Hall Ticket No Question Paper Code: AHS003
INSTITUTE OF AERONAUTICAL ENGINEERING
(Autonomous) Dundigal, Hyderabad - 500 043
B.Tech I Semester End Examinations, May - 2018 Regulations: IARE - R16
COMPUTATIONAL MATHEMATICS AND INTEGRAL CALCULUS
(Common to AE / ME / CE)
Time: 3 hours Max. Marks: 70
Answer ONE Question from each Unit All Questions Carry Equal Marks
All parts of the question must be answered in one place only
UNIT – I
1. a) Find the real root of the equation 𝑥 𝑙𝑜𝑔10𝑥 = 1.2 by Regula-Falsi method upto 3 decimal places.
[7M]
b) Given that sin450 = 0.7071, sin500 = 0.7660, sin550 = 0.8192 𝑎𝑛𝑑 sin600 = 0.8660, find sin520 using Newton interpolation formula. Estimate the error.
[7M]
2. a) Find by Newton-Raphson method correct to 4 places of decimals of the equation 3𝑥 − 𝑐𝑜𝑠𝑥 − 1 = 0.
[7M]
b) Given u1= 22, u2 = 30, u= = 82, u7 = 206 then find u8 by Lagrange’s interpolation formulae.
[7M]
UNIT – II
3. a) Derive the Normal Equations of the second degree parabola y = ax2 + bx + c by the method of least squares.
[7M]
b) Solve 𝑦′= 𝑥 + 𝑦 , given y(1) = 0 to find y(1.1) and y(1.2) by Taylor’s series method. [7M]
4. a) Using the principle of least squares fit an equation of the form 𝑦 = 𝑎𝑒𝑏𝑥 (𝑎 > 0) to the following data
x 1 2 3 4
y 1.65 2.70 4.50 7.35
[7M]
b) Given 𝑦′= 𝑥 + 𝑠𝑖𝑛𝑦, y(0) = 1. Compute y(0.2) and y(0.4) by Euler’s modified method. [7M]
UNIT – III
5. a) Evaluate 0𝑙𝑜𝑔 2 0𝑥 0𝑥 +𝑙𝑜𝑔𝑦 𝑒𝑥 +𝑦 +𝑧𝑑𝑧 𝑑𝑦 𝑑𝑥 [7M]
b) Find the area of the loop of the curve r 1c o s [7M]
MODEL QUESTION PAPER
6. a) By changing the order of integration, evaluate 01 12−𝑥𝑥𝑦 𝑑𝑥 𝑑𝑦. [7M]
b) Find the volume common to the cylinders 𝑥2+ 𝑦2= 𝑎2 and 𝑥2+ 𝑧2= 𝑎2. [7M]
UNIT – IV 7. a)
Prove that 𝑑𝑖𝑣 𝑐𝑢𝑟𝑙 𝑓 = 0 [7M]
b) Evaluate 𝐹 . 𝑛 𝑑𝑠 where 𝐹 = 𝑧 𝑖 + 𝑥 𝑖 − 3𝑦2𝑧 𝑘 and S is the surface x2+ y2= 16 included in the first octant between 𝑧 = 0 and 𝑧 = 5
[7M]
8. a) Find the constants a, b, c so that the vector 𝐴 = 𝑥 + 2𝑦 + 𝑎𝑧 𝑖 + 𝑏𝑥 − 3𝑦 − 𝑧 𝑗 + 4𝑥 + 𝑐𝑦 + 2𝑧)𝑘 is irrotational. Also find ∅ such that 𝐴 = 𝛻∅.
[7M]
b) Verify Green’s theorem for [ (𝑥𝑦 + 𝑦𝑐 2)𝑑𝑥 + 𝑥2𝑑𝑦] where c is bounded by 𝑦 = 𝑥 𝑎𝑛𝑑 𝑦 = 𝑥2.
[7M]
UNIT – V 9. a) Show that 1
2 = 𝜋 [7M]
b) Prove the relation 𝑥 𝐽𝑛′ 𝑥 = 𝑛 𝐽𝑛 𝑥 − 𝑥 𝐽𝑛+1 𝑥 .. [7M]
10. a) Solve in series the equation 𝑑2𝑦
𝑑𝑥2− 𝑥𝑦 = 0 about 𝑥 = 0. [7M]
b) State and Prove Generating function of Bessel’s. [7M]
INSTITUTE OF AERONAUTICAL ENGINEERING
(Autonomous)
COURSE OBJECTIVES (COs):
The course should enable the students to:
I Enrich the knowledge of solving algebraic, transcendental and differential equation by numerical methods.
