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![Transforming product into sum :
sin x. cos y = ½ [sin (x + y) + sin (x – y)] cos x. sin y = ½ [sin (x + y) – sin (x – y)]
cos x. cos y = ½ [cos (x + y) + cos (x – y)] sin x. sin y = ½ [cos (x – y) – cos (x + y)]
Transforming sum into product:
sin 𝐶 + sin 𝐷 = 2 sin
𝐶+𝐷
2
𝑐𝑜𝑠
𝐶−𝐷
2
sin 𝐶 − sin 𝐷 = 2 cos
𝐶+𝐷
2
𝑠𝑖𝑛
𝐶−𝐷
2
cos 𝐶 + cos 𝐷 = 2 cos
𝐶+𝐷
2
𝑐𝑜𝑠
𝐶−𝐷
2
cos 𝐶 − 𝑐𝑜𝑠 𝐷 = −2 sin
𝐶+𝐷
2
𝑠𝑖𝑛
𝐶−𝐷
2](https://image.slidesharecdn.com/mathppt-240313162110-d958277a/85/Math-ppt-pptx-9-320.jpg)
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- 1. ENGINEERING MATHEMATICS-I MAT 1151 Dr.Shobha M E Assistant Prof. Snr Scale Dept of Mathematics, MIT Manipal Joint Secretary, Sports Council MAHE Mob:8050238237 Email: shobha.me@manipal.edu
- 2. First Semester-4 credits SecondSemester -4 credits First Semester Chapters: Matrix Algebra Linear Algebra Differential equations and applications. Numerical Methods
- 3. Differential equations andapplications: • First order differential equations, Basic applications. • Methods of solving first order differential equations • Higher order differential equations: Solution of homogeneous and nonhomogeneous linear equations. • Cauchy and Legendre’s differential equations. • Solution of system of differential equations References: ‘A short course in differential equations’ by Rainville E.D. and Bedient P.E ‘Advanced Engineering Mathematics’ by Kreyzig E ‘Higher Engineering Mathematics’ by B.S.Grewal
- 4. Numerical Methods-II: Solution ofAlgebraic and Transcendental equations: • Bisection method, Method of false position, Iteration method, Newton-Raphson method. • Solution of System of Non-linear equations using Newton-Raphson method. Numerical solution of ordinary differential equations: • Taylor’s series method, Euler’s method, Modified Euler’s method, Runge-Kutta methods. References: ‘Introductory methods of Numerical analysis’ by Sastry S.S ‘Higher Engineering Mathematics’ by B S Grewal
