(a) Two surfaces are called orthogonal at a point of intersection if their normal lines are perpendicular at that point. Show that surfaces with equations F( x, y, z) = 0 and G(x,y, z) = 0 are orthogonal at a point P where ∇F≠ 0 and ∇F≠ 0 if and only if FxGx +FyGy+FzGz=0 at P (b) Use part (a) to show that the surfaces z2 = x2 +y2 and x2 +y2 + z2= 12are orthogonal at every point of intersection. Can you see why this is true without using calculus?

Q: (a) Find an equation of the tangent plane to the surface at the given point. z = x2 - y2, (6, 2, 32)…

A:

Q: (a) Find an equation of the tangent plane to the surface at the given point. z = x2 - y2, (9, 2, 77)…

A:

Q: An equation of the tangent plane to the surface x = 3 y2 - z2 at the point (8, –2,2) is: (A) (x – 8)…

A:

Q: 3: For the surface f(x,y, z) = tan(xy + xz + yz), find: (1) the gradient of f; (2) the directional…

A: As per our guidelines we are supposed to answer only first three parts. kindly repost others part as…

Q: 3. (a) Show that the two surfaces S1 : z = xy and S2 : z =3x^2/4 - y^2 perpendicularly at the point…

A: As per our guideline, we are supposed to solve only first subpart. Kindly repost other subpart as…

Q: please solve

A:

Q: (a) Find an equation of the tangent plane to the surface at the given point. x + y +z = 11, (5, 4,…

A:

Q: 2. A surface S in R®(r,y, :) is defined parametrically in the form u + -1 2u (a) Find points on the…

A:

Q: 3) Let h(x, y) = xye-(x+y<), Classify all local extrema of the surface using the second derivative…

A:

Q: The plane y = 1 intersects the surface z = x4 + 5xy – yt in a certain curve. Find the slope of this…

A:

Q: Question* An equation of the tangent plane to the surface x² + xy +22²-3-0 at the point P, (1,1,1)…

A: Basic differentiation is used along with the stated formula

Q: (a) Find an equation of the tangent plane to the surface at the given point. x + y +z = 11, (4, 5,…

A:

Q: 3. Give an example of a surface defined by a function F(r, y, 2) k such that the tangent plane to…

A: Equation of tangent plane of the function, F(x,y,z) at the point (x0,y0,z0) is  Fx =…

Q: 2. ) Consider the surface x² + y²-2xy-x+3y-z=-4 at the point Po(2, -3, 18). Determine the following:…

A: To find the tangent and normal.

Q: Consider the surface x2 + y2 − 2xy − x + 4y − z2 + 4 = 0 at the point Po (2, −2, 5). Determine the…

A: This is a multi-variable calculus.

Q: An equation of the tangent plane to the surface z = 2 y? - x² at the point (2,-2,4) is: (A) 4 (x –…

A: z=f(x,y)Let F(x,y,z) =f(x,y)-zEquation of tangent plane is given by,    Fxxo,yo,zo(x-xo) +…

Q: Given the surface f(x,y)=x² - y² –xy+2021. (a) Find the critical point(s) of the surface f(x, y).…

A:

Q: (a) Find an equation of the tangent plane to the surface at the given point. z = x2 - y2, (5, 4, 9)…

A:

Q: y y = 1 (1, 1) A X=1 y=x² с X (0,0) Region A is bounded by the parabola x = y², the line y = 1, and…

A:

Q: A parametric equation of a normal line at the point A(1.2,1)to the surface f(x.y,z) z-x +2xy-y 0 is:…

A:

Q: ) Consider the surface x² + y²-2xy-x+3y-z=-4 at the point Po(2, -3, 18). Determine the following:…

A:

Q: |Consider the surface z = x' + 2y? – 4ry. 1. (a) Graph the surface using technology. Make a rough…

A:

Q: (a) Find an equation of the tangent plane to the surface at the given point. xyz = 12, (1, 4, 3) %3D…

A: Apologies for the inconvenience.   #b) In part #a) we have found that, the direction ratio of the…

Q: (a) Find an equation of the tangent plane to the surface at the given point. xyz = 12, (1, 4, 3) 12x…

A:

Q: Consider the surface f(x,y) = 3y? – 2y – 3x² + 6xy. (a) Find all critical points on the surface. (b)…

A:

Q: 5. Let f(x, y) = (a) Find the slope of the surface z = direction at the point (4, 2). (b) Find the…

A: Let's find.

