MIDTERM EXAMINATION. Spring MTH301- Calculus II (Session - 3)

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1 ASSALAM O ALAIKUM All Dear fellows ALL IN ONE MTH3 Calculus II Midterm solved papers Created BY Ali Shah From Sahiwal BSCS th semester alaoudin.bukhari@gmail.com Remember me in your prayers MIDTERM EXAMINATION Spring MTH3- Calculus II (Session - 3). Every real number corresponds to on the co-ordinate line. Infinite number of points Two points (one positive and one negative) A unique point None of these. There is one-to-one correspondence between the set of points on co-ordinate line and. Set of real numbers Set of integers Set of natural numbers Set of rational numbers

2 3. Which of the following is associated to each point of three dimensional space? A real number An ordered pair An ordered triple A natural Number. All axes are positive in octant. First Second Fourth Eighth 5. The spherical co-ordinates of a point are 3,,.What are its cylindrical co-ordinates? 3 3 3,, 3 cos 3 sin 3, 3, 3 sin, 3, 3, 3 cos 3, 3 and y = t. Which 6. Suppose f ( x, y ) = xy y where x = t3+ one of the following is true? df dt df dt df dt df dt = t + = 6t t = 8t + = t + 8t + 7. Let w = f ( x, y, z ) and x = g ( r s ), w x w y w z + + x r y r z r y, = h ( r s ), z =, t ( r s ) then, by chain rule w = r

3 3 w x w y w z + + r r r r r r w x x w y y w z z + + x r s y r s z r s w r w r w r + + r x r y r z 8. Magnitude of vector is, magnitude of vector to tail is 5 degrees. What is is 3and angle between them when placed tail? Is the function f ( x, y ) continuous at origin? If not, why? f ( x, y ) is continuous at origin f (, ) is not defined f (, ) is defined but f (, ) is defined and lim f ( x, y ) does not exist lim f ( x, y ) exist but these two numbers are not equal. ( x, y ) (, ) ( x, y ) (, ). Is the function f ( x, y ) continuous at origin? If not, Why? f ( x, y ) is continuous at origin f (, ) is not defined f (, ) is defined but f (, ) is defined and lim f ( x, y ) does not exist lim f ( x, y ) exist but these two numbers are not equal. ( x, y ) (, ) ( x, y ) (, ). Let R be a closed region in two dimensional space. What does the double integral over R calculates? Area of R Radius of inscribed circle in R. Distance between two endpoints of R. None of these

4 . Which of the following formula can be used to find the volume of a parallelepiped with adjacent edges formed by the vectors, and? 3. Two surfaces are said to be orthogonal at appoint of their intersection if their normals at that point are. Parallel Perpendicular In opposite direction Same direction. By Extreme Value Theorem, if a function f ( x, y ) is continuous on a closed and bounded set R, then f ( x, y ) has both on R. Absolute maximum and absolute minimum value Relative maximum and relative minimum value Absolute maximum and relative minimum value Relative maximum and absolute minimum value 5. Let the function f ( x, y ) has continuous second-order partial derivatives (f xx, f yy and f xy ) in some circle centered at a critical point ( x, y ) and let D = f xx ( x, y ) f yy ( x, y ) f xy ( x, y ) if D > and f xx ( x, y ) < then f has. Relative maximum at ( x, y ) Relative minimum at ( x, y ) Saddle point at ( x, y ) No conclusion can be drawn. 6. Let the function f ( x, y ) has continuous second-order partial derivatives ( f xx, f yy and f xy ) in some circle centered at a critical point ( x, y ) and let D = f xx ( x, y ) f yy ( x, y ) f xy ( x, y ) if if D = then.

5 5 f has relative maximum at ( x, y ) f has relative minimum at ( x, y ) f has saddle point at ( x, y ) No conclusion can be drawn. 7. If R = R R where, R and R are no over lapping regions then f ( x, y )da + f ( x, y )da = R R f ( x, y )da R f ( x, y )da f ( x, y )da R R f ( x, y )dv R f ( x, y )dv f ( x, y )da R R 8. If R = {( x y,) / x and y } then, ( 6 x + xy 3 )da = R (6x + xy 3 ) dydx + xy 3 ) dydx + xy 3 ) dydx + xy 3 ) dydx (6x (6x (6x 9. If R = {( x y, ) / x and y } then, ( xe )da = y R ( xe ) dydx y