- 5. Numerical Methods-I: Interpolation: • Finitedifferences and divided differences. • Newton-Gregory and Lagrange’s interpolation formulae. • Newton’s divided difference interpolation formula. • Numerical differentiation. • Numerical integration: Trapezoidal rule, Simpson’s one third rule and Simpson’s three eighth rule. References: Introductory methods of Numerical analysis’ by Sastry S.S ‘Higher Engineering Mathematics’ by B S Grewal
- 6. Matrix Algebra: • Matrices:Elementary column and row transformations, Inverse of a matrix by elementary row operations, Echelon form and rank of a matrix. • System of linear equations: Consistency, Solution by Gauss elimination, Gauss Jordon, Gauss Jacobi and Gauss Seidel methods. • Eigen values and Eigen vectors: Elementary properties, Computation of largest eigen value by power method. References: ‘Higher Engineering Mathematics’ by B.S.Grewal ‘Introductory methods of Numerical analysis’ by Sastry S S ‘Advanced Engineering Mathematics’ by Kreyzig E
- 7. LIST OF FORMLAE TRIGONOMERTY: FundamentalIdentities : (i) cos2 + sin2 = 1 (ii) 1 + tan2 = sec2 (iii) cot2 + 1 = cosec2 Addition and Subtraction formulae: sin(𝑥 ± 𝑦) = sin 𝑥. cos 𝑦 ± cos 𝑥. sin 𝑦 c𝑜𝑠 𝑥 ± 𝑦 = 𝑐𝑜𝑠𝑥. 𝑐𝑜𝑠𝑦 ∓ 𝑠𝑖𝑛𝑥. 𝑠𝑖𝑛𝑦 tan (𝑥 ± 𝑦) = 𝑡𝑎𝑛𝑥±𝑡𝑎𝑛𝑦 1∓𝑡𝑎𝑛𝑥.𝑡𝑎𝑛𝑦
- 8. Linear Algebra: • Generalizationof vector concept to higher dimensions, Generalized vector operations, Vector spaces and sub spaces, Linear independence and dependence, Basis. • Gram- Schmidt process of orthogonalization. References: ‘Linear Algebra’ by G. Hadley