II Apply multiple integration to evaluate mass, area and volume of the plane.
III Analyze gradient, divergence and curl to evaluate the integration over a vector field.
IV Understand the Bessels equation to solve them under special conditions with the help of series solutions
COURSE LEARNING OUTCOMES (CLOs):
Students, who complete the course, will have demonstrated the asking to do the following:
CAHS003.01 Solve the algebraic and transcendental equations using bisection method, method of false position and Newton-Raphson method.
CAHS003.02 Apply numerical methods to interpolate the functions of values for equal intervals using finite differences.
CAHS003.03 Understand the Newton rapson method to the real-world problem for a finite barrier quantum well.
CAHS003.04 Evaluate the functional value by using lagranges interpolation formula for unequal intervals.
CAHS003.05 Understand the Lagrange’s interpolation in real-world problem for neural network learning.
CAHS003.06 Apply method of least squares to fit linear and non linear curves.
CAHS003.07 Solve differential equation using single step method- Taylor’s series.
CAHS003.08 Solve differential equation using multi step methods- Euler’s, Modified Euler’s and Runge-Kutta methods.
CAHS003.09 Understand the multistep methods in real-world problem for real time Aircraft dynamics.
CAHS003.10 Understand the Runge-kutta method in real-world problem for embedding the sensor signals into the iterative computation
CAHS003.11 Evaluate double integral and triple integrals.
CAHS003.12 Utilize the concept of change order of integration to evaluate double integrals.
CAHS003.13 Determine the area and volume of a given curve.
CAHS003.14 Understand transformation of co-ordinate system from plane to plane.
CAHS003.15 Analyze scalar and vector fields and compute the gradient, divergence and curl.
CAHS003.16 Understand integration of vector function.
CAHS003.17 Evaluate line, surface and volume integral of vectors.
CAHS003.18 Use Vector integral theorems to facilitate vector integration.
CAHS003.19 Analyze the concept of vector calculus in real-world problem for fluid dynamics.
CAHS003.20 Solve the Differential Equations by series solutions.
CAHS003.21 Understand Gamma function to evaluate improper integrals.
CAHS003.22 Analyze Bessel’s function and study its properties.
CAHS003.23 Analyze Bessel’s function as a Solution to Schrödinger equation in a cylindrical function of the second kind.
CAHS003.24 Understand gamma function to finds application in such diverse areas as quantum physics, astrophysics and fluid dynamics.
CAHS003.25 Possess the knowledge and skills for employability and to succeed in national and international level competitive examinations.
MAPPING OF SEMESTER END EXAMINATION (SEE) TO COURSE LEARNING OUTCOMES (CLOs):
SEE Question
No.
Course Learning Outcomes (CLOs)
Blooms Taxonomy
Level
1
a CAHS003.01
Solve the algebraic and transcendental equations using bisection method, method of false position and Newton- Raphson method.
Understand b CAHS003.02 Apply numerical methods to interpolate the functions of
values for equal intervals using finite differences. Remember
2
a CAHS003.01
Solve the algebraic and transcendental equations using bisection method, method of false position and Newton- Raphson method.
Understand b CAHS003.04 Evaluate the functional value by using lagranges
interpolation formula for unequal Understand
3
a CAHS003.06 Apply method of least squares to fit linear and non linear
curves. Understand
b CAHS003.07 Solve differential equation using single step method-
Taylor’s series. Understand
4
a CAHS003.06 Apply method of least squares to fit linear- non linear
curves. Understand
b CAHS003.08 Solve differential equation using multi step methods-
Euler’s, Modified Euler’s and Runge-Kutta methods. Understand 5 a CAHS003.11 Evaluate double integral and triple integrals. Understand b CAHS003.13 Determine the area and volume of a given curve. Understand 6 a CAHS003.12 Utilize the concept of change the order of integration to
evaluate double integrals. Remember
b CAHS003.13 Determine the area and volume of a given curve. Understand 7 a CAHS003.15 Analyze scalar and vector fields and compute the
gradient, divergence and curl. Remember
b CAHS003.17 Evaluate line, surface and volume integral of vectors Understand 8 a CAHS003.15 Understand integration of vector function. Remember
b CAHS003.18 Verify vector integral theorems. Remember
9 a CAHS003.21 Understand Gamma function to evaluate improper
integrals. Understand
b CAHS003.22 Analyze Bessel’s function and study its properties. Understand 10 a CAHS003.20 Solve the Differential Equations by series solutions. Understand b CAHS003.22 Analyze Bessel’s function and study its properties. Remember
