric functions dx Let parametric equation is given by x f (t ) and y g (t ), t [a, b] then f (t ) and dt 1 dy dy dy dt g (t ) g (t ) therefore , f (t ) 0 dt dx dx f (t ) dt Higher Order Derivatives d2 f Second order derivative of f ( x) is denoted as f ( x) and defined as dx 2 2 d2 f d f ( x) 2 f ( x) and when y f ( x) then second derivative is dented as dx dx 2 d y d 2 y d dy y y2 2 and defined as y y2 2 dx dx dx dx 45 | P a g e Applied Mathematics (4320001) Q.1 Do as directed (ONE MARK QUESTIONS): dy A 1 If x a cos , y a sin , then ___. dx (a) cot (b) cot (c) tan (d) cos dy A 2 If x tan , y sec , then ___. dx (a) cot x (b) sin (c) sec (d) cos dy A 3 If x t 2 3, y 3t 2 5 , then ___. dx (a) 1 (b) 3 (c) 2 (d) 6 d2y U 4 y sin 3 x then ____. dx 2 (a) 9sin 3x (b) 9sin 3x (c) 3sin 3x (d) 3cos3x 5 f ( x) x3 2 x 9 then f (2) _____. A (a) 1 (b) 0 (c) 6 (d) 12 d2 U 6 cos x ___ dx 2 (a) 1 (b) sin x (c) sin x (d) cos x d2 U 7 log x ___ dx 2 1 1 1 1 (a) (b) (c) (d) x x x2 x2 d2 U 8 5sin x ___. dx 2 (a) 5 (b) 5cos x (c) 5sin x (d) 5cos x d2 U 9 (sin 2 x cos 2 x) _____ 2 dx (a) 1 (b) 0 (c) sin x (d) cos x d2 U 10 2023 ___ dx 2 (a) 1 (b) 0 (c) 2021 (d) 2022 46 | P a g e Applied Mathematics (4320001) Q.2 Do as directed (3 0R 4 MARKS QUESTIONS): 1 a dy y A For x at , y , prove that t dx x 2 dy A For x a( sin ), y a(1 cos ) find dx 47 | P a g e Applied Mathematics (4320001) 3 dy A For x a(1 cos ), y b( sin ) find dx 4 dy b A For x a cos2 , y b sin 2 , prove that . dx a 48 | P a g e Applied Mathematics (4320001) 5 dy b A For x a sin 3 t , y b cos3 t , prove that cot t. dx a 6 d A x x Find dx 49 | P a g e Applied Mathematics (4320001) 7 d A sin x x Find dx 8 d sin x A Find x dx 50 | P a g e Applied Mathematics (4320001) 9 d A sin x tan x Find dx 10 d 2 5x A Find 2 (e 11) dx 51 | P a g e Applied Mathematics (4320001) 11 d2y A For x at 2 , y 2at , then find dx 2 12 d 2 y dy A For y e , prove that 2x 2 y 0. dx 2 dx 52 | P a g e Applied Mathematics (4320001) 13 2 d 2 y dy A 2 x For y 2e 3e 3x , prove that 6 y 0. dx 2 dx 14 d2y A For y A cos pt B sin pt , prove that 2 p 2 y 0. dx 53 | P a g e Applied Mathematics (4320001) 15 2 d 2 y dy A For y log(sin x) , prove that 1 0. dx 2 dx Answer Key: Q-1: Answers 1) (a) 2) (b) 3) (b) 4) (a) 5) (d) 6) (d) 7) (d) 8) (c) 9) (b) 10) (b) Q-2: Answers 2) dy sin 3) dy b tan. cot. dx 1 cos 2 dx a 2 x ( x log x) sin x ( x cot x log sin x) 6) x 7) x sin x sin x 8) 9) tan x (1 sec 2 x log sin x) xsin x log