Mathematics I (Calculus) 2070

Attempt all questions.

Group A (10×2=20)

1. Define odd and even function, with example.

2. Show that the series  Converges to -1.

3. Test the convergence of the series 

4. Find the eccentricity of the curve 2x² + y² = 4.

5. Find the angle between the planes 3x − 6y − 2z = 15 and 2x + y − 2z = 5

6. Find the velocity and acceleration of a particle whose position is 

7. Evaluate 

8. Find the Jacobean j(u,v,w) if x=u+v, y=2 u,z=3w.

9. Show that y = x² + 5 is the solution of 

10.Find  and  at (1,2) of f(x, y) = x² + 2xy + 5.

Group B (5×4=20)

11. State Rolles’s theorem and verify it for the functionf(x) = sinx in [0, π].

12. Find the Taylors series and the Taylor polynomials generated by f(x) = e^x at x = 0.

13. Find the length of the cardioids r = 1 + cosθ.

14. Find the gradient vector of f(x,y) at a pointP(x₀, y₀).Find an equation for the tangent to the ellipse x² + 4y² = 4 at point (−2,1).

15. Find the general solution of 

Group C (5×8=40)

16. Find the area of the region bounded by x = 2y². , x = 0 and y = 3.

Or

Investigates the convergence of the integrals 

17. Find the torsion ,normal and curvature for the space curve 

18.Evaluate 

19. Find the local maximum , minimum and saddles point of 6x² − 2x³ + 3y² + 6xy.

OR

Find the greatest and smallest values that the function f(x,y) =xy takes on the ellipse

20. Define the wave equation by the modeling of vibrating string.