8. Calculate fo F • dr, where F(x, y) = y² i + (6x + 6y)j, and C is the portion of the parabola 2y² – x = 1 that goes from (1, -1) to (7,2).

Q: xxS y dS S is the part of the paraboloid y= x 2 + z2 that lies inside the cylinder x 2 + z2 =4

A: And the projection of S onto the xz-plane is given by the disk So using the parameterization in x…

Q: Find the point(s) of intersection of the line x = 1 – t, y = t, z =1– 2t - and the paraboloid z = 8…

A:

Q: (a) Let F = 4x27 + y} - (z – 1)k. Compute I7. dA where S is a square of side 5 units centered at…

A: Given that, the vector is F→=4x2i→+yj→-z-1k→. Determine the value of ∫SF→·dA→.…

Q: At what point does the normal to y=4x2−(2+3x)y=4x2−(2+3x) at (1,−1)(1,−1) intersect the parabola a…

A:

Q: (4) The equation of the tangent plane to the surface In = z²(x – 2y) + 3z +3 at (4, 2, –1) is 2y

A:

Q: Q4/ Find the abs.max cand abs, min Valuse of the Runctia Kcx, y) = I+xy-3x on the closed regen…

A: We have f(x, y) = 1+xy-3x ∂f∂x = y-3, ∂f∂x = 0, y-3 = 0, y = 3 ∂f∂y = x, ∂f∂y = 0, x=0 So one…

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

A: The solution are next step

Q: Find the points on the ellipse 2x + 4y = 1 where f(x.y) = xy has its extreme values.

A:

Q: Find the points on the surface x² −yz = 5 that are closest to the origin.

A: Let the point P be (x,y,z). The distance between the point P and origin is given by the formula…

Q: Evaluate SF dR given that F (x, y) = 2yî + (xy – 2x) %3D and C is the curve in the third quadrant…

A:

Q: Evaluate the line integral 32xy ds, where C is the portion of the parabola y = x² from (-1,1) to…

A:

Q: At what points does the curve  r(t) = ti + (5t − t2)k  intersect the paraboloid  z = x2 + y2

A: We have to find the intersection point

Q: Show that line d:3x-4y-5=0 is tangent to hyperbola H:x2-4y2=5 and find the point of contact of the…

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Q: Find the minimum of f(x,y) = 8x + 2y on the ellipsoid 9x² + 3y² = 2. Leave your answer correct to…

A:

Q: Find the minimum of f(x,y) = 3x + 3y on the ellipsoid 3x2 + 3y = 2. Leave your answer correct to two…

A:

Q: The orthogonal trajectories to the family of hyperbolas xy= C is given by the family 1)y=cx 2)y2 -x2…

A: We need to find the orthogonal trajectory to the family of hyperbola xy=c

Q: Find the equation of the tangent plane to the paraboloid z = 2x2 +y? at the point (1,2,6). Select…

A:

Q: Find the minimum of f(x,y) = 3x + 4y on the ellipsoid 6x2 + 2y² = 3. Leave your answer correct to…

A: Solution of the problem as follows

Q: Find an equation of the tangent plane to the surface 4x^2 + 9y^2 − z^2 = 14 at P = (2, 1, 4).

A: Given,4x2+9y2-z2=14at P=(2,1,4)

Q: A surface in R³ is defined by the equation 4z = x² + 4y².

A: Given:-                 The surface in  is given by  . Sketch the intersections (traces) of this…

Q: Find the equation of the tangent to the hyperbola x² – 3y² = 1 at the point S(2,-1). A) 2x + 3y – 1…

A:

Q: What is the shortest distance from the surface xy +12x +z^2 =144 to the origin?

A: To find the shortest distance from surface xy +12x +z2 =144  to origin.

Q: 1. Compute SF · dr where F(x, y) = (y²,–x²) and C is the part of th parabola u = 12 from (-1 1) to…

A: First take dot product of f and dr Then put value of y in terms of x Integrate and raise limit of…

Q: At what point on the paraboloid  y= x^2 +z^2is the tangent plane parallel to the plane x+2y+3z=1 ?