x cos x x 10) 25e5 x 11) d2 y 2y dx 2 x 2 54 | P a g e Applied Mathematics (4320001) Link of BISAG Lectures YouTube Channel name: DTEGUJ (Link: https://www.youtube.com/@dteguj8385 ) (Directorate of Technical Education Department, Government of Gujarat) Differentiation and Its applications 1 https://bit.ly/2QvIULE 2 https://bit.ly/2QsO4bk 3 https://bit.ly/2U0S44F Suggested Activities and website list for aspiring students https://math.libretexts.org/Bookshelves/Calculus/Calculus_3e_(Apex)/09%3A_Curves_i n_the_Plane/9.02%3A_Parametric_Equations https://wiki.geogebra.org/en/Curves https://www.whitman.edu/mathematics/calculus_online/chapter03.html https://www.whitman.edu/mathematics/calculus_online/chapter06.html 55 | P a g e Applied Mathematics (4320001) Date: …………… Tutorial No. 7 (Unit No. 2: Differentiation and its Applications) Use concept of derivative to solve the problems related to velocity, acceleration and Maxima-Minima of given simple functions. Use suitable graphical software to visualize the concept of maxima-minima of function. COURSE OUTCOME Demonstrate the ability to solve engineering related problems based on applications of differentiation List of main formulas/working rules: Velocity and Acceleration ds Equation of motion of a moving particle is given by s f (t ) , then velocity v and 1 dt dv d 2 s acceleration a . dt dt 2 Maximum and Minimum values of function. Steps to find maximum and minimum value of function y f ( x) dy d2y 1) Find f ( x ) and f ( x ) dx dx 2 2 dy 2) Solve the equation f ( x) 0. Let x1 , x2 ,..., xn are solution of f ( x) 0. dx 3) Find values of f ( x1 ), f ( x2 ),..., f ( xn ) (4) If f ( xi ) 0 , then y f ( x) has maximum value at x xi and maximum value is f ( xi ). (5) If f ( xi ) 0 , then y f ( x) has minimum value at x xi and minimum value is f ( xi ). 56 | P a g e Applied Mathematics (4320001) Q.1 Do as directed (ONE MARK QUESTIONS): 1 Equation of motion of a moving particle is given by s f (t ) then velocity v ___. R ds d 2s (a) f (t ) (b) 0 (c) (d) dt dt 2 2 Equation of motion of a moving particle is given by s f (t ) then acceleration a U ___. ds d 2s (a) f (t ) (b) 0 (c) (d) 2 dt dt 3 Equation of motion of a moving particle is given by s t 11 , then velocity at t=6 A is __ (a) 1 (b) 11 (c) 6 (d) 5 4 f ( x) has maxima at x a if _____ R (a) (b) (c) (d) f (a) 0, f (a) 0 f (a) 0, f (a) 0 f (a) 0, f (a) 0 f (a) 0, f (a) 0 5 The maximum value of a function f ( x) cos x