Course: Mathematics for Computing–II (3403) Semester: Spring, 2012
Level: BS (CS) Total Marks: 100
Pass Marks: 50

ASSIGNMENT No. 1
(Units: 1–3)
Note: All questions carry equal marks.
Q. 1 a) Sketch the region whose area is , and use your sketch to show that .
b) i) Given that , show that .
ii) Use the result in part (i) to make guess at the value of the limit:
iii) Use L’Hopitals’s rule to substantiate your guess.

Q. 2 a) Verify that in L’Hopital’s rule is of no help in finding the limit. Then find the limit by some other method.
b) A ball is dropped from a height of 10m. Each time the ball strikes the ground it bounces vertically to a height that is of the preeceding height. Find the total distance the ball will travel if it is allowed to bounce indefinitely.

Q. 3 a) The Fibonacci sequence is defined by
i) List the first eight terms of the sequence.
ii) Find assuming that it exists.
b) Prove that the series and diverges for
Q. 4 a) Find the radius and interval of convergence of the series
b) i) Use the Maclaurin series for Cos x to find the Machaurin series for
ii) Use the result in part (i) to help find

Q. 5 a) Suppose that the base of solid is elliptical with a major axis of length 9 and a minor axis of length 4. Find the volume of the solid if;
i) the cross sections perpendicular to the major axis are squares.
ii) the cross sections perpendicular to the minor axis are equilateral triangles.
b) Consider the conic whose equation is
;
i) Use the discriminant to identify the conic.
ii) Graph the equation by solving for y in terms of x and graphing both solutions.

Assignment No.2
(Units 4–7) Total Marks: 100

Note: All questions carry equal marks.

Q. 1 a) Locate the relative extreme values for the function f(x, y) = 2xy3 – 3xy2 + 10. Also, what will be the maximum area of a rectangle of perimeter p.
b) Derive the general polar equation of a conic and using it sketch the graph of a particular parabola, ellipse and a hyperbola.

Q. 2 a) Define the arc-length formulae in for parametric and polar curves and using these formulae find the arc-length of one parametric curve and one polar curve.
b) Define scalar triple product of three non-zero vectors and show that this give the volume of a parallelepiped with three vectors as adjacent sides. Also give a particular example of scalar triple product.

Q. 3 a) Define the rectangular, cylindrical and spherical coordinates of a point and write their conversion formulae. Apply these conversion formulae for three different examples.
b) Derive the formulae for the curvature and torsion of a regular curve using these formulae find the curvature and torsion of the circular helix and the ratio of that torsion to curvature.

Q. 4 a) Derive the Frenet-Serret formulae for a unit speed curve and apply them to a particular unit speed curve.
b) The temperature at a point (x, y) on a metal plate in xy-plane is T(x,y) = x3+2y2+x degree. Find the rate at which temperature changes with distance if we start at the point (1, 2) and move (i) to the right and parallel to x-axis and (ii) upward and parallel to the y-axis.

Q. 5 a) State the second partial test and apply this test to two different examples.
b) Define Lagrange multiplier and using it find the dimentions of a rectangle having parameter and maximum area.



3403 Mathematics for Computing–II Credit Hours: 4 (4 + 0)

Recommended Book:
Anton Howard, Calculus, John Wiley & Sons, Inc

Course Outline:
Unit–1: Improper Integrals; L’ Hospital Rule
Improper Integrals (Different Kinds), L’ Hospital Rule, Indeterminate Forms

Unit–2: Infinite Series
Sequences (Convergent and Divergent), Monotone Sequences, Infinite Series, Convergence and Divergence of Infinite Series, convergence Tests (The Comparison Tests, The Integral Test, The Ratio Test, The Root Test), Alternating Series, Absolute and Conditional Convergence, Power Series, Taylor and Maclaurin Series.

Unit–3: Topics in Analytic Geometry
Conic Sections, the Parabola, The Ellipse, The Hyperbola, Rotation of Axes, Second Degree Equations

Unit–4: Polar Coordinates and Parametric Equations
Polar Coordinates, Graphs in Polar Coordinates, Area in Polar Coordinates, Parametric Equations, Tangent Lines and Arc-Length in Polar Coordinates.

Unit–5: Three Dimensional Space; Vectors
Rectangular Coordinates In 3-Space, Spheres, cylindrical Surfaces, Vectors, Norm of A Vector, Dot Product, Projections, Cross Product, Parametric Equations of Lines, Planes In 3-Space, Cylindrical and Spherical Coordinates.

Unit–6: Vector-Valued Functions
Parametric Curves In 3-Space, Parametric Equations in Vector Form, Calculus of Vector-Valued Functions, Unit Tangent and Normal Vectors, Curvature

Unit–7: Partial Derivatives
Functions of Two or More Variables, Limits and Continuity, Partial Derivatives, Differentiability and Chain Rules For Functions of Two Variables, Tangent Planes, Total Differentials For Functions of Two Variables, Directional Derivatives, Gradient For Functions of Two Variables, Maxima and Minima of Functions of Two Variables, Lagrange Multiplier.

Unit–8: Multiple Integral
Double Integrals, Double Integrals Over Rectangular Regions, Double Integrals in Polar Coordinates, Surface Area, Triple Integrals, Volume by Triple Integral, Change of Variables.

Unit–9: Topics in Vector Calculus
Vector Fields, Gradient Fields, Divergence and Curl, the Ñ Operator, The Laplacian Line Integrals, Independence of Path, Conservative Vector Fields, Green’s Theorem, Surface Integrals, The Divergence Theorem, Surface Integrals, The Divergence Theorem, The Stoke’s Theorem.

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