- 9. Transforming product intosum : sin x. cos y = ½ [sin (x + y) + sin (x – y)] cos x. sin y = ½ [sin (x + y) – sin (x – y)] cos x. cos y = ½ [cos (x + y) + cos (x – y)] sin x. sin y = ½ [cos (x – y) – cos (x + y)] Transforming sum into product: sin 𝐶 + sin 𝐷 = 2 sin 𝐶+𝐷 2 𝑐𝑜𝑠 𝐶−𝐷 2 sin 𝐶 − sin 𝐷 = 2 cos 𝐶+𝐷 2 𝑠𝑖𝑛 𝐶−𝐷 2 cos 𝐶 + cos 𝐷 = 2 cos 𝐶+𝐷 2 𝑐𝑜𝑠 𝐶−𝐷 2 cos 𝐶 − 𝑐𝑜𝑠 𝐷 = −2 sin 𝐶+𝐷 2 𝑠𝑖𝑛 𝐶−𝐷 2
- 10. Formulae for multipleangles: sin 2𝐴 = 2 sin 𝐴 cos 𝐴 = 2𝑡𝑎𝑛𝐴 1+tan2 𝐴 𝑐𝑜𝑠2𝐴 = cos2 𝐴 − sin2 𝐴 = 1 − 2 sin2 𝐴 = 2 cos2 𝐴 − 1 = 1−tan2 𝐴 1+tan2 𝐴 tan 2𝐴 = 2 tan 𝐴 1−tan2 𝐴 sin 3𝐴 = 3 sin 𝐴 −4sin3 𝐴 sin 3𝐴 = 4 cos3 𝐴 −3𝑐𝑜𝑠 𝐴 𝑡𝑎𝑛3𝐴 = 3𝑡𝑎𝑛𝐴−tan3 𝐴 1−3 tan2 𝐴
- 11. Differential Calculus: • Rulesof differentiation: 1) 𝑑 𝑑𝑥 𝑢𝑣 = 𝑢 𝑑𝑣 𝑑𝑥 + 𝑣 𝑑𝑢 𝑑𝑥 (Product Rule) 2) 𝑑 𝑑𝑥 𝑢 𝑣 = 𝑣 𝑑𝑢 𝑑𝑥 −𝑢 𝑑𝑣 𝑑𝑥 𝑣2 (Quotient Rule) 3) 𝐼𝑓 𝑢 = 𝑓 (𝑧) 𝑎𝑛𝑑 𝑧 = 𝑔 (𝑥), then 𝑑𝑢 𝑑𝑥 = 𝑑𝑢 𝑑𝑧 . 𝑑𝑧 𝑑𝑥 (Chain Rule)
- 12. 𝑑 𝑑𝑥 𝑥𝑛 = 𝑛𝑥𝑛−1 𝑑 𝑑𝑥 𝑒𝑥 =𝑒𝑥 𝑑 𝑑𝑥 𝑎𝑥 = 𝑎𝑥 𝑙𝑜𝑔𝑒𝑎 𝑑 𝑑𝑥 (log𝑒 𝑥) = 1 𝑥 𝑑 𝑑𝑥 (log𝑎 𝑥) = 1 𝑥 𝑙𝑜𝑔𝑎 𝑑 𝑑𝑥 sin 𝑥 = 𝑐𝑜𝑠𝑥 𝑑 𝑑𝑥 tan 𝑥 = sec2 𝑥 𝑑 𝑑𝑥 𝑐𝑜𝑠𝑥 = −𝑠𝑖𝑛𝑥 𝑑 𝑑𝑥 cot 𝑥 = −𝑐𝑜𝑠𝑒𝑐2𝑥 𝑑 𝑑𝑥 s𝑒𝑐 𝑥 = 𝑠𝑒𝑐𝑥 𝑡𝑎𝑛𝑥
- 13. 𝑑 𝑑𝑥 cosec 𝑥 =−𝑐𝑜𝑠𝑒𝑐𝑥 𝑐𝑜𝑡𝑥 𝑑 𝑑𝑥 sin−1 𝑥 = 1 1−𝑥2 𝑑 𝑑𝑥 cos−1 𝑥 = − 1 1−𝑥2 𝑑 𝑑𝑥 tan−1 𝑥 = 1 1+𝑥2 𝑑 𝑑𝑥 cot−1 𝑥 = − 1 1+𝑥2 𝑑 𝑑𝑥 sec−1 𝑥 = 1 𝑥 𝑥2−1 𝑑 𝑑𝑥 cosec−1 𝑥 = − 1 𝑥 𝑥2−1
- 14. Integral Calculus: 𝑥𝑛𝑑𝑥 = 𝑥𝑛+1 𝑛+1 (𝑛≠ −1) +c 1 𝑥 𝑑𝑥 = log𝑒 𝑥 +c 𝑒𝑥𝑑𝑥 = 𝑒𝑥 +c 𝑎𝑥 𝑑𝑥 = 𝑎𝑥 log𝑒 𝑎 +c 𝑠𝑖𝑛𝑥 𝑑𝑥 = −𝑐𝑜𝑠𝑥+c 𝑐𝑜𝑠𝑥 𝑑𝑥 = 𝑠𝑖𝑛𝑥 +c 𝑐𝑜𝑡𝑥 𝑑𝑥 = log |𝑠𝑖𝑛𝑥 |+c 𝑠𝑒𝑐𝑥 𝑑𝑥 = log |𝑠𝑒𝑐𝑥 + 𝑡𝑎𝑛𝑥 | 𝑐𝑜𝑠𝑒𝑐𝑥 𝑑𝑥 = log |𝑐𝑜𝑠𝑒𝑐𝑥 − 𝑐𝑜𝑡𝑥| sec2 𝑥 𝑑𝑥 = 𝑡𝑎𝑛𝑥 cosec2 𝑥 𝑑𝑥 = −𝑐𝑜𝑡𝑥 𝑡𝑎𝑛𝑥 𝑑𝑥 = − log |𝑐𝑜𝑠 𝑥|+c
- 15. 𝑑𝑥 𝑎2+𝑥2 = 1 𝑎 tan−1 𝑥 𝑎 +𝑐 𝑑𝑥 𝑎2−𝑥2 = 1 2𝑎 𝑙𝑜𝑔 | 𝑎+𝑥 𝑎−𝑥 | +c 𝑑𝑥 𝑥2−𝑎2 = 1 2𝑎 𝑙𝑜𝑔 𝑥−𝑎 𝑥+𝑎 + 𝑐 𝑑𝑥 𝑎2−𝑥2 = sin−1 𝑥 𝑎 + 𝑐 𝑎2 − 𝑥2 𝑑𝑥 = 𝑥 𝑎 𝑥2 − 𝑎2 + 𝑎2 2 sin−1 𝑥 𝑎 + 𝑐 𝑒𝑎𝑥𝑠𝑖𝑛𝑏𝑥 𝑑𝑥 = 𝑒𝑎𝑥 𝑎2+𝑏2 (𝑎𝑠𝑖𝑛𝑏𝑥 − 𝑏𝑐𝑜𝑠𝑏𝑥)+c 𝑒𝑎𝑥𝑐𝑜𝑠𝑏𝑥 𝑑𝑥 = 𝑒𝑎𝑥 𝑎2+𝑏2 (𝑎𝑐𝑜𝑠𝑏𝑥 + 𝑏𝑠𝑖𝑛𝑏𝑥)+c