is ___ U (a) 1 (b) 0 (c) 1 (d) 2 6 The maximum value of a function f ( x) sin x is ___ U (a) 1 (b) 0 (c) 1 (d) 2 7 f ( x) has minima at x a if _____. R (a) (b) (c) (d) f (a) 0, f (a) 0 f (a) 0, f (a) 0 f (a) 0, f (a) 0 f (a) 0, f (a) 0 8 Equation of motion of a moving particle is given by s f (t ) , particle change its A direction at t 3 seconds then velocity at t 3 seconds is ____ (a) 0 (b) 3 (c) 6 (d) 5 Equation of motion of a moving particle is given by s 3t 15 ,then acceleration at A 9 t=4 is __. (a) 1 (b) 0 (c) 4 (d) 15 10 Equation of motion of a moving particle is given by s f (t ) , v is velocity then U acceleration a ___. dv ds (a) f (t ) (b) (c) (d) v dt dt 57 | P a g e Applied Mathematics (4320001) Q.2 Do as directed (3 0R 4 MARKS QUESTIONS): 1 Equation of motion of a moving particle is given by s 2t 3 3t 2 12t 5 , find A velocity at t=1second and acceleration at t=2 second. 2 Equation of motion of a moving particle is given by s t 3 3t 2 4t 3 , find velocity A and acceleration at t=2 second. 58 | P a g e Applied Mathematics (4320001) 3 Equation of motion of a moving particle is given by s t 3 5t 2 3t , when will A particle stop? Find the acceleration of particle at that time. 4 Equation of motion of a moving particle is given by s t 3 3t , t 0 ,when the A velocity and acceleration will be equal? 59 | P a g e Applied Mathematics (4320001) 5 Equation of motion of a moving particle is given by s t 3 6t 2 9t 4 , when will A particle change its direction? Find the s and a of particle at that time. 6 Find maximum and minimum of the function f ( x) 2 x3 15x 2 36 x 10. A 60 | P a g e Applied Mathematics (4320001) 7 Find maximum and minimum of the function f ( x) x3 3x 11. A 8 Find maximum and minimum of the function f ( x) x3 4 x 2 5x 7. A 61 | P a g e Applied Mathematics (4320001) 9 Find maximum and minimum of the function f ( x) 3x3 4 x 2 x 5. A 10 Find maximum and minimum of the function f ( x) x loge x A 62 | P a g e Applied Mathematics (4320001) Answer Key: Q-1: Answers 1) (c) 2) (d) 3) (a) 4) (d) 5) (a) 6) (a) 7) (b) 8) (a) 9) (b) 10) (b) Q-2: Answers 1) 6,18 2) 4,6 3) t 1/ 3 or t 3 and t 1 at 1/3 = 8 or at 3 =8 5) t 1 or t 3 , st 1 =8 , st 3 =4 and 6) Maximum=38, Minimum=37 at 1 = 6 , at 3 =6 7) Maximum=9, Minimum=13 8) Maximum=9, Minimum=239/27 9) Maximum=3, Minimum=1229/243 10) 1 Minimum = e Link of BISAG Lectures YouTube Channel name: DTEGUJ (Link: https://www.youtube.com/@dteguj8385 ) (Directorate