6 6 ( xe ) dxdy y ( xe ) dxdy y ( xe ) dydx y. If R = {( x y,) / x and y 9, ( 3 x x } then ( 3x x xy dydx ) ( 3x x xy dxdy ( 3x x xy dxdy ( 3x x xy dydx 9 ) 9 ) 9 9 ) xy da = R ). Suppose that the surface f ( x, y, z ) has continuous partial derivatives at the point ( a, b, c ) Write down the equation of tangent plane at this point.. Evaluate the following double integral ( xy 8 x 3 ) dydx. 3. Evaluate the following double integral ( 3 + x 3 y ) dxdy. Let f ( x, y, z ) = xy e z Find the gradient of f. 5. Find, Equation of Tangent plane to the surface f ( x, y, z ) = x + y + z 9 at the point (,, ). 6. Use the double integral in rectangular co-ordinates to compute area of the region bounded by the curves y = x and y = x.

7 7 MIDTERM EXAMINATION Fall 9 MTH3- Calculus II Question No: i. Let x be any point on co-ordinate line. What does the inequality -3 < x < means? a. The set of all inte gers between -3 and b. The set of all natural numbers between -3 and. c. The set of all rational numbers between -3 and d. The set of all real numbers between -3 and Question No: Which of the following number is associated to each point on a co-ordinate line? An integer A real number A rational number

8 8 A natural number Question No: 3 Which of the following set is the union of set of all rational and irrational numbers? Set of rational numbers Set of integers Set of real numbers Empty set. Question No: is an example of Irrational numbers Rational numbers Integers Natural numbers Question No: 5 Which of the following is associated to each point on a plane? A real number A natural number

9 9 An ordered pair An ordered triple Question No: planes intersect at right angle to form three dimensional space. Three Four Eight Twelve Question No: 7 At each point of domain, the function Is defined Is continuous Is infinite Has a limit Question No: 8 What is the general equation of parabola whose axis of symmetry is parallel to y-axis? y = ax + b (a )

10 x = ay + b (a ) y = ax + bx + c (a ) x = ay + by + c (a ) Question No: 9 The spherical co-ordinates (,, ), of a point are,,. What are the rectangular co-ordinates of this point? (,, ) (,, ) (,, ) (,, ) Question No: Let f ( x, y ) = y x e x +. 5 f y 3 x Which method is best suited for evaluation of? Normal method of finding the higher order mi xed partial derivatives Chain Rule

11 Laplacian Method Euler s method for mixed partial derivative Question No: Suppose f ( x, y ) = x 3e xy. Which one of the following is correct? f = 3 x e xy + x 3 ye xy x f = 3 x e xy + x e xy x f = 3 x e xy x f = 3 x ye xy x Question No: Suppose f ( x, y ) = x 3e xy f = 3 x 3e xy y f = x3e xy y f = x e xy y. Which one of the statements is correct?

12 f = x 3 ye xy y Question No: 3 Suppose f ( x, y ) = xy where x = t + and y = 3 t. Which one of the following is true? df = 6t t dt df = 6t dt df = t 3 + 6t 6 dt df = 6t + t dt Question No: Let i, j and k be unit vectors in the direction of x-axis, y-axis and z-axis respectively. Suppose that a = i + 5 j k 6 3. What is the magnitude of vector a?

13 3 3 8 Question No: 5 Is the function f ( x, y ) = f ( x, y ) continuous at origin? If not, why? xy x + y f ( x, y ) is continuous at origin lim f ( x, y ) ( x, y ) (, ) does not exist lim f (, ) ( x, y ) (, ) f ( x, y ) is defined and Question No: 6 exists but these two numbers are not equal. Let R be a closed region in two dimensional space. What does the double integral over R calculates? Area of R. Radius of inscribed circle in R. Distance between two endpoints of R. None of these Question No: 7 What is the relation between the direction of gradient at any point on the surface to the tangent plane at that point?

14 parallel perpendicular opposite direction No relation between them. Question No: 8 Two surfaces are said to intersect orthogonally if their normals at every point common to them are perpendicular parallel in opposite direction Question No: 9 the function f ( x, y ) has continuous second-order partial derivatives (f xx, f yy and f xy ) If D> and f xx ( x, y ) < then f has Relative maximum at Relative minimum at Saddle point at ( x, y ) ( x, y ) ( x, y ) No conclusion can be drawn. Question No: in some ( x, y ) and let D = f xx ( x, y ) f yy ( x, y ) f xy ( x, y ) circle centered at a critical point Let

15 5 Which of the following are direction ratios for the line joining the points (, 3, 5) and (,, )? 3, and 9, - and -, -3 and.5, -3 and 5/ Question No: If R = R R, where R and R are no overlapping regions then f (x, y)da+ f (x, y)da= R R f (x, y)da R f (x, y)da f (x, y)da R R R f (x, y)dv

16 6 f (x, y)da f (x, y)da R R Question No: If R = {( x, y ) / x and y }, then ( xe y )da = R ( xe y )dydx ( xe y )dxdy ( xe y )dxdy ( xe y )dydx Question No: 3

17 7 If R = {( x, y ) / x and y 3}, then ( ye xy )da = R 3 3 ( ye xy )dydx 3 ( ye xy )dxdy 3 ( ye xy )dxdy ( xe y )dydx Question No: Which of the following is geometrical representation of the equation Parabola Straight line Half cylinder Cone y = x, in three dimensional space?

18 8 Question No: 5 ( Marks: 3 ) 3,, 3 Consider the point point. in spherical coordinate system. Find the rectangular coordinatesof this Question No: 6 ( Marks: 5 ) Consider a function f ( x, y ) = xy x y.one of its critical point is (, ). Find whether (, ) is relative maxima, relative minima or saddle point of f ( x, y ). Question No: 7 Find f ( x, y ) = x y y Directional derivative of the function, u= 3, at the point (, ) 3 i+ j MIDTERM EXAMINATION Spring MTH3- Calculus II Question No: in the direction of vector,

19 9 Which of the following is the interval notation of real line? (-, + ) (-, ) (, + ) Question No: What is the general equation of parabola whose axis of symmetry is parallel to y-axis? y = ax + b (a ) x = ay + b (a ) y = ax + bx + c (a ) x = ay + by + c (a ) Question No: 3 Which of the following is geometrical representation of the equation dimensional space? A point on y-axis Plane parallel to xy-plane Plane parallel to yz -axis Plane parallel to xz-plane y=, in three

20 Question No: Suppose f ( x, y ) = x 3e xy. Which one of the statements is correct? f = 3 x 3e xy y f = x3e xy y f = x e xy y f = x 3 ye xy y Question No: 5 If f ( x, y ) = x y y + ln x then f x 3 = xy + y + x xy x x

21 y x Question No: 6 Let w = f(x, y, z) and x = g(r, s), y = h(r, s), z = t(r, s) then by chain rule w = r w x w y w z + + x r y r z r w x w y w z + + r r r r r r w x x w y y w z z + + x r s y r s z r s w r w r w r + + r x r y r z Question No: 7 Is the function f ( x, y ) continuous at origin? If not, why? 3x y if ( x, y ) f ( x, y ) = x + y if ( x, y ) =

22 f ( x, y ) f (, ) is continuous at origin is not defined lim f (, ) ( x, y ) (, ) is defined but does not exist lim equal. f (, ) f ( x, y ) ( x, y ) (, ) f ( x, y ) is defined and Question No: 8 exists but these two numbers are not Let R be a closed region in two dimensional space. What does the double integral over R calculates? Area of R. Radius of inscribed circle in R. Distance between two endpoints of R. None of these (not sure) Question No: 9 Two surfaces are said to be orthogonal at a point of their intersection if their normals at that point are Parallel Perpendicular In opposite direction Question No:

23 3 Two surfaces are said to intersect orthogonally if their normals at every point common to them are perpendicular parallel in opposite direction Question No: Let the function (f xx f ( x, y ), f yy and f xy ) has continuous second-order partial derivatives in some circle centered at a critical point ( x, y ) and let D = f xx ( x, y ) f yy ( x, y ) f xy ( x, y ) If D> and f xx ( x, y ) > then f has Relative maximum at Relative minimum at Saddle point at ( x, y ) ( x, y ) ( x, y ) No conclusion can be drawn. Question No: Let the function (f xx f ( x, y ), f yy and f xy ) has continuous second-order partial derivatives in some circle centered at a critical point D = f xx ( x, y ) f yy ( x, y ) f xy ( x, y ) ( x, y ) and let

24 If D> and f xx ( x, y ) < then f has Relative maximum at Relative minimum at Saddle point at ( x, y ) ( x, y ) ( x, y ) No conclusion can be drawn. Question No: 3 Let the function (f xx f ( x, y ), f yy and f xy ) has continuous second-order partial derivatives in some circle centered at a critical point ( x, y ) and let D = f xx ( x, y ) f yy ( x, y ) f xy ( x, y ) If D< then f has Relative maximum at Relative minimum at Saddle point at ( x, y ) ( x, y ) ( x, y ) No conclusion can be drawn Question No: Let ( x, y, z ) and ( x, y, z ) be any two points in three dimensional space. What does the formula ( x x ) + ( y y ) + ( z z ) calculates?

25 5 Distance between these two points Midpoint of the line joining these two points Ratio between these two points Question No: 5 The function f ( x, y ) = y x is continuous in the region and discontinuous elsewhere. x y x y x > y Question No: 6 Plane is an example of Curve Surface Sphere Cone Question No: 7 If R = R R, where R and R are no overlapping regions then

26 6 f (x, y)da+ f (x, y)da= R R f (x, y)da R f (x, y)da f (x, y)da R R f (x, y)dv R f (x, y)da f (x, y)da R R Question No: 8 If R = {( x, y ) / x and y }, then (6 x + xy 3 )da = R (6 x + xy 3 )dydx

27 7 (6 x + xy 3 )dxdy (6 x + xy 3 )dxdy (6 x + xy 3 )dxdy Question No: 9 If R = {( x, y ) / x and y 3}, then ( ye xy )da = R 3 3 ( ye xy )dydx 3 ( ye xy )dxdy 3 ( ye xy )dxdy ( xe y )dydx

28 8 Question No: If R = {( x, y ) / x and y 9}, then (3 x x xy )da = R (3 x x xy )dydx 9 9 (3 x x xy )dxdy (3 x x xy )dxdy (3 x x xy )dydx Question No: ( Marks: ) Following is the graph of a function of two variables

29 9 In its whole domain, state whether the function has relative maximum value or absolute maximum value at point B. Also, justify your answer Question No: ( Marks: ) Let the function f ( x, y ) is continuous in the region R, where R is bounded by graph of functions g and g (as shown below). In the following equation, replace question mark (?) with the correct value.? f ( x, y ) da = R? Question No: 3? f ( x, y )?? ( Marks: 3 ) Evaluate the following double integral. ( 3 + x 3 y ) dx dy Question No: ( Marks: 3 )

30 3 Let f ( x, y, z ) = yz 3 x Find the gradient of f. Question No: 5 ( Marks: 5 ) Find, Equation of Normal line (in parametric form) to the surface f ( x, y, z ) = xy + yz xz + Question No: 6 at the point ( 5, 5, ) ( Marks: 5 ) Use double integral in rectangular co-ordinates to compute area of the region bounded by the curves y = x and y = x, in the first quadrant.

31 3 MIDTERM EXAMINATION Spring MTH3- Calculus II (Session - 3) Question No: Which of the following number is associated to each point on a co-ordinate line? An integer A real number A rational number A natural number Question No: If a >, then the parabola Positive y = ax + bx + c opens in which of the following direction? x - direction Negative x - direction Positive y Negative Question No: 3 y - direction - direction Rectangular co-ordinate of a point is 3,, 3 (, 3, ). What is its spherical co-ordinate?

32 3 3,, 3,, 3 3,, 3 Question No: If a function is not defined at some point, then its limit exist at that point. Always Never May Question No: 5 Suppose f ( x, y ) = x 3e xy. Which one of the statements is correct? f = 3 x 3e xy y f = x3e xy y f = x e xy y f = x 3 ye xy y Question No: 6 If f ( x, y ) = x y y 3 + ln x

33 33 f x then = xy + x y + x xy x y x Question No: 7 Suppose f ( x, y ) = xy y where x = 3t + and y = t df = t + dt df = 6t t dt df = 8t + dt df = t + 8t + dt Question No: 8. Which one of the following is true?

34 3 Is the function f ( x, y ) f ( x, y ) = f ( x, y ) f (, ) continuous at origin? If not, why? If x and y Otherwise is continuous at origin is not defined lim f (, ) ( x, y ) (, ) is defined but does not exist lim f (, ) f ( x, y ) ( x, y ) (, ) f ( x, y ) is defined and Question No: 9 exists but these two numbers are not equal. What is the relation between the direction of gradient at any point on the surface to the tangent plane at that point? parallel perpendicular opposite direction No relation between them. Question No: Two surfaces are said to intersect orthogonally if their normals at every point common to them are perpendicular parallel in opposite direction Question No: By Extreme Value Theorem, if a function R, then f ( x, y ) f ( x, y ) has both on R. is continuous on a closed and bounded set

35 35 Absolute maximum and absolute minimum value Relative maximum and relative minimum value Question No: Let the function (f xx f ( x, y ), f yy and f xy ) has continuous second-order partial derivatives in some circle centered at a critical point ( x, y ) and let D = f xx ( x, y ) f yy ( x, y ) f xy ( x, y ) If D> and f xx ( x, y ) < then f has Relative maximum at Relative minimum at Saddle point at ( x, y ) ( x, y ) ( x, y ) No conclusion can be drawn. Question No: 3 Let the function (f xx f ( x, y ), f yy and f xy ) has continuous second-order partial derivatives in some circle centered at a critical point D = f xx ( x, y ) f yy ( x, y ) f xy ( x, y ) If D= then f has relative maximum at ( x, y ) has relative minimum at ( x, y ) f f has saddle point at ( x, y ) No conclusion can be drawn. Question No: ( x, y ) and let

36 36 f ( x, y ) = y x The function is continuous in the region and discontinuous elsewhere. x y x y x> y Question No: 5 Plane is an example of Curve Surface Sphere Cone Question No: 6 If R = R R, where R and R are no overlapping regions then f (x, y)da+ f (x, y)da= R R f (x, y)da R f (x, y)da f (x, y)da R R

37 37 f (x, y)dv R f (x, y)da f (x, y)da R R Question No: 7 If R = {( x, y ) / x and y }, then (6 x + xy 3 )da = R (6 x + xy 3 )dydx (6 x + xy 3 )dxdy (6 x + xy 3 )dxdy (6 x + xy 3 )dxdy Question No: 8 If R = {( x, y ) / x and y }, then R ( x + y )da =

38 38 ( x + y )dydx ( x + y )dxdy ( x + y )dxdy ( x + y )dxdy Question No: 9 If R = {( x, y ) / x and y 3}, then ( ye xy )da = R 3 3 ( ye xy )dydx 3 ( ye xy )dxdy 3 ( ye xy )dxdy ( xe y )dydx

39 39 Question No: If R = {( x, y ) / x and y 9}, then (3 x x xy )da = R 9 9 (3 x x xy )dydx 9 (3 x x xy )dxdy 9 (3 x x xy )dxdy (3 x x xy )dydx Question No: ( Marks: ) Evaluate the following double integral. ( xy + y ) dx dy 3 Question No: ( Marks: ) y Find the gradient of f Let f ( x, y ) = + x + Question No: 3 ( Marks: 3 ) Evaluate the following double integral.

40 ( 3 + x 3 y ) dx dy Question No: ( Marks: 3 ) Let f ( x, y, z ) = yz 3 x Find the gradient of f. Question No: 5 ( Marks: 5 ) Find Equation of a Tangent plane to the surface (,, ) Question No: 6 ( Marks: 5 ) Evaluate the iterated integral ( xy ) dy dx x x f ( x, y, z ) = x + 3 y + z 3 9 at the point

41 MIDTERM EXAMINATION MTH3 Suppose f ( x, y ) = x 3e xy. Which one of the following is correct? f = 3x e xy + x3 ye xy x f = 3 x ye xy x f = 3 x e xy + x e xy x f = 3x e xy x Let R be a closed region in two dimensional space. What does the double integral over R calculates? Area of R. Radius of inscribed circle in R. Distance between two endpoints of R. None of these What is the distance between points (3,, ) and (6,, -)? planes intersect at right angle to form three dimensional space. Three 8 There is one-to-one correspondence between the set of points on co-ordinate line and Set of real numbers Set of integers

42 Set of natural numbers Set of rational numbers f ( x, y ) has continuous second-order partial derivatives ( f xx, f yy and f xy ) ( x, y ) Let the function in some circle centered at a critical point and let D = f xx ( x, y ) f yy ( x, y ) f xy ( x, y ) If D = then f has relative maximum at ( x, y ) has relative minimum at ( x, y ) f f (x, y ) has saddle point at No conclusion can be drawn. If R = {( x, y ) / x and y 3}, then ( ye xy )da = R ( ye xy )dydx ( ye xy )dxdy ( ye xy )dxdy ( xe y )dydx Suppose f ( x, y ) = xy where x = t + and y = 3 t df = 6t t dt. Which one of the following is true?

