S3x² = 16-4z E B S1:2x - 4y +z = 4 DS=aQ S2:2y=6-x S4: x = 0 S5: y = 0 S6: z = 0 0(0,0,0) is the origin and is hidden g) Draw surfaces S1, S2, and S3 in Geogebra as trimmed surfaces as seen on the solid.
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- Does the sphere x2+y2+z2=100 have symmetry with respect to the a x-axis? b xy-plane?7. Graph the surface z = f (x, y) = x ^ 2 + 2 y ^ 2 - 2x + 4y + 2. Also write the reduced equation of the intersection curve of the surface with the z = 0 plane.(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?
- Determine the points of the surface M: x²+y²+z²-4x-4y+6=0 for which the distance from the axis Z has an extreme value.Consider the solid Q bounded by the surfaces S1: z - 1 = x2 , S2: x + y = 2, S3: x = 0, S4: y = 0, S5: z = 0 Let C be the boundary of the surface S1, oriented as shown in the following figure: the figure is in the first attached image the answers are in the second image(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 ∇G ≠ 0 if and only if FxGx + FyGy + FzGz = 0 atP(b) Use part (a) to show that the surfaces z2 = x2 + y2 and x2 + y2 + z2 = r2 are orthogonal at every point of intersection. Can you see why this is true without using calculus?
- Find the points on the surface x² −yz = 5 that are closest to the origin.Suppose the temperature at the point (x, y, z) on the sphere x2 + y2 + ?2 = 16 is T= 800xy2z3 . Locate the lowest and highest temperatures on the sphere.The surface S formed by the point P(x,y,z) such that the distance from P to the point A(1.0,-1) is equal to 1/3 of the distance from P to the plane π : z+3=0, is a quadratic.(a) Determine your center (or vertex) (___, ____, ____)(b) 12 being the coefficient of the term in z, what is the coefficient of the term in x of the general equation of this quadratic?(c) This quadratic represents which surface?
- Suppose that the Celsius temperature at the point (x, y, z) on the sphere x2 + y2 + z2 = 1 is T = 400xyz2. Locate the highest and lowest temperatures on the sphere.3. (a) Show that the two surfaces S1 : z = xy and S2 : z =3x^2/4 - y^2 perpendicularly at the point (2, 1, 2).b) Show that every tangent plane to the cone z^2 = x^2+y^2 must pass through the origin.Given the surface $x^3+axyz+z^3=1$ then for what value of 'a' is the surface a smooth submanifold of $R^3$. show it in step by step manner? I don't like the solution from chatgpt.