of Technical Education Department, Government of Gujarat) Differentiation and Its applications 1 https://bit.ly/2x9G8oe 2 https://bit.ly/3b5aTJV Suggested Activities and website list for aspiring students https://www.whitman.edu/mathematics/calculus_online/chapter06.html 63 | P a g e Applied Mathematics (4320001) Date: …………… Tutorial No. 8 (Unit No. 3: Integration and its Applications) Solve examples of integration using working rules, standard forms of integration and method of substitution. COURSE OUTCOME Demonstrate the ability to solve engineering related problems based on applications of integration. List of main formulas/working rules: 1 Some standard formulas: 𝑥 𝑛+1 1)∫ 1 𝑑𝑥 = 𝑥 + 𝑐 2)∫ 𝑥 𝑛 𝑑𝑥 = +𝑐 𝑛+1 1 3)∫ 𝑥 𝑑𝑥 = 𝑙𝑜𝑔|𝑥| + 𝑐 4)∫ 𝑒 𝑥 𝑑𝑥 = 𝑒 𝑥 + 𝑐 𝑎𝑥 5)∫ 𝑎 𝑥 𝑑𝑥 = 𝑙𝑜𝑔𝑎 + 𝑐 6)∫ 𝑠𝑖𝑛𝑥 𝑑𝑥 = −𝑐𝑜𝑠𝑥 + 𝑐 7)∫ 𝑐𝑜𝑠𝑥 𝑑𝑥 = 𝑠𝑖𝑛𝑥 + 𝑐 8)∫ 𝑠𝑒𝑐 2 𝑥 𝑑𝑥 = 𝑡𝑎𝑛𝑥 + 𝑐 9)∫ 𝑐𝑜𝑠𝑒𝑐 2 𝑥 𝑑𝑥 = −𝑐𝑜𝑡𝑥 + 𝑐 10)∫ 𝑠𝑒𝑐𝑥⦁𝑡𝑎𝑛𝑥 𝑑𝑥 = 𝑠𝑒𝑐𝑥 + 𝑐 𝑓΄(𝑥) 11)∫ 𝑡𝑎𝑛𝑥 𝑑𝑥 = 𝑙𝑜𝑔|𝑠𝑒𝑐𝑥| + 𝑐 12)∫ 𝑑𝑥 = 𝑙𝑜𝑔|𝑓(𝑥 )| + 𝑐 𝑓(𝑥) 13)∫ 𝑐𝑜𝑡𝑥 𝑑𝑥 = 𝑙𝑜𝑔|𝑠𝑖𝑛𝑥| + 𝑐 14)∫ 𝑐𝑜𝑠𝑒𝑐𝑥⦁𝑐𝑜𝑡𝑥 𝑑𝑥 = −𝑐𝑜𝑠𝑒𝑐𝑥 + 𝑐 15)∫ 𝑠𝑒𝑐𝑥 𝑑𝑥 = 𝑙𝑜𝑔|𝑠𝑒𝑐𝑥 + 𝑡𝑎𝑛𝑥| + 𝑐 16)∫ 𝑐𝑜𝑠𝑒𝑐𝑥 𝑑𝑥 = 𝑙𝑜𝑔|𝑐𝑜𝑠𝑒𝑐𝑥 − 𝑐𝑜𝑡𝑥| + 𝑐 𝑒 𝑘𝑥 −𝑐𝑜𝑠𝑘𝑥 17)∫ 𝑒 𝑘𝑥 𝑑𝑥 = +𝑐 18)∫ 𝑠𝑖𝑛𝑘𝑥 𝑑𝑥 = 𝑘 +𝑐 𝑘 1 𝑥 1 19)∫ √𝑎2 𝑑𝑥 = 𝑠𝑖𝑛−1 (𝑎) + 𝑐 20)∫ 𝑑𝑥 = 𝑙𝑜𝑔| 𝑥 + √𝑥 2 ± 𝑎2 | + 𝑐 −𝑥 2 √𝑥 2±𝑎 2 1 1 𝑥 1 1 𝑥−𝑎 21)∫ 𝑥 2 +𝑎2 𝑑𝑥 = 𝑎 ⦁𝑡𝑎𝑛 −1 (𝑎) + 𝑐 22)∫ 𝑥 2 −𝑎2 𝑑𝑥 = 2𝑎 ⦁𝑙𝑜𝑔 |𝑥+𝑎 | + 𝑐 𝑛+1 1 1 𝑥+𝑎 𝑛 (𝑓(𝑥)) 23)∫ 𝑎2 −𝑥 2 𝑑𝑥 = 2𝑎 ⦁𝑙𝑜𝑔 |𝑥−𝑎 | + 𝑐 24)∫(𝑓 (𝑥 )) ⦁𝑓΄(𝑥 ) 𝑑𝑥 = +𝑐 𝑛+1 64 | P a g e Applied Mathematics (4320001) Q.1 Do as directed (ONE MARK QUESTIONS): 1 ∫ 𝑑𝑥 = __________ R (a) 0 (b) 1 (c) 𝑥 + 𝑐 (d) 1 + 𝑐 2 ∫ 𝑒 −5𝑥 𝑑𝑥 = ____________ U 𝑒 −5𝑥 (a) 𝑒 −5𝑥 + 𝑐 (b) −5𝑥 + 𝑐 (c) −5𝑒 −5𝑥 + 𝑐 (d)− +𝑐 5 3 ∫ 5𝑥 4 𝑑𝑥 = ____________ + 𝑐 U (a) 𝑥 4 (b)4𝑥 3 (c) 25𝑥 5 (d) 𝑥 5 4 ∫ 𝑡𝑎𝑛2 𝑥 𝑑𝑥 = ____________ + 𝑐 A (a) 2 𝑡𝑎𝑛𝑥 𝑠𝑒𝑐 2 𝑥 (b)𝑡𝑎𝑛𝑥 + 𝑥 (c) 𝑡𝑎𝑛𝑥 − 𝑥 (d)𝑠𝑒𝑐𝑥 𝑡𝑎𝑛𝑥 5 1 R ∫ 𝑑𝑥 = ___________ 𝑥2 +1 (a) 𝑠𝑖𝑛−1 𝑥 + 𝑐 (b) 𝑡𝑎𝑛−1 𝑥 + 𝑐 (c) 𝑐𝑜𝑡 −1 𝑥 + 𝑐 (d)− 𝑡𝑎𝑛−1 𝑥 + 𝑐 𝑥 6 ∫ 𝑑𝑥 = ____________ + 𝑐 A 𝑥2 +1 1 (a) 𝑙𝑜𝑔 (𝑥 2 + 1) (b)2⦁𝑙𝑜𝑔 (𝑥 2 + 1) (c) 2 ⦁𝑙𝑜𝑔 (𝑥 2 + 1) (d) −2⦁𝑙𝑜𝑔 (𝑥 2 + 1) 1 7 ∫ 𝑑𝑥 = ____________ + 𝑐 U √𝑥 √𝑥 (c) 2√𝑥 (d)𝑙𝑜𝑔 1 (a) 𝑙𝑜𝑔√𝑥 (b) √𝑥 2 8 ∫(𝑠𝑖𝑛−1 𝑥 + 𝑐𝑜𝑠 −1 𝑥 ) 𝑑𝑥 = ____________ + 𝑐 A 𝜋 (a) 2 𝜋 (b) 2 ⦁𝑥 (c) 𝜋⦁𝑥 (d) 𝜋 9 𝑙𝑜𝑔𝑥 A ∫ 𝑑𝑥 = ____________ + 𝑐 𝑥 1 (a) 𝑙𝑜𝑔𝑥 1 (b)2 𝑙𝑜𝑔𝑥 (c) 2 (𝑙𝑜𝑔𝑥 )2 (d) 𝑒 𝑥 10 ∫ 𝑒 − 𝑙𝑜𝑔(𝑠𝑒𝑐𝑥) 𝑑𝑥 = ____________ + 𝑐 A (a) 𝑠𝑒𝑐𝑥 𝑡𝑎𝑛𝑥 (b) 𝑐𝑜𝑠𝑥 (c) 𝑡𝑎𝑛𝑥 (d) 𝑠𝑖𝑛𝑥 65 | P a g e Applied Mathematics (4320001) Q.2 Do as directed (3 0R 4 MARKS QUESTIONS): 2 1 1 A