Q: Tangent of x²2/3 + y2/3 + z²/3 = a²/3 surface at any point (Xo , Yo ,Zo ) Show that the sum of the…

A:

Q: This problem concerns the elliptic cone. Consider the equation x²/a² + y²/b² – z²/c² = 0. (a)…

A:

Q: 3. Consider the surface z = xln y – x²y. (a) Show that Q = (1,1, –1) lies on this surface. (b) Find…

A: The given surface is z=xlny-x2y (a) Substitute (x,y,z)=(1,1,−1) in z=xlny-x2y.…

Q: (a) Find an equation of the tangent plane to the surface at the given point. z = x2 - y2, (6, 3, 27)…

A:

Q: (a) Find an equation of the tangent plane to the surface at the given point. хуz %3D 10, (1, 2, 5)…

A: Given:  a)   The equation of surface is   F(x, y, z) = xyz – 10 = 0   given point =1,2,5 On…

Q: (a) Find an equation of the tangent plane to the surface at the given point. xyz = 8, (1, 4, 2) (b)…

A:

Q: (a) Two surfaces are called orthogonal at a point of intersection if their normal lines are…

A: a Given surfaces Fx,y,z=0 and Gx,y,z=0 Then their partial derivatives are Fx,Fy,Fz and Gx,Gy,Gz Let…

Q: Show that the surfaces are tangent to each other at the given point by showing that the surfaces…

A: The equation of the given surfaces:…

Q: (a) Find an equation of the tangent plane to the surface at the given point. z = x2 - y?, (7, 3, 40)…

A:

Q: Consider the surface f(x.y) = 7-2x - 4y (a paraboloid) and the point P(1,1,1) on the surface. a.…

A: Given surface f(x,y)=7-2x^2-4y^2

Q: Solve the problem. Find the equation for the tangent plane to the surface z = In(10x²+ 3y² + 1) at…

A:

Q: 14. Let II be the plane tangent to the implicit surface x? + y° – e*+2y = 4 + x sin(z) at the point…

A: According to question given that implicit surface x2+y3-ex+2y=4+xsin(z)Point are=(-2,1,0 )

Q: (a) Find an equation of the tangent plane to the surface at the given point. хуz %3D 6, (1, 2, 3)…

A:

Q: (a) Find the point of intersection of R(t) = ti+t²j+2t²k (0<t< 3) and the surface z = 16-x2 – y. %3D…

A:

Q: 7. Consider the surface S defined by the equation x³ +y2 – 3x – 2y – 4z = -2. Find all points P =…

A:

Q: (a) Find an equation of the tangent plane to the surface at the given point. x+y+z= 12, (4, 3, 5)…

A: Our Aim is to find :-(a) An equation of the tangent plane to the surface at the given point.Here…

Q: The equation of the tangent plane to the surface f (x,y,z) = xecosz - z - 1= 0 at the point…

A: The answer is as follows :

Q: Graph the surface z = f (x, y) = x ^ 2 + 2 y ^ 2 - 2x + 4y + 2. Also write the reduced equation of…

A:

Q: (a) Find an equation of the tangent plane to the surface at the given point. x2 + y2 + z2 = 26, (1,…

A:

Q: The state of stress at a point with respect to a Cartesian coordinates system „X1X2X3is given by: 2…

A: Given - The state of stress at a point with respect to a Cartesian coordinates system x1x2x3 is…

Q: A parametric equation of a normal line at the point A(1,2,1)to the surface f(x, y,z) = z- x? +…

A: The given function is f(x,y,z)=z-x2+2xy-y2=0. We have to calculate the parametric equation of normal…

Q: The curve C consists of the portion of the parabola y =x^2-1 joining (1,0) to (2,3); followed by the…

A: Work done Work done=∫F.dr                 =∫abF.drdtdt Consider the curve C Work done=∫F.dr…