### APJ ABDUL KALAM TECHNOLOGICAL UNIVERSITY FIRST SEMESTER B.TECH DEGREE EXAMINATION, DECEMBER 2015

### MA101 CALCULUS

Max. Marks: 100 Duration: 3 Hours

PART A

Answer ALL Questions

1. Find the derivative of tanh 2

2. Examine the convergence of the series ∑∞ 3 !

3. Convert the rectangular coordinates (0,4,√3) to cylindrical and spherical coordinates 2

4. Find equations of the paraboloid = + in cylindrical and spherical coordinates 3

5. If U= , . Find +

2

6. The length, width, and height of a rectangular box are measured with an error of at most 5%. Use a total differential to estimate the maximum percentage error that results if these quantities are used to calculate the diagonal of the box. 3

7. Find z,if , z 4x 10y. 2

8. A particle moves on the curves x=2 , y= -4t,z=3t-5 where t is the time .Find the component of acceleration at the time t=1 in the direction ̂-3 ̂+2 . 3

9. Evaluate∫∫ 2

10. Find the Jacobian of the transformations x=uv and y= 3

11. Find curl F⃗ at the point (1,-1,1) where F⃗ =xz3 ̂−2x2yz +2̂yz4 2 →

12. The function ɸ(x,y,z) = xy+yz+xz is a potential for the vector field , find the vector →

field . 3

PART B

MODULE 1

Answer ANY TWO Questions

13. Find the Maclaurin series for cosx and also find cos 1,calculate the absolute error 5

14. Prove that tanh 1 x 1 ln 1 x,-1< x< 1 5 2 1 x

15. Show the series ∑∞ ( ) converges and∑∞ (−1) diverges 5

MODULE 2

Answer ANY TWO Questions

16. Find the natural domain of the following functions.

i. (,)= 3 −1

ii. (,)=log (−) 5

x

2 y

17. Evaluate (x,y)Lt ( 1,2) x2 y 2 . State the properties used in the evaluation.

5

18. Find the traces of the surface x2 y 2 z 2 0 in the planes x=2 and y=1 and

identify the same. 5

MODULE 3

Answer ANY TWO Questions

19. Find maximum and minimum values of

f(x, y)=x3+3xy2

-15x2

-15y2+72x 5

20. Let L(x, y,z) denote the local linear approximation to f (x, y,z) x y

at the point

y z

P(-1,1,1). Compare the error in approximating f by L at Q(-0.99,0.99, 0.01)with

the distance between P and Q . 5

3

21. z 3xy2z3 ; y 3x2 2 ;z x 1 Find dw and dw

dx dy 5

MODOULE 4

Answer ANY TWO Questions

22. Given a circular helix r(t)=acost ̂+ ̂+ ,a, b>0 ,0 t ,find its arc length and unit

tangent vector. 5

23. The position vector at any time t of a particle moving along a curve is r⃗(t)=t ̂+

+̂.

Find the scalar and vector tangential and normal component of the acceleration at

time t=1 5

24. Find the parametric equation of the tangent line to the curve of intersection of the

paraboloid = + and the ellipsoid 3 +2 + =9 (1,1,2) 5

MODULE 5

Answer ANY THREE Questions

25. Evaluate∫∫(+ ) over the region in the positive quadrant

for which x+ y ≤ 1 5

26. Change the order of integration in∫∫ and hence evaluate the same. 5

27. Find the area bounded by the Parabolas y2=4xand x2=-(y/2). 5

28. Find the volume bounded by the cylinder x2+y2=4 the planes y+ z=3 and z=0

5

29. Evaluate∫

∞

∫

∞ ( )si (n) by means of the transformation u = x+y, v=y 5

MODULE 6

Answer ANY THREE Questions

4

30. Use Green’s theorem to evaluate ∮( − ) where C is the square with vertices

(0, 0), (π, 0), (π, π) and (0, π) 5

31. Use Stoke's theorem to evaluate the integral ∮⃗. , where ⃗= ̂+ ̂+ ;

C is the triangle in the plane x+y+z = 1 with vertices (1,0,0) ,(0,1,0) and (0,0,1)with a

counter clockwise orientation looking from the first octant towards the origin. 5

32. Use Gauss Divergence Theorem to find the outward flux of vector field

F x y z( , , ) x i3y j3 z k3 across the surface of the region enclosed by circular

cylinder x2 y

2 9 and the plane z 0 and z 2 5

33. Use Gauss Divergence Theorem to find the outward flux of vector field

F x y z( , , ) x i3y j3 z k3 across the surface of the region enclosed by circular

cylinder x2 y

2 9 and the plane z 0 and z 2 5

34. Find the work done by the force field ⃗(,)=(− )̂+(cos+ ) ̂on a

particle that travels once around the unit circle + =1 in the counter clockwise

direction. 5

PART A

Answer ALL Questions

1. Find the derivative of tanh 2

2. Examine the convergence of the series ∑∞ 3 !

3. Convert the rectangular coordinates (0,4,√3) to cylindrical and spherical coordinates 2

4. Find equations of the paraboloid = + in cylindrical and spherical coordinates 3

5. If U= , . Find +

2

6. The length, width, and height of a rectangular box are measured with an error of at most 5%. Use a total differential to estimate the maximum percentage error that results if these quantities are used to calculate the diagonal of the box. 3

7. Find z,if , z 4x 10y. 2

8. A particle moves on the curves x=2 , y= -4t,z=3t-5 where t is the time .Find the component of acceleration at the time t=1 in the direction ̂-3 ̂+2 . 3

9. Evaluate∫∫ 2

10. Find the Jacobian of the transformations x=uv and y= 3

11. Find curl F⃗ at the point (1,-1,1) where F⃗ =xz3 ̂−2x2yz +2̂yz4 2 →

12. The function ɸ(x,y,z) = xy+yz+xz is a potential for the vector field , find the vector →

field . 3

PART B

MODULE 1

Answer ANY TWO Questions

13. Find the Maclaurin series for cosx and also find cos 1,calculate the absolute error 5

14. Prove that tanh 1 x 1 ln 1 x,-1< x< 1 5 2 1 x

15. Show the series ∑∞ ( ) converges and∑∞ (−1) diverges 5

MODULE 2

Answer ANY TWO Questions

16. Find the natural domain of the following functions.

i. (,)= 3 −1

ii. (,)=log (−) 5

x

2 y

17. Evaluate (x,y)Lt ( 1,2) x2 y 2 . State the properties used in the evaluation.

5

18. Find the traces of the surface x2 y 2 z 2 0 in the planes x=2 and y=1 and

identify the same. 5

MODULE 3

Answer ANY TWO Questions

19. Find maximum and minimum values of

f(x, y)=x3+3xy2

-15x2

-15y2+72x 5

20. Let L(x, y,z) denote the local linear approximation to f (x, y,z) x y

at the point

y z

P(-1,1,1). Compare the error in approximating f by L at Q(-0.99,0.99, 0.01)with

the distance between P and Q . 5

3

21. z 3xy2z3 ; y 3x2 2 ;z x 1 Find dw and dw

dx dy 5

MODOULE 4

Answer ANY TWO Questions

22. Given a circular helix r(t)=acost ̂+ ̂+ ,a, b>0 ,0 t ,find its arc length and unit

tangent vector. 5

23. The position vector at any time t of a particle moving along a curve is r⃗(t)=t ̂+

+̂.

Find the scalar and vector tangential and normal component of the acceleration at

time t=1 5

24. Find the parametric equation of the tangent line to the curve of intersection of the

paraboloid = + and the ellipsoid 3 +2 + =9 (1,1,2) 5

MODULE 5

Answer ANY THREE Questions

25. Evaluate∫∫(+ ) over the region in the positive quadrant

for which x+ y ≤ 1 5

26. Change the order of integration in∫∫ and hence evaluate the same. 5

27. Find the area bounded by the Parabolas y2=4xand x2=-(y/2). 5

28. Find the volume bounded by the cylinder x2+y2=4 the planes y+ z=3 and z=0

5

29. Evaluate∫

∞

∫

∞ ( )si (n) by means of the transformation u = x+y, v=y 5

MODULE 6

Answer ANY THREE Questions

4

30. Use Green’s theorem to evaluate ∮( − ) where C is the square with vertices

(0, 0), (π, 0), (π, π) and (0, π) 5

31. Use Stoke's theorem to evaluate the integral ∮⃗. , where ⃗= ̂+ ̂+ ;

C is the triangle in the plane x+y+z = 1 with vertices (1,0,0) ,(0,1,0) and (0,0,1)with a

counter clockwise orientation looking from the first octant towards the origin. 5

32. Use Gauss Divergence Theorem to find the outward flux of vector field

F x y z( , , ) x i3y j3 z k3 across the surface of the region enclosed by circular

cylinder x2 y

2 9 and the plane z 0 and z 2 5

33. Use Gauss Divergence Theorem to find the outward flux of vector field

F x y z( , , ) x i3y j3 z k3 across the surface of the region enclosed by circular

cylinder x2 y

2 9 and the plane z 0 and z 2 5

34. Find the work done by the force field ⃗(,)=(− )̂+(cos+ ) ̂on a

particle that travels once around the unit circle + =1 in the counter clockwise

direction. 5

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