Mathematics I (Calculus) 2073

Attempt all questions.

Group A (10×2=20)

1. If f(x) = sin x and g(x) = -x/2. Find f(f(x)) and g(f(x)).

2. Define critical point. Find the critical point of f(x) = 2x2. 

3. Evaluate 

4. Find the equation of the parabola with vertex at the origin and directrix at x= 7.

5. Find a vector parallel to the line of intersection of the planes 3x + 6y – 2z = 5.

6. Evaluate 

7. Find  and if f(x,y) = x² + y²

8. Evaluate 

9. Show that y = ax² + b is the solution of xy’’ + y’ = 0.

10.Solve 

Group B (5×4=20)

11. Verify Rolle’s theorem for f(x) = x3, x ∈ [-3,3].

12. Find the Taylor series expansion of the case at ex, at x=0.

13. Find a Cartesian equivalent of the polar equation r cos (θ-π/3) = 3.

14. Evaluate it 

15. Obtain the general solution of 

Group C (5×8=40)

16.Evaluate the integrals and determine whether they converge or diverge 

OR

Find the area bounded on the parabola y = 2 – x2 and the line y = -x.

17. Find the curvature of the helix 

18.Find the volume enclosed between the surfaces z = x² + 3y² and z = 8 – x² – y²

19. Find the extreme values of the function F(x,y) = xy –x² –y² -2x -2y + 4

OR

Find the extreme values of f(x,y) = xy subject to g(x,y) = x² + y² – 10 = 0.

20. Define second order partial differential equation. Define initial boundary value problem. Derive the heat equation or wave equation in one dimension.