b) If we are at the point (1, 2, 4) on the surface z = xy², and we move a short distance in the direction ( ), my new z - value will still be %3D approximately 4.

Q: 4. Find the equations of the line tangent to the intersection of the surfaces z = x´ + 2y and

A:

Q: Find an equation of the plane tangent to the following surface at the given point. z=6-3x- 3y; (2,3,…

A:

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

A: The solution are next step

Q: Find the equation of the plane tangent to the following surface at the given points. x² +…

A:

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

A:

Q: 4. Where does the curve r(t) = ti + t²j+ (3 t2)k intersect the paraboloid z = x2 +y² ?

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Q: B Suppose we want to find all the points on the surface gx, y) x+ 4y - 4 where the tangent plane is…

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Q: Find the points on the surface x2 - zy = 4 closest to the origin.

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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: 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: s Compute yz dz + zzdy + zydz along the curve (t, t2, t³), 0 st<1

A:

Q: Find the equation of the tangent plane to the surface z = 4 +y? at the point (2,-3, 10) Ox+ 4y- z =…

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 orthogonal trajectories of the family of hyperbolas xy = c

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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 the points on the surface z2 = xy + 4 closest to the origin.

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Q: 8. Calculate fo F • dr, where F(x, y) = y² i + (6x + 6y)j, and C is the portion of the parabola 2y²…

A: Given curve is 2y2-x=1 and points are (1,-1) to (7,2)

Q: Find the equation of the plane tangent to the surface of Z= x2+ Y2 at point P(125)

A:

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: Identify a generating curve on the xz-plane for the surface of revolution with equation x2 + y? + 22…

A: Solution :-

Q: Find a generating curve and the axis of revolution for the surface                             x2 +…

A: Find a generating curve and the axis of revolution for the surface   x2+3y2+z2 =9 If the equation of…

Q: 2. For the surface z = In(y+ Vx² + y²). (b) Find the linear approximation at the point (3,-4).

A:

Q: (8) Find the point(s) where the line r(t) = ti+(1+t)j+5k intersects the paraboloid z = x² + y?.

A: A detailed solution is given below.

Q: If the equation of the tangent plane to x2+y2-11z2=0 at (1,1, /2/11) is x+ay+Bz+y=0, then a+B+y=D

A: Given query is to find the equation of the tangent to the plane.

Q: 17. Find the area of one side of the "winding wall" standing orthogonally on the curve x = y°, 0 < y…

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: 4. Find an equation of the tangent plane to the surface z=- xy+8 at the point (1,1,-10) ху — 2х

A: Equation of tangent plane is: {df(x0,y0)/dx} (x-x0)+{df(x0,y0)/dy} (y-y0)-(z-z0)=0

Q: 9. Show that the sum of the T-, y-, and z- intercepts of any tangent plane to the surface VI + Vỹ +…

A: Given, Gradient of the surface at any point (x0,y0,z0) is given by,

Q: If the equation of the tangent plane to x2+y2-147z²=0 at (1,1,/2 / 147) is x+ay+Bz+y=0, then a+ß+y=

A:

Q: If the equation of the tangent plane to x2+y²-83z2=0 at (1,1,/2/83) is x+ay+Bz+y=0, then a+ß+y=

A:

Q: The surface of a hill above the xy-plane is described implicitly by 3x² + y² + (z + 1)² = 2, z ≥ 0.…

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 an equation of the plane tangent to the following surface at the given points. z =6 cos (x- y)+…

A:

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

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Q: Find the equation for the tangent plane to the surface z= In (7x + 10y + 1) at the point (0,0,0).

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Q: Find the points on the surface 1x^2+2y^2+5z^2=1 at which the tangent plane is parallel to the plane…

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Q: Find the equation of the tangent plane to the surface 9x +36y +4z? = 108 at the point (-2,1,-3).…

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Q: Find an equation of the plane tangent to the surface z2 - xy = 14 at (2, 1, 4).

A:

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: Find an equation of the plane tangent to the surface f(x, y) = xy² – 5x at the poin (1, 2, –1).

A: Consider the provided question, Hello. Since your question has multiple parts, we will solve first…

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

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

A:

Q: Are there any points on the hyperboloid r² – y² – 2² =1 where the tangent plane is parallel to the…

A:

Q: Find a generating curve on the xz-plane for the surface of revolution with equation x²+ y²+ z²= e2x

A: Solution: Given equation is x^2+y^2+z^2=e^(2x)

Q: At which points does the curve r(t)=ti+(2t -t² )k intersect the paraboloid z = x² +y? ?

A: Given: r→t=ti^+2t-t2k^ and z=x2+y2

Q: If the two spheres (x-4)^2+(y-1)^2+z^2=27 and (x-13)^2+(y-10)^2+(z+9)^2=a are externally tangent to…

A: Consider the given two spheres x-42+y-12+z2=27 and x-132+y-102+z+92=a are externally tangent to each…

Q: Let F(x, y) that starts at (0, 0)and ends at (1, 2). (6æ? + 4y) 7 + (4x + 3) j. Calculate la F . dř,…

A:

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

A: