APJ ABDUL KALAM TECHNOLOGICAL UNIVERSITY FIRST SEMESTER B.TECH DEGREE (SUPPLEMENTARY) EXAMINATION, FEBRUARY 2017 (2015 ADMISSION)

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1 B116S (015 dmission) Pages: RegNo Name PJ BDUL KLM TECHNOLOGICL UNIVERSITY FIRST SEMESTER BTECH DEGREE (SUPPLEMENTRY) EXMINTION, FEBRURY 017 (015 DMISSION) MaMarks : 100 Course Code: M 101 Course Name: CLCULUS PRT Duration : Hours (nswer all questions Each question carries marks) n 1 1) Show that the series converges n 1 ) Classify the surface z ( 1) ( y ) ) Find the Maclaurin series for cos y 4) Lt, y ( 1, ) y 5) Convert the cylindrical co-ordinate into rectangular co ordinate of ( 4, / ) 6) Find the slope of the surface z = y in the direction at the point (,) 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,,) a b 9) 1 1 y ddy 10) Find the area of the region R enclosed by y 1, y, 0, y PRT B (nswer 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/ 1 5 1) Test the convergence of Page 1 of

2 B116S (015 dmission) Pages: (nswer 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 (nswer any questions Each question carries 7 marks) 17) Find the local linear approimation L of f(,y,z) = yz at the point P(1,,) Compare the error in approimating f by L at the point Q(1001, 00, 00) with the distance PQ 18) Find the relative etrema of f (, y) 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 (nswer any questions Each question carries 7 marks) 0) Suppose that a particle moves along a curve in -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 1) 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 lso 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 (nswer any questions Each question carries 7 marks) 4 ) 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 1 1 5) Change the order of integration and evaluate *** 0 y ddy Page of

3 B100 Pages: (016 DMISSIONS) Reg No: Name: PJ BDUL KLM TECHNOLOGICL UNIVERSITY FIRST SEMESTER BTECH DEGREE EXMINTION, FEBRURY 017 M101: CLCULUS Ma Marks: 100 Duration: Hours PRT (nswer ll Questions and each carries 5 marks) 1 a) Test the convergence of! () b) Test the convergence of () 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) = y- -y () 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(,y) = e y -ye at the point P(0,0) in the direction of 5i j () 4 a) Change the order of integration in (, ) () 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 z j z k () b) Find the work done by = y i j on a particle that moves along the curve y = from (0,0) to (0,1) () 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 = (-y)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 () Page 1 of

4 B100 Pages: (016 DMISSIONS) PRT B MODULE I (nswer ny Two Questions) 7 a) Test the convergence of the following series (5) 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 = (5) MODULE II (nswer ny Two Questions) 10 Find the local linear approimation L to f(,y) = ln (y) at P(1,) and compare the error in approimating f by L at Q(101, 01) with the distance between P and Q (5) 11 Show that the function f (,y) = tan -1 (y/) satisfies the Laplace s equation (5) = 0 (5) 1 Find the relative minima of f(,y) = -yy -8y (5) MODULE III (nswer ny Two Questions) 1 Find the unit tangent vector and unit normal vector to = 4 cos ti 4 sin tj t k at t = (5) 14 Suppose a particle moves through - space so that its position vector at time t is = t i t j t k Find the scalar tangential component of acceleration at the time t=1 (5) 15 Given that the directional derivative of f(,y,z) at (,-, 1) in the direction of i j - k is -5 and that (,,1) = 5 Find (,,1) (5) MODULE IV (nswer ny Two Questions) 16 the integral ddy by reversing the order of integration (5) 17 d dy (5) 18 Find the volume of the solid in the first octant bounded by the co-ordinate planes and the plane yz = 6 (5) Page of

5 B100 Pages: (016 DMISSIONS) MODULE V (nswer ny Three Questions) 19 Let = i yj zk and let r = and f be a differentiable function of one variable show that ( )= ( ) (5) 0 the line integral [ ] along y = from (,) to (0,0) (5) 1 Show that (,y) = (cosy y cos ) i ( sin siny) j is a conservative vector field Hence find a potential function for it (5) Show that the integral ( e y d e y dy ) is independent of the path and hence evaluate the integral from (0,0) to (,) (5) 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 k where 0 t 1 (5) MODULE VI (nswer ny Three Questions) 4 Use Green s theorem to evaluate the integral ( cos y d y sin dy) where C is the square with vertices (0,0), (π,0), (π,π) and (0,π) (5) 5 the surface integral where σ is the portion of the cone z = between the planes z = 1 and z = (5) 6 Use divergence theorem to find the outward flu of the vector field = i y j z k across the unit cube = 0, = 1,y=0, y = 1, z = 0 and z = 1 (5) 7 Use Stoke s theorem to evaluate the integral where = (-y) i (y-z) j (z-)k and C is the boundary of the portion of the plane yz = 1 in the first octant (5) 8 Use Stoke s theorem to evaluate the integral where = z i j 5y k and C is the boundary of the paraboloid y z = 4 for which z 0 and C is positively oriented (5) *** Page of

6 B115S (015 dmission) Total No of pages: Reg No Name: PJ BDUL KLM TECHNOLOGICL UNIVERSITY FIRST SEMESTER BTECH DEGREE EXMINTION, DECEMBER 016 (015 DMISSION) Course Code: M 101 Course Name: CLCULUS Ma Marks: 100 Duration: Hours PRT (nswer all questions Each question carries marks) 1 1 Find the derivative of y (1 cosh n Test the convergence of n 1 n 1 n ) Classify the surface 4 4 y z 8 y 4 z 4 4 Convert the rectangular co-ordinate into spherical co-ordinate of (,, 4 ) 5 Prove that 6 Find the velocity, acceleration and speed of a particle moving along the curve z z y y where f = y 1 t, y 4t, z 1 t at u z z Find and u v v 7 Given z e y, u v, 8 Find the unit tangent vector and unit normal vector to the curve y = et cos t, y = et sin t, z = et at t = y 0 0 yddy Find the area of the region R enclosed between the parabola y and the line y = (10*=0 Marks) Page 1 of

7 B115S (015 dmission) Total No of pages: PRT B 11 (nswer any questions each question carries 7 marks) n Find the radius of curvature and interval of curvature of n 1 n 1 Test the convergence of Determine the Taylor s series epansion of f() = sin at = π/4 (nswer any questions each question carries 7 marks) 14 Find the nature of domain of the following function 1 f (, y) y f (, y) ln( y) 15 Show that the function f (, y ) y 6 y approaches zero as (, y) (0,0) along the line y = m 16 Find the trace of the surface y z 0 in the plane = and y = 1 y z 0 (nswer any questions each question carries 7 marks) 17 Find the local linear approimation of f (, y ) the error y at (,4) and compare in approimation by L(04,98) with the distance between the points Find the relative etrema of f (, y ) y y 8 y z u z If z e y, u v, y Find and v v u (nswer any questions each question carries 7 marks) 0 If r (t ) e t i e t j tk 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

8 B115S (015 dmission) Total No of pages: line to the surface z = 4 y y at the point P (1, -, 10) Find the directional derivative of f y y z z at (1,-,0) in the direction of a i j k (nswer any questions each question carries 7 marks) y d where R is the region in the first quadrant enclosed between the R circle y 5 and the line y=5 4 y Change the order of integration and evaluate y ddy 1 y 5 Find the volume bounded by the cylinder y = 4 the planes y z = and z = 0 Page of

9 B1004 (016) Reg No: Name PJ BDUL KLM TECHNOLOGICL UNIVERSITY FIRST SEMESTER BTECH DEGREE EXMINTION, DEC 016 (016 DMISSION) Course Code: M 101 Course Name: CLCULUS Ma Marks: 100 Duration: Hours PRT nswer LL questions 1 Determine whether the series converges and if so, find its () sum Find the Maclaurin series for the function () If = () Compute the differential Find the domain of Find the directional derivative of, then find = direction of 4 ) () = 1 and ( ) () (, )= (5,0 ), in the () 4 5 Use double integration to find the area of the plane region enclosed by the () = sin ( )= = (,, )= Confirm that () (,, )= 6, ( given curves 5 ( ) = 5 1, 5 = of the function 6 1 where (, ) = for 0 4 is a potential function for () cos where C is the curve (), 0 Using Green s theorem evaluate, where C is the unit () circle oriented counter clockwise If is any closed surface enclosing a volume V and =, Using Divergence theorem show that Page 1 of =7 ()

10 B1004 (016) PRT B (Each question carries 5 Marks) nswer 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 ( ) nswer any TWO questions = (,, ), prove that =0 10 If 11 function (, ) = 1 (, ) = 4 5 to (, )at a point P Determine the point P Find the absolute etrema of the function (, ) = 4 on R ; is given with a local linear approimation where R is the triangular region with vertices (0,0) (0,4) and (4,0) nswer any TWO questions ( ) 1 the definite integral 14 Find the velocity, acceleration, speed, scalar tangential and normal components of acceleration at the given t of ( )= 15 ; = Find the equation of the tangent plane and parametric equation for the normal line to the surface =4 at the point (1,-,10) nswer any TWO questions 16 the integral by first reversing the order of integration Find the volume of the solid in the first octant bounded by the co-ordinate Page of

11 B1004 (016) planes and the plane =1 PRT C (Each question carries 5 Marks) nswer any THREE questions (,, )= ( 1) 19 Find div F and curl F of 0 Show that ( 1 Find the work done by the force field (,, )=( along the curve ) = where = ) on a particle that moves ) ( : =, =,1 where (, ) = along the triangle joining the vertices (0,0), (1,0), and (0,1) Determine whether (, ) = 4 4 is a conservative vector field If so, find the potential function and the potential energy nswer any THREE questions 4 5 ) ( where C is the boundary of the region between = and = Using Green s theorem evaluate the surface integral over the ) surface represented by the vector valued function (, )= 6 7,1 Using Divergence Theorem evaluate, 0 where (,, ) = ( ) ( ) ( ), is the surface of the cylindrical solid bounded by =, = 0, = 1 (,, ) = 4( ) Determine whether the vector field 4( ) 4( ) is free of sources and sinks If it is not, locate them 8 Using Stokes theorem evaluate (,, )= 4 C is the rectangle: 0, 1, 0 in the plane = Page of where