APJ ABDUL KALAM TECHNOLOGICAL UNIVERSITY FIRST SEMESTER B.TECH DEGREE EXAMINATION, FEBRUARY 2017 MA101: CALCULUS PART A

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1 A B1A003 Pages:3 (016 ADMISSIONS) Reg. No:... Name:... APJ ABDUL KALAM TECHNOLOGICAL UNIVERSITY FIRST SEMESTER B.TECH DEGREE EXAMINATION, FEBRUARY 017 MA101: CALCULUS Ma. Marks: 100 Duration: 3 Hours PART A (Answer All Questions and each carries 5 marks) 1. a) Test the convergence of! () b) Test the convergence of. (3). a) Find the slope of the sphere +y +z =1 in the y- direction at (,, ). () b) Find the critical points of the function f(, = y- 3 -y 3. (3) 3. a) Find the velocity at time t= of a particle moving along the curve (t) = e t sint i + e t cost j + t k. () b) Find the directional derivative of f(, = e y -ye at the point P(0,0) in the direction of 5i j. (3) 4. a) Change the order of integration in (, ). (3) b) Find the area of the region enclosed by y= and y =. () 5. a) Find the divergence of the vector field f (,y,z) = y i + y 3 z j+ 3z k. () b) Find the work done by = y i + 3 j on a particle that moves along the curve y = from (0,0) to (0,1). (3) 6. a) Using Green s theorem to evaluate + ( + ) where C is the triangle with vertices (0,0), (1,0) and (1,1). () b) Use Stoke s theorem to evaluate. dr where = (-i + (y-z) j + (z-) k and C is the circle +y = a in the y plane with counter clockwise orientation looking down the positive z- ais. (3) Page 1 of 3

2 A B1A003 Pages:3 (016 ADMISSIONS) PART B MODULE I (Answer Any Two Questions) 7. a) Test the convergence of the following series i) ( )!!! ii) ( ) k 8. Use the alternating series test to show that the series ( 1) ( ) ( ) converge. 9. Find the Taylor s series of f() = sin about the point =. MODULE II (Answer Any Two Questions) 10. Find the local linear approimation L to f(, = ln ( at P(1,) and compare the error in approimating f by L at Q(1.01,.01) with the distance between P and Q. 11. Show that the function f (, = tan -1 (y/) satisfies the Laplace s equation + = Find the relative minima of f(, = 3 -y+y -8y. MODULE III (Answer Any Two Questions) 13. Find the unit tangent vector and unit normal vector to = 4 cos ti + 4 sin tj + t k at t =. 14. Suppose a particle moves through 3- space so that its position vector at time t is = t i+ t j + t 3 k. Find the scalar tangential component of acceleration at the time t= Given that the directional derivative of f(,y,z) at (3,-, 1) in the direction of i j - k is -5 and that (3,,1) = 5. Find (3,,1). MODULE IV (Answer Any Two Questions) 16. Evaluate the integral ddy by reversing the order of integration. 17. Evaluate d dy. 18. Find the volume of the solid in the first octant bounded by the co-ordinate planes and the plane +y+z = 6. Page of 3

3 A B1A003 Pages:3 (016 ADMISSIONS) MODULE V (Answer Any Three Questions) 19. Let = i +yj+ zk and let r = and f be a differentiable function of one variable show that ( )= ( ). 0. Evaluate the line integral [ + ] along y = 3 from (3,3) to (0,0). 1. Show that (, = (cosy + y cos ) i + ( sin sin j is a conservative vector field. Hence find a potential function for it.. Show that the integral (3 e y d + 3 e y dy ) is independent of the path and hence evaluate the integral from (0,0) to (3,). 3. Find the work done by the force field = y i + yz j + z k on a particle that moves along the curve C: (t) = t i + t j + t 3 k where 0 t 1. MODULE VI (Answer Any Three Questions) 4. Use Green s theorem to evaluate the integral ( cos y d y sin d where C is the square with vertices (0,0), (π,0), (π,π) and (0,π). 5. Evaluate the surface integral where σ is the portion of the cone z = + between the planes z = 1 and z = Use divergence theorem to find the outward flu of the vector field = i + 3y j + z k across the unit cube = 0, = 1,y=0, y = 1, z = 0 and z = Use Stoke s theorem to evaluate the integral. where = (- i + (y-z) j + (z-)k and C is the boundary of the portion of the plane +y+z = 1 in the first octant. 8. Use Stoke s theorem to evaluate the integral. where = z i + 3 j + 5y k and C is the boundary of the paraboloid +y +z = 4 for which z 0 and C is positively oriented. *** Page 3 of 3

4 A B1A16S (015 Admission) Pages: Reg.No... Name... APJ ABDUL KALAM TECHNOLOGICAL UNIVERSITY FIRST SEMESTER B.TECH DEGREE (SUPPLEMENTARY) EXAMINATION, FEBRUARY 017 (015 ADMISSION) Ma.Marks : 100 Course Code: MA 101 Course Name: CALCULUS PART A Duration : 3 Hours (Answer all questions. Each question carries 3 marks) n 1 1) Show that the series converges. n 1 ) Classify the surface z ( 1) ( y ) 3 3) Find the Maclaurin series for cos y 4) Evaluate Lt, y ( 1, ) y 5) Convert the cylindrical co-ordinate into rectangular co ordinate of ( 4, / 3 3). 6) Find the slope of the surface z = y in the direction at the point (,3). 3 7) Find the directional derivative of f y y z z a i j k at (1,-,0) in the direction of 8) Find the unit normal to the surface y + z + yz = c at (-1,,3) a b 9) Evaluate 1 1 y ddy 10) Find the area of the region R enclosed by y 1, y, 0, y. PART B (Answer any questions. Each question carries 7 marks) 4 ( 1) 11) Test the absolute convergence of n n n n 1 4 1) Determine the Taylor s series epansion of f() = sin at = π/ ) Test the convergence of Page 1 of

5 A B1A16S (015 Admission) Pages: (Answer any questions. Each question carries 7 marks) 14) Find the equation of the paraboloid coordinates. 15) Find F ( f ( ),g( y ),h( z )) if F(,y,z ) z y in the cylindrical and spherical y e yz 16) By converting into polar coordinate evaluate, f ( ), y Lt,g( ( 0,0 ) y ) y 1, h( z ) z y ln y (Answer any questions. Each question carries 7 marks) 17) Find the local linear approimation L of f(,y,z) = yz at the point P(1,,3). Compare the error in approimating f by L at the point Q(1.001,.00, 3.003) with the distance PQ. 18) Find the relative etrema of f (, 3 y y 8y 19) If f is a differentiable function of three variables and suppose that w w w w f y, y z, z Show that 0 y z (Answer any questions. Each question carries 7 marks) 0) Suppose that a particle moves along a curve in 3-space so that its position vector at time t is r(t) = 4cos πt i +4sin πt j + t k. Find the distance travelled and the displacement of the particle during the time interval 1 t 5 3 1) A particle is moving along the curve, r ( t t) i ( t 4) j where t denotes the time. Find the scalar tangential and normal components of acceleration at t = 1. Also find the vector tangential and normal components of acceleration at t = 0. ) Find the arc length of the parametric curve =5cos t, y = 5sin t, z = t ; 0 t π (Answer any questions. Each question carries 7 marks) 4 3) Evaluate the integral by converting into polar co ordinates ( 0 0 4) Using triple integral to find the volume bounded by the cylinder y ) dyd y 4and the planes z 0 and y z ) Change the order of integration and evaluate *** 0 y ddy Page of

6 A A7001 Reg No.: Total Pages: 3 Name: APJ ABDUL KALAM TECHNOLOGICAL UNIVERSITY FIRST SEMESTER B.TECH DEGREE EXAMINATION, DECEMBER 017 Course Code: MA101 Course Name: CALCULUS Ma. Marks: 100 Duration: 3 Hours PART A Answer all questions, each carries5 marks. Marks 1 a) Test the convergence of the series 1 (). k 1 k 1 b) Find the radius of convergence of n (3). n1 n 3 a) y Find the Slope of the surface z e 5y in the y-direction at the point (4,0). () b) Find the derivative of z 4 1 y with respect to t along the path (3) logt, y t. 3 a) 3 Find the directional derivative of f y yz z at (1,,0) in the direction of () a i j k. b) Find the unit tangent vector and unit normal vector to r( t) 4costi 4sin tj tk at t. (3) 4 a) log 3 log Evaluate e y dyd. () b) Evaluate R y. 0 0 y da,where R is the region bounded by the curves y and 5 (a) Find the divergence and curl of the vector F(, y, z) yz i y j yz k. () (b) Evaluate (3 y ) d y dy along the circular arc C given by (3) C cost, y sin t for 0 t. 6 (a) Use line integral to evaluate the area enclosed by the ellipse y 1. () b a (b) Evaluate ( 3 d 3dy,where C is the circle y 4. (3) 7 C PART B Module 1 Answer any two questions, each carries 5 marks. n n Test the convergence or divergence of the series ( ). n 1 n1 (3) Page 1 of 3

7 A A ( )! Test the absolute convergence of k k ( 1). k1 (3k )! 9 1 Find the Taylor series for at. 1 Module 1I Answer any two questions, each carries 5 marks. 10 Find the local linear approimation L to f (, log( at P(1,) and compare the error in approimating f by L at Q(1.01,.01) with the distance between P and Q. 11 Let w 4 4y z, sin cos, y sin sin, z cos. Find w w w, and Locate all relative etrema and saddle points of f (, 4y y. Module 1II Answer any two questions, each carries 5 marks. 13 Find the equation of the tangent plane and parametric equation for the normal line to the surface y z 5 at the point ( 3,0, 4) A particle is moving along the curve r ( t) ( t t) i ( t 4) j where t denotes the time. Find the scalar tangential and normal components of acceleration at t 1. Also find the vector tangential and normal components of acceleration at t The graphs of r1 ( t) t i tj 3t k and r ( t) ( t 1) i t j (5 t) k are 4 intersect at the point P (1,1,3 ).Find, to the nearest degree, the acute angle between the tangent lines to the graphs of r 1( t) & r ( t) at the point P (1,1,3 ). Module 1V Answer any two questions, each carries5 marks e y dyd. Change the order of integration and evaluate 17 Use triple integral to find the volume bounded by the cylinder y 9 and between the planes z 1 and z Find the area of the region enclosed between the parabola y and the line y. Module V Answer any three questions, each carries5 marks. 19 Determine whether F (, (cos y y cos ) i (sin sin j is a conservative vector field. If so find the potential function for it. 0 (3,3) y e e y Show that the integral ( e log y ) d ( e log ) dy,where and y y (1,1) are positive is independent of the path and find its value. 1 Find the work done by the force field F(, y, z) yi yz j z k on a particle 3 that moves along the curve C : r( t) ti t j t k (0 t 1). 0 4 Page of 3

8 A A7001 Let r i yj zk and r r,let f be a differentiable function of one variable, f ( r) then show that f ( r) r. r 3 z y z Find.( F) and ( F) where F(, y, z) e i 4e j e y k. 4 Module VI Answer any three questions, each carries5 marks. y Use Green s Theorem to evaluate log(1 d dy,where C is the (1 y C ) triangle with vertices (0,0), (,0) and (0,4). 5 Evaluate the surface integral zds,where is the part of the plane y z 1 that lies in the first octant. 6 Using Stoke s Theoremevaluate F. dr where F(, y, z) zi 4 y j yk, C C is the rectangle 0 1,0 y 3in the plane z y. 7 Using Divergence Theorem evaluate. n ds where F(, y, z) 3 i y 3 F 3 j z k, is the surface of the cylindrical solid bounded by y 4, z 0 and z 4. 8 Determine whether the vector fields are free of sources and sinks. If it is not, locate them 3 (i) ( y z) i z j sin y k (ii) y i y j y k **** Page 3 of 3

9 A B1A005 Pages: 3 Reg. No. Name: APJ ABDUL KALAM TECHNOLOGICAL UNIVERSITY FIRST SEMESTER B.TECH DEGREE EXAMINATION, JUNE/JULY 017 Course Code: MA 101 Course Name: CALCULUS (For 015 Admission and 016 Admission) Ma. Marks :100 Duration: 3 hours PART A Answer all questions. Each question carries 5 marks. 1. (a) Find the interval of convergence and radius of convergence of the infinite series! () (b) Determine whether the series converges or not (3). (a) Find the slope of the surface = 3 + in the -direction at the point (4,) () (b) Find the derivative of = + with respect to along the path =, = (3) 3. (a) Find the directional derivative of (, )= at (1,1) in the direction of the vector () (b) If ( )has a constant direction, then prove that = 0 (3) 4. (a) Evaluate ( )( ) ddy () (b) Evaluate ddy where is the triangaular region bounded by the -ais, = and = 1. (3) 5. (a) Show that ( + ) d + dy + 3z dz is independent of the path joining the points A and B. () (b) If = i + yj + zk and r =, then prove that = ( + 1) (3) 6. (a) Using line integral evaluate the area enclosed by the ellipse + = 1 () (b)evaluate (e d + y dy dz) where is the curve + = 4, =. (3) PART B Answer any two questions each Module I to IV Module I 7. Determine whether the series converge or diverge Page 1 of 3 ( )! 8. Check the absolute convergence or divergence of the series ( 1) ( )! (!)

10 A B1A005 Pages: 3 9. Find the Taylor series epansion of log cos about the point Module II 10. If u = log ( 3 + y 3 + z 3 3yz), Show that + + = ( ) 11. The length, width and height of a rectangular bo are measured with an error of atmost 5%. Use a total differential to estimate the maimum percentage error that results if these quantities are used to calculate the diagonal of the bo. 1. Locate all relative etrema and saddle points of (, )= Module III 13. Find the angle between the surfaces + + = 9 and = + 3at the point (, 1,) 14. Let = i + yj + zk and r =, then prove that ( )= ( ). 15. Find an equation of the tangent plane to the ellipsoid = 9at the point (,1,1) and determine the acute angle that this plane makes with the plane. Module IV 16. Change the order of integration and hence evaluate y dy d 17. Evaluate ( ) ( + ) d dy using polar co-ordinates 18. Find the volume of the paraboloid of revolution + = 4 cut off by the plane = 4 Module V Answer any 3 questions. 19. Evaluate the line integral ( + ) from (1,0,0) to ( 1,0, )along the heli that is represented by the parametric equations = cos t, y = sin t, z =t 0. Evaluate the line integral ( ) d + y dy along the curve C, = from (1, - 1) to (1,1) 1. Find the work done by the force field = ( + ) + along the line segment from (0,0,0) to(1,3,1) and then to (, 1,5).. Show that = ( + ) is a conservative vector field. Also find its scalar potential. 3. Find the values of constants,, so that = (ay + bz 3 )i + (3 cz)j + (3z k may be irrotational. For these values of,, find the scalar potential of Page of 3

11 A B1A005 Pages: 3 Module VI Answer any 3 questions. 4. Verify Green s theorem for (y + y )d + dy where C is bounded by = and = 5. Apply Green s theorem to evaluate ( ) + ( + ) where C is the boundary of the area enclosed by the -ais and the upper half of the circle + = 6. Apply Stokes theorem to evaluate ( + ) + ( ) + ( + ) where C is the boundary of the triangle with vertices (0,0,0),(,0,0) and (0,3,0) 7. Use Divergence theorem to evaluate where = i + zj + yzk and S is the surface of the cube bounded by = 0, = 1, = 0, = 1, = 0 and = 1.Also verify this result by computing the surface integral over S 8. State Divergence theorem. Also evaluate where = ai + byj + czk and S is the surface of the sphere + + = 1 **** Page 3 of 3

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14 A B1A004 (016) Reg. No.:... Name. APJ ABDUL KALAM TECHNOLOGICAL UNIVERSITY FIRST SEMESTER B.TECH DEGREE EXAMINATION, DEC 016 (016 ADMISSION) Ma. Marks: 100 Course Code: MA 101 Course Name: CALCULUS Duration: 3 Hours PART A Answer ALL questions 1 (a) Determine whether the series converges and if so, find its () sum. (b) Find the Maclaurin series for the function (3) (a) If =, then find () (b) Compute the differential of the function = (). (3) 3 (a) Find the domain of () = 5 + 1,, = 1 and ( ) () (b) Find the directional derivative of (, ) = (5,0 ), in the direction of = (3) 4 (a) Evaluate (b) Use double integration to find the area of the plane region enclosed by the given curves = sin = for 0 () (3) 5 (a) Confirm that (,, ) = is a potential function for (,, ) = (b) Evaluate. where (, ) = + cos where C is the curve () = +, 0 6 (a) Using Green s theorem evaluate +, where C is the unit circle oriented counter clockwise. (b) If is any closed surface enclosing a volume V and = + + 3, Using Divergence theorem show that. = 7 () (3) () (3) Page 1 of 3

15 A B1A004 (016) PART B (Each question carries 5 Marks) Answer any TWO questions 7 Test the nature of the series! 8 Check whether the series ( 1) is absolutely convergent or not. 9 Find the radius of convergence and interval of convergence of the series () Answer any TWO questions 10 If = (,, ), prove that + + = 0 11 A function (, ) = + ; is given with a local linear approimation (, ) = to (, )at a point P. Determine the point P. 1 Find the absolute etrema of the function (, ) = 4 on R where R is the triangular region with vertices (0,0) (0,4) and (4,0). Answer any TWO questions 13 Evaluate the definite integral ( + + ). 14 Find the velocity, acceleration, speed, scalar tangential and normal components of acceleration at the given t of () = 3 + ; = 15 Find the equation of the tangent plane and parametric equation for the normal line to the surface = 4 + at the point (1,-,10) 16 Evaluate the integral integration. Answer any TWO questions 17 Evaluate by first reversing the order of 18 Find the volume of the solid in the first octant bounded by the co-ordinate Page of 3

16 A B1A004 (016) planes and the plane + + = 1 PART C (Each question carries 5 Marks) Answer any THREE questions 19 Find div F and curl F of (,, ) = Show that ( ) = ( + 1) where = Find the work done by the force field (,, ) = ( + ) + ( ) on a particle that moves along the curve : =, =,1 3 Evaluate. where (, ) = along the triangle joining the vertices (0,0), (1,0), and (0,1). 3 Determine whether (, ) = is a conservative vector field. If so, find the potential function and the potential energy. Answer any THREE questions 4 Using Green s theorem evaluate + ) + ( + ) where C is the boundary of the region between = and =. 5 Evaluate the surface integral over the surface represented by the vector valued function (, ) = + +, 1 3, 0 6 Using Divergence Theorem evaluate. where (,, ) = ( ) + ( ) + ( ), is the surface of the cylindrical solid bounded by + =, = 0, = 1. 7 Determine whether the vector field (,, ) = 4( ) + 4( ) + 4( ) is free of sources and sinks. If it is not, locate them. 8 Using Stokes theorem evaluate. (,, ) = + 4 +, where C is the rectangle: 0 1, 0 3 in the plane = Page 3 of 3

17 A B1A15S (015 Admission) Total No. of pages: 3 Reg No.. Name:. APJ ABDUL KALAM TECHNOLOGICAL UNIVERSITY FIRST SEMESTER B.TECH DEGREE EXAMINATION, DECEMBER 016 (015 ADMISSION) Ma. Marks: 100 Course Code: MA 101 Course Name: CALCULUS PART A Duration: 3 Hours (Answer all questions. Each question carries 3 marks) 1 Find the derivative of y n n Test the convergence of n 1 n 1 1 ( 1 cosh 3 Classify the surface 4 4 y z 8 y 4 z 4 4 Convert the rectangular co-ordinate into spherical co-ordinate of (, 3, 4 ) z z 5 Prove that y y where f = y. 6 Find the velocity, acceleration and speed of a particle moving along the curve 1 3t, y 3 4t, z 1 3t at 7 Given y u z e, u v, y v Find z z u and v 8 Find the unit tangent vector and unit normal vector to the curve = e t cos t, y = e t sin t, z = e t at t = 0. 9 y 9 Evaluate yddy Find the area of the region R enclosed between the parabola y and the line y = ) (10*3=30 Marks) Page 1 of 3

18 A B1A15S (015 Admission) Total No. of pages: 3 PART B (Answer any questions each question carries 7 marks) n 11 Find the radius of curvature and interval of curvature of n 1 n 3 1 Test the convergence of Determine the Taylor s series epansion of f() = sin at = π/4. (Answer any questions each question carries 7 marks) 14 Find the nature of domain of the following function 1. f (,. f (, ln( y y 15 Show that the function f (, 6 approaches zero as (, (0,0) y along the line y = m Find the trace of the surface y z 0 in the plane = and y = 1. y z 0 (Answer any questions each question carries 7 marks) 17 Find the local linear approimation of f (, y at (3,4) and compare the error in approimation by L(3.04,3.98) with the distance between the points. 18 Find the relative etrema of f (, 3 y y 8y y u z z 19 If z e, u v, y Find and v u v 0 If r( t) e i e j tk t (Answer any questions each question carries 7 marks) t 1) Find the scalar tangential and normal component of acceleration at t = 0 ) Find the vector tangential and normal component of acceleration at t = 0. 1 Find the equation of the tangent plane and parametric equations of the normal Page of 3

19 A B1A15S (015 Admission) Total No. of pages: 3 line to the surface z = 4 3 y + y at the point P (1, -, 10). 3 Find the directional derivative of f y y z z of a i j k at (1,-,0) in the direction 3 Evaluate R (Answer any questions each question carries 7 marks) y da where R is the region in the first quadrant enclosed between the circle y 5and the line +y=5 y 4 Change the order of integration and evaluate 1 y ddy 5 Find the volume bounded by the cylinder +y = 4 the planes y + z = 3 and z = 0. y Page 3 of 3