Mathematics I (Calculus) 2066

Attempt all questions.

Group A (10×2=20)

1. Find the length of the curve y = x^3/2 from x=0 to x =4.

2. Find the critical points of the function f(x) = x^3/2 (x-4).

3. Does the following series converge? 

4. Find the polar equation of the circle (x+2)² + y² = 4.

5. Find the area of the parallelogram where vertices are A(0,0), B(7,3), C(9,8) and D(2,5).

6. Evaluate the integral 

7. Evaluate the limit 

8. Find  if ω = x² + y – z + sin t and x + y = t.

9. Solve the partial differential equation p + q = x.

10. Find the general integral of the linear partial differential equation z(xp – yq) = z² – x² .

Group B (5×4=20)

11. State and prove Rolle ’s Theorem.

12. Find the length of the cardioid r = 1 + cosθ.

13. Define unit tangent vector of a differentiable curve. Find the unit tangent vector of the curve r(t) = (cos t + t sin t)i + (sin t – t cos t)j, t > 0.

14. What do you mean by critical point of a function f(x,y) in a region? Find local extreme values of the function f(x,y) = xy – x²– y² – 2x – 2y + 4.

15. Find a particular integral of the equation  = 2y – x²

Group C (5×8=40)

16. Graph the function y = x^4/3– 4x^1/3

17. What do you mean by Taylor’s polynomial of order n? Obtain Taylor’s polynomial and

Taylor’s series generated by the function f(x) =cos x at x =0.

18. Find the volume of the region enclosed by the surface z = x²+ 3y² and z = 8 – x²– y².

19. Obtain the absolute maximum and minimum values of the function. f(x,y) = 2 + 2x + 2y – x²– y² on the triangular plate in the first quadrant bounded by lines x = 0, y = 0, y = 9 – x.

OR

Evaluate the integral 

20. Show that the solution of the wave equation  and deduce the result if the velocity is zero.

OR

Find a particular integral of the equation (D² − D¹) = A cos(lx + my) where A, l, m are constants.