43 3 df = 6t dt df = t 3 + 6t 6 dt df = 6t + t dt Let i, j and k be unit vectors in the direction of x-axis, y-axis and z-axis respectively. Suppose a = i + 5 j k that. What is the magnitude of vector a? A straight line is geometric figure. One-dimensional Two-dimensional Three-dimensional Dimensionless If R = {( x, y ) / x and y }, then (6 x + xy 3 )da = R (6 x + xy 3 )dydx (6 x + xy 3 )dxdy (6 x + xy 3 )dxdy (6 x + xy 3 )dxdy

44 Which of the following formula can be used to find the Volume of a parallelepiped with adjacent a, b and c edges formed by the vectors a a a a b c b c b c b c ( ) ( ) ( ) ( )? f ( x, y ) = y x The function is continuous in the region and discontinuous elsewhere. x y x y x> y What is the relation between the direction of gradient at any point on the surface to the tangent plane at that point? parallel perpendicular opposite direction No relation between them. f ( x, y ) = x 3e xy Suppose. Which one of the statements is correct? f = 3 x 3e xy y f = x 3e xy y f = x e xy y f = x 3 ye xy y Two surfaces are said to intersect orthogonally if their normals at every point common to them are perpendicular parallel in opposite direction

45 5 f ( x, y ) has continuous second-order partial derivatives ( f xx, f yy and f xy ) ( x, y ) Let the function in some circle centered at a critical point D = f xx ( x, y ) f yy ( x, y ) f xy ( x, y ) If D > and f xx ( x, y ) < then f has Relative maximum at Relative minimum at ( x, y ) ( x, y ) (x, y ) Saddle point at No conclusion can be drawn. If R = {( x, y ) / x and y }, then ( x + y )da = R ( x + y )dydx ( x + y )dxdy ( x + y )dxdy ( x + y )dxdy f ( x, y, z ) = x y + xyz z If then what is the value of f (,, )? f (,, ) = f (,, ) = and let

46 6 f (,, ) = 3 f (,, ) = If R = {( x, y ) / x and y 9}, then (3 x x xy )da = R (3 x x xy )dydx (3 x x xy )dxdy (3 x x xy )dxdy (3 x x xy )dydx y Let f ( x, y ) = + x + Find the gradient of f Q- MARKS Q - Let the function f ( x, y ) is continuous in the region R, where R is a rectangle as shown below.complete the following equation f ( x, y ) da = R f ( x, y ) MARKS Q.Find all critical points of the function f ( x, y ) = xy x 3 y Evaluate (6 x + xy 3 )dx dy

47 7 Q-Evaluate the following double integral. ( 3 + x 3 y ) dx dy 3MARKS y= x Q- Let. If x changes from 3 to 3.3, find the approximate change in the value of y using differential dy. 3MARKS MIDTERM EXAMINATION MTH3 That all is collected by removing the mistakes of the one paper file, hope it helpful! In that the answer is unique point

48 8 In that the correct answer is the Set of Real Numbers. In this the exact answer is an ordered triple.

49 9 It s right This one is also wrong as the answer is th one.because the central point of spherical is pie/3 that should be same in cylindrical so, that s only occur in last one

50 That s right Here s answer is the nd one! 5

51 5 That can t say exactly guys, its formula is following! a.b = a b cosø Ø=angle between the vectors That s correct answer!

52 5 But that s wrong the correc one is the st.. That s the first answer, as that was been define in the lecture!

53 53 Here s the answer is the last one, you can check that on the start of lecture th, handout! Its answer is the perpendicular!

54 That s right answer! This is also right one..! 5

55 55 No conclusion can be drawn is the answere! Its answer is the first one option!

56 Its answer is the 3rd option It s right one! 56

57 This one s also right Its equation is f x (a )( x x ) + f y (b)( y y ) + f z (c)( z z ) = 57

58 58 These both can be solve by the following example the me gonna paste! Evaluate the following double integral. ( 3 + x 3 y ) dx dy Solution: ( 3 + x 3 y ) dx dy (3x + x 3xy )dy 3 xy + x y xy 3 I didn t got this question dear s!

59 59 You can use the above tangent equation to sove this! Its answer is the under mentioned! as our require is area of the following region V = da = da we can compute area by keeping f(x,y) const R R x = dydx x x = [ x] x = ( x x ) 3 x x =[ ] 3 = 3 3 = 6 = that's exactly what we require...! 6