(a) Between 1 and the disk x²+ (1+x² +y*) (b) Between % = 1 and the entire xy-plane. (1+x* + y?)*

Q: (a) Find the solution of the limit using polar coordinate system. x² + y² x2 + y2 lim(x.y)-(0,0)

A: To find the solution of the limit using polar coordinate system lim(x,y)→(0,0)x2+y2x2+y2 We have the…

Q: 13. Find the area of the surface generated when the curve y = -, on [2,4] is rotated about the…

A:

Q: Find a point on the surface x+4y--z² =1 where the tangent plane is parallel to the plane x+4y- z =…

A: we need to find gradient of given surface to know about direction of normal vector to tangent plane

Q: Find the absolute extrema of the function f(x, y) = x + y + 3.ry on the closed region bounded by the…

A: Absolute Maximum and Absolute Minimum

Q: Bounded by the curve y = (x + 3)², the x axis, and the lines x =-3 and x= 0. %3D

A:

Q: 7. Derive the orthogonal trajectories of the family of hyperbolas xy = c. %3D

A: The solution are next step

Q: 21 コミメ-2 of the with the Parabola y=z Find the hく line area region bet ween the

A: Let us consider the double integration I=∫D∫f(x,y) dxdy where  D=(x,y)|a<x<b,…

Q: The tangent plane to the graph of the function f(x,y)=x+xy+y2, which is parallel to the z=1+2x+3y…

A:

Q: 5. Use traces to sketch and identify the surface x2- y? - z2 = 4.

A:

Q: rea under the curve y = 2x – x² from x = 1 to x = 2 with n = 4 %3D tonglon

A:

Q: What is the largest vertical distance between the graphs of the functions y = Va and y x2 in the…

A:

Q: 1. Find the minimum distance from the point (0, 0, 0) to the plane x- y+z = 3.

A: perpendicular distance

Q: 5. A cuboid with the dimensions of 30cm, 40cm and 50cm was heated and each of 1 the sides expanded…

A: Solve

Q: 2. The second order partial derivatives Fxx(x,y) of the function ( F(x,y) =x'y- In (x'-y)) equal .

A:

Q: 10 Draw the graph of f and its tangent plane at the given point. (Use your computer algebra system…

A: Consider the function. Graph the surface and the tangent plane at the given point. To sketch the…

Q: 4. Graph the surface 1 = y² + z² – by first sketching at least three cross sections in each of the…

A: Graph the surface by using a 3D graphing calculator like desmos. Three cross sections in xy plane…

Q: Tangent to the parabola z = x² + 3y² , x = 1 , at point ( 1, 2, 13

A: “Since you have asked multiple question, we will solve the first question for you. If you want any…

Q: (6) Find the orthogonal trajectories of the family of curves r = 3C sin 0, where C being parameter.

A: Orthogonal trajectory curve of a family of curve is defined as a curve which intersects every member…

Q: Consider a circle of radius 2 centered at the origin. Use the Lagrange multiplier method to find the…

A: It is given that circle is of radius 2 and centered at origin. The function isfx,y=3x-y+5.

Q: Find the particular soluti y' + 3x²y = 3x² %3D Initial Condition: y = 6 %3D

A:

Q: Suppose that the Second Derivatives Test is applied to a real-valued function z = f(x, y), whose…

A: Given real value function is z=f(x, y) and second derivatives are continuous on the disk D centered…

Q: Find the minimum distance from the point (−2, −2, 0) to the surface                     z = √(1 − 2x…

A: We have to find the minimum distance from the point (-2,-2,0) to thesurface z=1-2x-2y

Q: 3. Calculate the centroid of the given illustration below. y-axis 7.5ft- 4ft 2ft 5ft- х-ахis 2ft

A:

Q: Vx2 +1 – Vx² – 1 y = Vx² +1+ vx² – 1 2.x3 dy = 2x - d.x - - show that Vx4 – 1 -

A: Given: y=x2+1-x2-1x2+1+x2-1      ......(1) To Prove:  dydx=2x-2x3x4-1      ......(2)

Q: A heart-shaped graph in a 2D plane is defined by the equation (x² + y² - 1)³= 4x²y³ Determine dy/dx…

A:

Q: What is the largest vertical distance between the graphs of the functions y = v and y = x in the…

A:

Q: 9. Determine the position of the centroid of a sheet of metal formed by the curve y = 4x - x² which…

A:

Q: A bug crawls on the surface z = x2 - y2 directly above a path in the xy-plane given by x = ƒ(t) and…

A: We are given that the bug crawls on the surface z = x2 - y2 and x = ƒ(t) and y = g(t).  So,…

Q: Find the appropriate parametrization for the given piecewise smooth-curvein R^2, with the implied…

A: Here, a semi-circle is of radius 3.

Q: 4. Find the local maximum. minimum and saddle points of f(r,y) =1-2y+ In Vr2+y2 +3 arctan . %3D

A: Given: fx,y=x-2y+lnx2+y2+3 arctanyx

Q: 3 Find the Minimum distance from the origin to the plane X+2y-z=5

A:

Q: The tangent plane to the graph of the function f(x,y)=x+xy+y2, which is parallel to the z=1+2x+3y…

A: Here given surface is fx,y=x+xy+y2 or x+xy+y2-z=0......1 Then, consider Fx,y,z=x+xy+y2-z

Q: 9. Find all the second partial derivatives of Z; given: a) Z = xy In(3xe) b) Z = Arctan (x² /y®) %3D

A: Given functions for Z are: (a) Z=xy3ln3xey (b) Z=Arctanx2y3=tan-1x2y3. We have to find second order…

Q: 1_2 Can you please help me with this example in a step by step guide. Thank you very much!!

A: To calculate the coordinates (x,y,z) and also define the absolute extremum for the provided surface…

Q: The slope of the surface z = xy² in the x- direction at the point (2, 3) is O 12 O 8 O 1 O 9 O 10

A: The slope of the surface z = f(x,y) in the x-direction at (x0,y0) is given by [∂z/∂x](x0,y0) =…

Q: 7) Let Z.= 3x2 - y² . Find all points at which |VZ| = 6 %3D

A: There is an ellipse,  all points on the ellipse are the solutions of given equation.

Q: 2.Proof that any tangent plane for the surface F(- point = 0 passses through a fixed

A: Given below clear explanation

Q: 2. Using the method of Lagrange multiplier, evaluate the max and min of the function: f (x, y) =xy…

A:

Q: The curve y = (64x – 4x³ )m bounds the shaded section on top and the x-axis on the bottom. Determine…

A: Given curve, y=64x-4x3 The shaded region is shown as: Formula for the centroid x, y in a region is…

Q: 1. Tangent to the parabola z = x + 3y?, x = 1, at point (1,2, 13

A:

Q: 5. Find the orthogonal trajectories of the family of hyperbolas x² - 2y = ax. %3D

A:

Q: y" + 3y' = 0, y(0) = -2, y'(0) = 3

A: The solution is given as

Q: A bee was flying upward along the curve that is the intersection of z = x⁴+xy³+12 with the plane…

A:

Q: 5. Are there any points on the graph of x2 – y² – 22 = 1 where the tangent plane is parallel to the…

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Q: 5. Find the moment of inertia of a plate covering the first-quad rant region bounded by y2- x, x-:…

A: Plot the given conditions for finding the boundary limits.

Q: -8x If the infinite curve y = e8x, x 2 0, is rotated about the x-axis, find the area of the…

A: Given curve is y=e-8x, x≥0 is rotated about x axis.

Q: 4 Determine the surface area of a funnel that is generated by revolving the graph of 1 y = f(x) =…

A: Given  y=fx=x3+112xand interval 1,2

Q: 10. Find a tangent plane to z² – x² +y² = 1 at (7, –5, 5).

A: To find a tangent plane to the given curve at the point (7,-5,5)

Q: (6) Find the orthogonal trajectories of the family of curves r= 2C sin 0, where C being parameter.

A: See the detailed solution below.

Q: 14. Find a point on the surface z = 3x- y at which the tangent plane is parallel to the plane 6x +…

A: