Mathematics I (Calculus) 2071

Attempt all questions.

Group A (10×2=20)

1. If f(x) = x + 2 and g(x) = x³ − 3 find g(f(3)).

2. Show that the area under the arch of the curve y = sin x is.

3. Test the convergence of the series 

4. Find the equation of the parabola with vertex at the origin and focus at (0,2).

5. Find the angle between the planes 3x − 6y − 2z = 7 and 2x + y − 2z = 5

6. Evaluate 

7. Find and if f(x, y) = 10 − x² − y².

8. Prove that 

9. Show that 

10.Solve 

Group B (5×4=20)

11. Verify Rolles’s theorem for the function f(x) = x² − 5x + 7 in the interval [2,3].

12. Find the Taylors series expression of f(x) = sin x at x = 0.

13. Obtain the polar equations for circles through the origin centered on x and y axis ,with radius a.

14. Evaluate 

15. Obtain the general solution of 

Group C (5×8=40)

16. State Lagranhes’s mean value theorem and verify the theorem for x = x³ − x² − 5x + 3in [0,4].

Or

Investigates the convergence of the integrals 

17. Define curvature of a curve .Show that the curvature of a (a) straight line on zero and (b) a circle of a radius a is l/a .

18.Find the volume enclosed between the surfaces Z = x² + 3y² and Z = 8 − x² − y²

19. Find the maximum and minimum of the function f(x, y) = x³ + y³ − 12x + 20.

OR

Find the Point on the ellipse x² + 2y² = 1 where f(x, y) = xy has its extreme values.

20. Define second order partial differential equation .What is initial boundary values problem ?Solve :u_t = u_xx = u_tt = u_xx