Mathematics I (Calculus) 2065

Attempt all questions.

Group A

1. Verify Rolle’s theorem for the function  on the interval [-3, 3].

2. Obtain the area between two curves y = sec2x and y = sin x from x = 0 to x = π/4.

3. Test the convergence of p – series  for p > 1.

4. Find the eccentricity of the hyperbola 9x² – 16y² = 144.

5. Find a vector perpendicular to the plane of P(1, -1, 0), C(2, 1, -1) and R(-1, 1, 2).

6. Find the area enclosed by the curve r² = 4cos2θ.

7. Obtain the values of  and  at the point (4, -5) if f(x,y) = x²+ 3xy + y -1.

8. Using partial derivatives , find if x² + cos y – y²= 0.

9. Find the partial differential equation of the function (x – a)² + (y – b)² + z²= c².

10. Solve the partial differential equation x²p + q = z² .

Group B

11. State and prove the mean value theorem for a differential function.

12. Find the length of the Asteroid x = cos³t, y = sin³t for 0 ≤ t ≥ 2π.

13. Define a curvature of a curve. Prove that the curvature of a circle of radius a is 1/a.

14. What is meant by direction derivative in the plain? Obtain the derivative of the function f(x,y) = x²+ xy at P(1, 2) in the direction of the unit vector 

15. Find the center of mass of a solid of constant density δ, bounded below by the disk: x² + y² = 4 in the plane z = 0 and above by the paraboid z = 4 – x² – y².

16. Graph the function f(x) = -x³ + 12x + 5 for -3 ≤x ≤ 3.

Group C

17. Define Taylor’s polynomial of order n. Obtain Taylor’s polynomial and Taylor’s series generated by the function f(x) = e^x at x = 0.

18. Obtain the centroid and the region in the first quadrant that is bounded above by the line y = x and below by the parabola y = x².

19. Find the maximum and the minimum values of f(x, y) = 2xy – 2y²– 5x² + 4x – 4. Also find the saddle point if it exists.

OR

Evaluate the integral 

20. What do you mean by d’ Alembert’s solution of the one-dimensional wave equation? Derive it.

OR

Find the particular integral of the equation (D² – D¹)z =2y-x² where