Mathematics I (Calculus) 2072

Attempt all questions.

Group A (10×2=20)

1. If f(x) = (x − 1) + x,then prove that f(x). f(1 − x) = 1

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

3. Evaluate: 

4. Find the equation of the parabola with vertex at the origin and directrix at y=2

5. Find the angle between the planes x − 2y − 2z = 5 and 5x − 2y − z = 0

6. Evaluate 

7. Find  and  if f(x, y) = ye².

8. Find the equation for the tangent plane to the surfaces Z = f(x, y) = g − x² − y² at the point (1,2,3).

9. Show that y = c₁xe^−2x + c₂e^−2x is the solution of y′′ + y′ − 2y = 0.

10.Solve 

Group B (5×4=20)

11. Verify Rolles’s theorem for f(x) = x², x ∈ [−1,1].

12. Find the Taylors series expression of f(x) = cos θ at x = 1.

13. Find the Cartesian equation of the polar equation 

14. Show that the function  is continuous at every point except the origin .

15. Solve 

Group C (5×8=40)

16. Find the area bounded on right by the line y=x-2 on the left by the parabola x=y² and below by the x-axis

Or

What is an improper integral? Evaluate 

17. Define curvature of a curve .find that the curvature of a helix 

18.Find the area enclosed by r² = 2a² cos 2θ

19. Find the extreme values of Z = x³ − y³ − 2xy + 6.

OR

Find the extreme value of function F(x, y) = xy takes on the ellipse 

20. Define initial boundary values problems .Derive the heat equation or wave equation in one dimension .