Mathematics I (Calculus) 2069

Attempt all questions.

Group A (10×2=20)

1. Verify the men value theorem for the function f(x) = √x(x − 1) in the interval [0, 1].

2. Find the length of the curve   for 0 ≤ x ≤ 1.

3. Test the convergence of the series  By comparison test.

4. Obtain the semi-major axis ,semi-minor axis,foci,vertices 

5. Find the angle between the vectors 2i+j+k and -4i+3j+k.

6. Obtain the area of the region R bounded by y=x and y= x² in the first quadratic .

7. Show that the function  Is continuous at every point in the plane except the origin.

8. Using partial derivatives ,find if 2xy + tany − 4y² = 0.

9. Verify that the partial differential equation  is satisfied by .

10.Find the general solution of the equation 

Group B (5×4=20)

11. State and prove mean value theorem for definite integral.

12. Find the area of the region that lies in the plane enclosed by the cardioid r = 2(i + cosθ).

13. What do you mean by principle unit normal vector? Find unit tangent vector and principle unit vector for the circular motion 

14. Define partial derivative of a function f(x,y) with respect to x at the point (x₀y₀).State Euler’s theorem ,verify if it for the function .f(x, y) = x² + 5xy + sinx + 7e^x,

15. Find the particular integral of the equation 

Group C (5×8=40)

16. Graph the function 

17. What is mean by maclaurin series? Obtain the maclaurin series for the function 

18.Evaluate the double integral   by applying the transformation  and integrating over an appropriate region in the uv-plane.

19. Define maximum and minimum of a function at a point .Final the local maximum and local minimum of the function f(x, y) = 2xy − 5x² − 2y² + 4x + 4y − 4.

20. Find the solution of the equation 

Or

Find the particular integral of the equation