Mathematics I (Calculus) 2067

Attempt all questions.

Group A (10×2=20)

1. Define a relation and a function from a set into another set. Give suitable example.

2. Show that the series  converges by using integral test.

3. Investigate the convergence of the series 

4. Find the foci, vertices, center of the ellipse 

5. Find the equation for the plane through (-3,0,7) perpendicular to 

6. Define cylindrical coordinates (r, v, z). Find an equation for the circular cylinder 4x² + 4y² = 9 in cylindrical coordinates.

7. Calculate  for f(x,y) = 1 – 6x²y, R : 0 ≤ x ≤ 2, -1 ≤ y ≤ 1.

8. Define Jacobian determinant for x = g(u, v, w), y = h(u, v, w), z = k(u, v, w).

9. What do you mean by local extreme points of f(x,y)? Illustrate the concept by graphs.

10. Define partial differential equations of the first index with suitable examples.

Group B (5×4=20)

11. State the mean value theorem for a differentiable function and verify it for the function

f(x) =  on the interval [-1,1].

12. Find the Taylor series and Taylor polynomials generated by the function f(x) = cos x at x = 0.

13. Find the length of the cardioid r = 1 – cosθ.

14. Define the partial derivative of f(x,y) at a point (x₀, y₀) with respect to all variables. Find the derivative of f(x,y) = xe^y = cos(x, y) at the point (2, 0) in the direction of A = 3i – 4j.

15. Find a general solution of the differential equation 

Group C (5×8=40)

16. Find the area of the region in the first quadrant that is bounded above by y = √x and below by the x-axis and the line y = x – 2.

OR

Investigate the convergence of the integrals

17. Calculate the curvature and torsion for the helix r(t) = (a cos t)i + (a sin t)j + btk,a,b ≥ 0, a² + b² ≠ 0.

18. Find the volume of the region D enclosed by the surfaces z = x² + 3y² and z = 8 – x² – y².

19. Find the absolute maximum and minimum values of f(x,y) = 2 + 2x + 2y – x²– y² on the triangular plate in the first quadrant bounded by lines x = 0, y = 0 and x + y =9.

OR

Find the points on the curve xy²= 54 nearest to the origin. How are the Lagrange multipliers defined?

20. Derive D’ Alembert’s solution satisfying the initials conditions of the one-dimensional wave equation.