A:

Q: Find the equation of the plane tangent to the grapl UI z = f(x, y) = x² + 2y³. + 1 at the points in…

A:

Q: 2 Find the points on the curve ax+ 2bxy + ay = c,ay = c, 0<a<b<c,whose distance from the origin is…

A:

Q: Find the points on the curve x2 + xy + y2 = 1 in the xy-plane that are nearest to and farthest from…

A:

Q: Find the equation of the plane that passes through the point (2, 1,2) and is tangent to the surface…

A:

Q: 2. Find the orthogonal trajectories of the family of parabolas y = cx O In(x² + y² ) = c O 2r? + y²…

A: Explanation of the answer is as follows

Q: Q4/Find the abs.max and abs.min valuse of the huctu f,y) = \+xy-3x on the cosol regen oounded by the…

A:

Q: Determine the equation of the normal line to the hyperbola x - 4y- = 9 at (5, 2). 8x + 5y – 50 = 0…

A: Given equation of hyperbola is  x2-4y2=9          ...(1) We have to find the equation of the normal…

Q: 3. (a) Find the equation of the plane tangent to the surface given by f(x, y) = 2xª+2y^+7xy at the…

A:

Q: 1. Compute (1/zy, 1/(z + y))- dr along the curve (t, t2). 1<ts4

A:

Q: Find the minimum of f(x,y) = 7x + 3y on the ellipsoid 7x2 + 4y2 = 8. Leave your answer correct to…

A:

Q: 4. (a) Use the trapezoidal rule to numerically integrate from a = 0 to b= 2 /(z)-0.2+25 +3r

A:

Q: Find the equation w=T(x,y,z) of the tangent hyperplane at p. F(x,y,z)=3x²-2y²+xz², p=(1,2,-1)

A: General equation for the tangent to the plane is expressed as:…

Q: Find the equation of the tangent plane to the surface z = x² +y² + 1 at the point (1,1,3). Write the…

A: use the equation of tangent  plane for the given surface.

Q: Suppose that the edge lengths x, y, and z of a closed rectangular box are changing at the following…

A: To determine the rate at which the box’s (a) Volume, (b) Surface area and (c) diagonal length are…

Q: Find the maximum value of f(x, y) = 4xy(9 – 72x – 9y) over the triangle with vertices -- - (0,0), (5…

A: Given: fx,y=4xy9-72x-9y vertices 0,0 18,00,1

Q: Obtain the Solution x, y > 0, xy > 1 of the PDE 2y on the x+y 1 Uxy such that u = x+y 0, ux…

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Q: Find the equation of the tangent plane to the surface z =e/ In (2y) at the point (2, 4,1.643). Note:…

A:

Q: Compute y ds over the part of the parabola y =2Px, for P > 0 starting at (0,0) at (2P,2P).

A:

Q: Find the point on the curve of intersection of the paraboloid x – y + 2z = 0 that is dosest to the…

A: Here we have to find the point of intersection of the paraboloid x2+y2+2z=4 and the plane x - y + 2z…

Q: 3. Find the equation of the plane tangent to the graph of f (x, y) = 3x + 4y – xy at the point (1,…

A:

Q: Find the equations of the tangent line in the given plane. Tangent to the parabola z = x² + 3y² , x…

A: Given :                    z = x2 + 3y2      at the point (1,2,13)

Q: 3. Find the equation of the tangent plane to the surface in R³ defined by z = 2x² + y² at the point…

A:

Q: O Evaluate S. F · dr where F = (x², (xy + 4)) and c is the parabola y = x² + 1 om (0, 1) to (2, 5).

A: F=&lt;x2,(xy+4)&gt;C: y=x2+1              (0,1) to (2,5)⇒dy=2xdx ∫cF.dr= ∫cx2dx+(xy+4)dy…

Q: 16. Evaluate (x + V9) ds where C consists of two segments: the part of the parabola y = x² from…

A: In the given question we have to evaluate the integral ∫C(x+y)ds where C is the regionconsist of two…