Evaluate: ∫ (√𝑥 + ) 𝑑𝑥 √𝑥 2 𝑥 2 − 5𝑥 − 24 A Evaluate: ∫ dx 𝑥 2 + 3𝑥 66 | P a g e Applied Mathematics (4320001) 3 2 + 3𝑠𝑖𝑛𝑥 A Evaluate: ∫ 𝑑𝑥 𝑐𝑜𝑠 2 𝑥 4 A Evaluate: ∫(𝑐𝑜𝑠𝑒𝑐𝑥 − 𝑐𝑜𝑡𝑥 ) 𝑐𝑜𝑠𝑒𝑐𝑥 𝑑𝑥 67 | P a g e Applied Mathematics (4320001) 5 𝑡𝑎𝑛𝑥 A Evaluate: ∫ 𝑑𝑥 𝑠𝑒𝑐𝑥 + 𝑡𝑎𝑛𝑥 6 12 A Evaluate: ∫ 𝑑𝑥 16 + 9𝑥 2 68 | P a g e Applied Mathematics (4320001) 7 2𝑥 A Evaluate: ∫ 𝑑𝑥 1 + 𝑥4 8 1 A Evaluate: ∫ 𝑑𝑥 𝑥 ⦁ 𝑙𝑜𝑔𝑥 69 | P a g e Applied Mathematics (4320001) 9 𝑠𝑖𝑛√𝑥 A Evaluate: ∫ 𝑑𝑥 √𝑥 10 A Evaluate: ∫ 𝑡𝑎𝑛3 𝑥 ⦁ 𝑠𝑒𝑐 2 𝑥 𝑑𝑥 70 | P a g e Applied Mathematics (4320001) 11 𝑥4 + 𝑥2 + 1 A Evaluate: ∫ 𝑑𝑥 𝑥2 + 1 12 A Evaluate: ∫ 𝑠𝑒𝑐 2 𝑥 ⦁ 𝑐𝑜𝑠𝑒𝑐 2 𝑥 𝑑𝑥 71 | P a g e Applied Mathematics (4320001) 13 𝑐𝑜𝑠2𝑥 A Evaluate: ∫ 𝑑𝑥 𝑠𝑖𝑛2 𝑥⦁ 𝑐𝑜𝑠 2 𝑥 14 A Evaluate: ∫ 𝑐𝑜𝑠5𝑥 ⦁ 𝑠𝑖𝑛3𝑥 𝑑𝑥 72 | P a g e Applied Mathematics (4320001) 15 A Evaluate: ∫ 𝑠𝑖𝑛3𝑥 ⦁ 𝑠𝑖𝑛𝑥 𝑑𝑥 Answer Key: Q-1: Answers 1) (c) 2) (d) 3) (d) 4) (c) 5) (b) 6) (c) 7) (c) 8) (b) 9) (c) 10) (d) 73 | P a g e Applied Mathematics (4320001) Q-2: Answers 1) 𝑥2 2) 𝑥 − 8 log|𝑥| + 𝑐 + 2𝑥 + log|𝑥| + 𝑐 2 3) 2𝑡𝑎𝑛𝑥 + 3𝑠𝑒𝑐𝑥 + 𝑐 4) 𝑐𝑜𝑠𝑒𝑐𝑥 − 𝑐𝑜𝑡𝑥 + 𝑐 5) 𝑠𝑒𝑐𝑥 − 𝑡𝑎𝑛𝑥 + 𝑥 + 𝑐 6) 3𝑥 𝑡𝑎𝑛 −1 ( ) + 𝑐 4 7) 𝑡𝑎𝑛 −1 (𝑥 2 ) + 𝑐 8) 𝑙𝑜𝑔(log|𝑥|) + 𝑐 9) −2𝑐𝑜𝑠√𝑥 + 𝑐 10) 𝑡𝑎𝑛4 𝑥 +𝑐 4 11) 𝑥3 12) 𝑡𝑎𝑛𝑥 − 𝑐𝑜𝑡𝑥 + 𝑐 + 𝑡𝑎𝑛 −1 𝑥 + 𝑐 3 13) – 𝑐𝑜𝑡𝑥 − 𝑡𝑎𝑛𝑥 + 𝑐 14) 𝑐𝑜𝑠2𝑥 𝑐𝑜𝑠8𝑥 − +𝑐 4 16 15) 𝑠𝑖𝑛2𝑥 𝑠𝑖𝑛4𝑥 − +𝑐 4 8 Link of BISAG Lectures YouTube Channel name: DTEGUJ (Link: https://www.youtube.com/@dteguj8385 ) (Directorate of Technical Education Department, Government of Gujarat) Topic Name: Integration & Its application 1 Lecture – 1: https://www.youtube.com/watch?v=TQuO5cqIHKQ 2 Lecture – 2: https://www.youtube.com/watch?v=m8ZZtY8s4Ws 3 Lecture – 3: https://www.youtube.com/watch?v=yKpE8e2q_eU Suggested Activities and website list for aspiring students https://www.mathsisfun.com/calculus/integration-introduction.html https://www.whitman.edu/mathematics/calculus_online/chapter08.html https://www.khanacademy.org https://www.accessengineeringlibrary.com/?implicit-login=true 74 | P a g e Applied Mathematics (4320001) Date: …………… Tutorial No. 9 (Unit No. 3: Integration and its Applications) Use the concept of integration by parts to solve related problems. Solve problems related to definite integral using properties. COURSE OUTCOME Demonstrate the ability to solve engineering related problems based on applications of integration. List of main formulas/working rules: Integration by parts: 1 𝑑 ∫ 𝑢⦁𝑣 𝑑𝑥 = 𝑢 ∫ 𝑣 𝑑𝑥 − ∫ ( (𝑢) ⦁ ∫ 𝑣 𝑑𝑥) 𝑑𝑥 𝑑𝑥 2 Some standard formulas: 𝑏 𝑏 𝑎 1) ∫ 𝑓 (𝑥 ) 𝑑𝑥 = 𝐹 (𝑏) − 𝐹(𝑎) 2)∫𝑎 𝑓 (𝑥 ) 𝑑𝑥 = − ∫𝑏 𝑓(𝑥 ) 𝑑𝑥 𝑎 𝑎 𝑎 𝑏 𝑏 3)∫0 𝑓 (𝑥 ) 𝑑𝑥 = ∫0 𝑓 (𝑎 − 𝑥 ) 𝑑𝑥 4)∫𝑎 𝑓 (𝑥 ) 𝑑𝑥 = ∫𝑎 𝑓(𝑎 + 𝑏 − 𝑥 ) 𝑑𝑥 𝑎 𝑎 𝑎 5)∫−𝑎 𝑓(𝑥 ) 𝑑𝑥 = 0 ; 𝐼𝑓 𝑓 (𝑥 ) 𝑖𝑠 𝑜𝑑𝑑 6)∫−𝑎 𝑓(𝑥 ) 𝑑𝑥 = 2 ∫0 𝑓(𝑥) 𝑑𝑥 ; 𝐼𝑓 𝑓(𝑥 ) 𝑖𝑠 𝑒𝑣𝑒𝑛 Q.1 Do as directed (ONE MARK QUESTIONS): 1 ∫ 𝑥 ⦁ 𝑙𝑜𝑔𝑥 𝑑𝑥 = __________ R (a) 𝑙𝑜𝑔𝑥 ∫ 𝑥 𝑑𝑥 + (b)𝑥 ∫ 𝑙𝑜𝑔𝑥 𝑑𝑥 − 𝑑 𝑑 ∫ ( (𝑥 ) ⦁ ∫ 𝑙𝑜𝑔𝑥 𝑑𝑥) 𝑑𝑥 𝑑𝑥 ∫ ( (𝑙𝑜𝑔𝑥 ) ⦁ ∫ 𝑥 𝑑𝑥) 𝑑𝑥 𝑑𝑥 (c)𝑙𝑜𝑔𝑥 ∫ 𝑥 𝑑𝑥 − (d)𝑥 ∫ 𝑙𝑜𝑔𝑥 𝑑𝑥 + 𝑑 𝑑 ∫ (𝑑𝑥 (𝑙𝑜𝑔𝑥 ) ⦁ ∫ 𝑥 𝑑𝑥) 𝑑𝑥 ∫ (𝑑𝑥 (𝑥 ) ⦁ ∫ 𝑙𝑜𝑔𝑥 𝑑𝑥) 𝑑𝑥 75 | P a g e Applied Mathematics (4320001) 𝑒 1 2 ∫ 𝑑𝑥 = ___________ U 𝑥 1 (a) 1 (b) 0 (c)𝑙𝑜𝑔(𝑒 − 1) (d)𝑒 1 3 ∫ 3𝑥 2 − 2𝑥 + 1 𝑑𝑥 = ___________ U −1 (a) 0 (b) 2 (c)4 (d)6 1 4 ∫ 𝑒 𝑥 𝑑𝑥 = ____________ U 0 (a) 𝑒 (b) 𝑒 − 1 (c)1 − 𝑒 (d)1 𝜋⁄ 6 5 ∫ 3 ⦁ 𝑠𝑖𝑛3𝑥 𝑑𝑥 = ___________ U 0 (a) 3 (b)0 (c)1 (d)9 𝜋⁄ 2 6 ∫ 𝑐𝑜𝑠2𝑥 𝑑𝑥 = ___________ U 0 𝜋 𝜋 (a)𝜋 (b) 2 (c) 4 (d)0 2 7 ∫ 𝑥 − 𝑠𝑖𝑛𝑥 𝑑𝑥 = ___________ U −2 (a)0 (b)2 − 𝑠𝑖𝑛2 (c)1 − 𝑠𝑖𝑛2 (d)2 𝜋⁄ 2 8 ∫ 𝑠𝑖𝑛𝑥 𝑑𝑥 = ___________ U −𝜋⁄2 π (a) 𝜋 (b)0 (c) 4 (d)−𝜋 𝜋⁄ 4 9 ∫ 𝑐𝑜𝑠2𝑥 𝑑𝑥 = ___________ U −𝜋⁄4 (a)0 (b)1 (c)2 (d)4 𝜋 10 ∫ 𝑐𝑜𝑠 2 𝑥 ⦁ 𝑠𝑖𝑛3 𝑥 𝑑𝑥 = ___________ U −𝜋 (a)2𝜋 (b) 1 (c)0 (d)−2𝜋 76 | P a g e Applied Mathematics (4320001) Q.2 Do as directed (3 0R 4 MARKS QUESTIONS): 1 A Evaluate: ∫ 𝑥 ⦁ 𝑒−𝑥 𝑑𝑥 2 A Evaluate: ∫ 𝑥 ⦁ 𝑐𝑜𝑠𝑥 𝑑𝑥 77 | P a g e Applied Mathematics (4320001) 3 A Evaluate: ∫ 𝑙𝑜𝑔𝑥 𝑑𝑥 4 3 A 3 Evaluate: ∫ 4𝑥 − 2𝑥 + 6 𝑑𝑥 2 78 | P a g e Applied Mathematics (4320001) 5 1 A 𝑥 Evaluate: ∫ 𝑑𝑥 1+𝑥 0 6 3 A 2𝑥 Evaluate: ∫ 𝑑𝑥 1 + 𝑥2 1 79 | P a g e Applied Mathematics (4320001) 𝜋 7 A Evaluate: ∫ 𝑡𝑎𝑛7 𝑥 𝑑𝑥 −𝜋 8 1 A 𝑥2 − 1 Evaluate: ∫ 𝑑𝑥 𝑥2 + 1 0 80 | P a g e Applied Mathematics (4320001) 𝜋⁄ 9 4 A Evaluate: ∫ 𝑠𝑒𝑐2 𝑥 ⦁ 𝑒𝑡𝑎𝑛𝑥 𝑑𝑥 0 𝜋⁄ 10 4 A Evaluate: ∫ √1 + 𝑐𝑜𝑠2𝑥 𝑑𝑥 0 81 | P a g e Applied Mathematics (4320001) 11 A Evaluate: ∫ 𝑥 2 ⦁ 𝑙𝑜𝑔𝑥 𝑑𝑥 12 1 𝑥2 A Evaluate: ∫ 𝑑𝑥 1 + 𝑥6 0 82 | P a g e Applied Mathematics (4320001) 𝜋⁄ 13 2 A 𝑠𝑒𝑐𝑥 Evaluate: ∫ 𝑑𝑥 𝑠𝑒𝑐𝑥 + 𝑐𝑜𝑠𝑒𝑐𝑥 0 𝜋⁄ 14 2 A √𝑠𝑖𝑛𝑥 Evaluate: ∫ 𝑑𝑥 √𝑐𝑜𝑠𝑥 + √𝑠𝑖𝑛𝑥 0 83 | P a g e Applied Mathematics (4320001) 15 2 A √2 − 𝑥 Evaluate: ∫ 𝑑𝑥 √2 − 𝑥 + √𝑥 0 Answer Key: Q-1: Answers 1) (𝒄) 2) (𝒂) 3) (𝒄) 4) (𝒃) 5) (𝒄) 6) (𝒅) 7) (𝒂) 8) (𝒃) 9) (𝒃) 10) (𝒄) Q-2: Answers 1) −𝑒 −𝑥 (𝑥 + 1) + 𝑐 2) 𝑥 𝑠𝑖𝑛𝑥 + 𝑐𝑜𝑠𝑥 + 𝑐 3) 𝑥 (𝑙𝑜𝑔𝑥 − 1) + 𝑐 4) 66 5) 1 − 𝑙𝑜𝑔2 6) 𝑙𝑜𝑔5 7) 0 8) 𝜋 1− 2 9) 𝑒−1 10) 1 11) 𝑥3 𝑥3 12) 𝜋 𝑙𝑜𝑔𝑥 − + 𝑐 12 3 9 13) 𝜋 14) 𝜋 4 4 15) 1 84 | P a g e Applied Mathematics (4320001) Link of BISAG Lectures YouTube Channel name: DTEGUJ (Link: https://www.youtube.com/@dteguj8385 ) (Directorate of Technical Education Education Department, Government of Gujarat) Topic Name: Integration & Its application 1 Lecture – 1: https://www.youtube.com/watch?v=0z_MUp_pFJM 2 Lecture – 2:https://www.youtube.com/watch?v=R3pShXzXBMU Suggested Activities and website list for aspiring students https://www.khanacademy.org https://www.geeksforgeeks.org/properties-of-definite-integrals/ 85 | P a g e Applied Mathematics (4320001) Date: …………… Tutorial No. 10 (Unit No. 3: Integration and its Applications) Apply the concept of definite integration to find area and volume. COURSE OUTCOME Demonstrate the ability to solve engineering related problems based on applications of integration. List of main formulas/working rules: 1 Area: Area of the region bounded between 𝑦 = 𝑓(𝑥 ), 𝑋 −axis, 𝑥 = 𝑎 line and𝑥 = 𝑏line is given by 𝒃 𝑨𝒓𝒆𝒂 (𝑨) = ∫ 𝒇(𝒙) 𝒅𝒙 𝒂 2 Volume: Volume of a solid is formed by revolving curve about 𝑋 − axis 𝒃 𝐕𝐨𝐥𝐮𝐦𝐞 ( 𝐕 ) = ∫ 𝝅𝒚𝟐 𝒅𝒙 𝒂 Volume of a solid is formed by revolving curve about 𝑌 −axis 𝒅 𝐕𝐨𝐥𝐮𝐦𝐞 ( 𝐕 ) = ∫ 𝝅𝒙𝟐 𝒅𝒚 𝒄 86 | P a g e Applied Mathematics (4320001) Q.1 Do as directed (3 0R 4 MARKS QUESTIONS): 1 Find area of the region bounded by the lines 𝑦 = 2𝑥, 𝑥 = 5 and 𝑋 −axis. A 2 Find area enclosed by curve 𝑦 = 𝑥 2 , 𝑋 −axis 𝑥 = 1 and 𝑥 = 2 A 87 | P a g e Applied Mathematics (4320001) 3 Find area of the region bounded by the curve 𝑦 = 𝑥 2 , line 𝑥 = 3, 𝑋 −axis 𝑌 −axis. A 4 Find area of the region bounded by the lines 𝑥 = 0, 𝑥 = 𝑎, 𝑦 = 0 and 𝑦 = 𝑏 A 88 | P a g e Applied Mathematics (4320001) 5 Find area enclosed by curve 𝑦 = 3𝑥 2 , line 𝑥 = 5 and 𝑋 −axis. A 6 Find area of the region bounded by the curve 𝑦 = 𝑥 2 , 𝑋 −axis and 𝑥 = 2 A 89 | P a g e Applied Mathematics (4320001) 7 Find area of the region bounded by the curve 𝑦 = 3𝑥 2 , 𝑋 −axis , 𝑥 = 2 & 𝑥 = 3 A 8 Find area of the region bounded by the curve 𝑦 = 𝑥 2 , 𝑋 −axis , 𝑥 = 2 & 𝑥 = 3 A 90 | P a g e Applied Mathematics (4320001) 9 Find volume of a solid obtained by revolving area enclosed by the curve 𝑦 2 = A 2𝑥 and straight line 𝑥 = 3 about 𝑋 −axis 10 Find area of the region bounded by the curve 𝑦 2 = 𝑥 and straight line 𝑥 = 2 A 91 | P a g e Applied Mathematics (4320001) 11 Find area enclosed by the curve 𝑦 = 𝑥 2 and straight line 𝑥 + 𝑦 = 2 A 12 Find area bounded by 𝑋−, 𝑌 −axis and straight line 𝑥 + 𝑦 = 1 A 92 | P a g e Applied Mathematics (4320001) 13 Find the volume of a sphere of radius 1. A 14 Find volume of a solid obtained by revolving area enclosed by 𝑦 2 = 4𝑎𝑥 and A 𝑥 = 𝑎 about 𝑋 −axis 93 | P a g e Applied Mathematics (4320001) 15 Find volume of a solid obtained by revolving area enclosed by straight lines A 𝑦 = 𝑟, 𝑥 = ℎ, 𝑥 = 0 and 𝑦 = 0 about 𝑋 −axis Answer Key: Q-1: Answers 1) 25 2) 7 3 3) 9 4) 𝑎𝑏 5) 125 6) 8 3 7) 19 8) 19 3 9) 9𝜋 10) 8√2 3 11) 8 12) 1 3 2 13) 4𝜋 14) 2𝜋𝑎3 3 15) 𝜋𝑟 2 ℎ 94 | P a g e Applied Mathematics (4320001) Link of BISAG Lectures YouTube Channel name: DTEGUJ (Link: https://www.youtube.com/@dteguj8385 ) (Directorate of Technical Education Education Department, Government of Gujarat) Topic Name: Integration & Its application 1 Lecture – 1: https://www.youtube.com/watch?v=YfQySoZXcPE Suggested Activities and website list for aspiring students https://www.khanacademy.org https://mathhints.com/applications-integration-area-volume/ 95 | P a g e Applied Mathematics (4320001) Date: …………… Tutorial No. 11 (Unit No. 4: Differential Equations) Solve problems of the order, degree of differential equations and Variable Separable Method. COURSE OUTCOME Develop the ability to apply differential equations to significant applied problems. List of main formulas/working rules: 1 Order The order of a differential equation is the order of the highest derivative that appears in the equation. 2 Degree The degree of a differential equation is the degree of the highest derivative that appears in the differential equation when all the derivatives appearing therein are free from radical signs and fractional powers. 3 Variable Separable Form: If a differential equation of first order and first degree can be reduced to the form 𝑓 (𝑥 ) 𝑑𝑥 = 𝑔(𝑦) 𝑑𝑦 then it is called separable equation. The general solution of above equation is given by ∫ 𝑓 (𝑥 ) 𝑑𝑥 = ∫ 𝑔(𝑦) 𝑑𝑦 + 𝑐 ; 𝑐 = Arbiter Constant 96 | P a g e Applied Mathematics (4320001) Q.1 Do as directed (ONE MARK QUESTIONS): 𝑑𝑦 2 1 The order of a differential equation ( ) + 2𝑦 = 𝑥 is ____________ R 𝑑𝑥 (a)0 (b)1 (c)2 (d)4 2 𝑑2 𝑦 𝑑𝑦 2 The degree of a differential equation + ( ) − 𝑦 = 0 is ____________ R 𝑑𝑥 2 𝑑𝑥 (a) 2 (b) 4 (c)1 (d)0 3 𝑑2𝑦 𝑑𝑦 3 R The order of a differential equation = (3 + ) is ___________ 𝑑𝑥 2 𝑑𝑥 (a) 3 (b) 2 (c)1 (d)0 2 4 𝑑4𝑦 𝑑 3 𝑦 𝑑𝑦 3 𝑑𝑦 5 R The order and degree of ( 4 ) + 3 − 𝑥 ( ) is _________respectively. 𝑑𝑥 𝑑𝑥 𝑑𝑥 𝑑𝑥 (a) 2 𝑎𝑛𝑑 4 (b)4 𝑎𝑛𝑑 5 (c)5 𝑎𝑛𝑑 4 (d)4 𝑎𝑛𝑑 2 4 3 5 𝑑3𝑦 𝑑2𝑦 R The order of a differential equation ( 3 ) + ( 2 ) − 2𝑦 = 0 is ____________ 𝑑𝑥 𝑑𝑥 (a) 1 (b) 2 (c)3 (d)4 4 3
