Consider the following. z = x2
+ y2, z = 36 − y, (6, -1, 37) (a) Find symmetric equations of the
tangent line to the curve of intersection of the surfaces at the
given point. x − 6 12 = y + 1 −2 = z − 37 −1 x − 6 1 = y + 1 12 = z
− 37 −12 x − 6 = y + 1 = z − 37 x − 6 12 = y + 1 12 = z − 37 12 x −
6 1 = y + 1 12 = z − 37 12 (b) Find the cosine of the angle between
the gradient vectors at this point. cos(θ) = State whether or not
the surfaces are orthogonal at the point of intersection.
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Consider the following. z = x2 + y2, z = 36 − y, (6, -1, 37) (a) Find symmetric equations of the tangent line to the curve of intersection of the surfaces at the given point. x − 6 12 = y + 1 −2 = z −...
2. Given the paraboloid: z + x2 + y2 = 6. a) Find the symmetric equations...
2. Given the paraboloid: z + x2 + y2 = 6. a) Find the symmetric equations of the normal line at the point (1, 2, 1). b) Find the equation of the tangent plane at the point (1, 2, 1). Simplify. c) Find the angle of inclination of the tangent plane to the xy-plane in degrees. Round to the tenths place.
2. We say that two curves intersect orthogonally if they intersect and their tangent lines are orthogonal at each point in the intersection. For example, the curve y = 0 intersects the curve x2 + y2-...
2. We say that two curves intersect orthogonally if they intersect and their tangent lines are orthogonal at each point in the intersection. For example, the curve y = 0 intersects the curve x2 + y2-1 orthogonally at (-1,0) and (1,0). Let H be the set of curves y2b with b ER. (a) Prove that the tangent line of each curve in H at a point (r, y) with y / 0 has slope (b) Let y -f(x) be a...
2. We say that two curves intersect orthogonally if they intersect and their tangent lines are orthogonal at each point in the intersection. For example, the curve y = 0 intersects the curve x2 + y2-...
2. We say that two curves intersect orthogonally if they intersect and their tangent lines are orthogonal at each point in the intersection. For example, the curve y = 0 intersects the curve x2 + y2-1 orthogonally at (-1,0) and (1,0). Let H be the set of curves y2-2.2-b with b є R. (a) Prove that the tangent line of each curve in H at a point (x, y) with y 0 has slope - (b) Let y-f(x) be a...
Question (2): (5 Marks) x-1-3-y x-1-6-y:+2 are (A) Determine intersecting or skew. If they intersect, find the point of intersection Given SI: x2-2y2 = 4z2-252 &s2: (0 Show that the tangent p...
Question (2): (5 Marks) x-1-3-y x-1-6-y:+2 are (A) Determine intersecting or skew. If they intersect, find the point of intersection Given SI: x2-2y2 = 4z2-252 &s2: (0 Show that the tangent planes to the two surfaces at P(2,0,-8) are perpendicular. whether the lines parallel, 2-z & 12 Marks] 4x2 +9y2-24. (B) Find the points on Si at which the tangent plane is parallel to the plane x+y+32-5 3 Marks]
Question (2): (5 Marks) x-1-3-y x-1-6-y:+2 are (A) Determine intersecting or...
Let z = x2 + y2 be the surface, and x = -1+t, y= 2+t, z...
Let z = x2 + y2 be the surface, and x = -1+t, y= 2+t, z = 2t + 7 be the line. Find the incorrect answer in the following 4 Select one: The acute angle between tangent to the surface and the given line at the -1 point (0,3,9) is į – cos V6 37 The normal to the surface at the point (0,3, 9) is 6 j-k. The line is normal to the surface. The line intersects at...
2. Find the slope of the tangent line to f(x, y) 6-x2 + xy - y2...
2. Find the slope of the tangent line to f(x, y) 6-x2 + xy - y2 at (4, 2), toward the point (7, 1). Then find the maximum slope and its direction
2. Find the slope of the tangent line to f(x, y) 6-x2 + xy - y2 at (4, 2), toward the point (7, 1). Then find the maximum slope and its direction
Uestion 7[value16jp (a) Find parametric equations for the tangent line to the curve of intersecti...
uestion 7[value16jp (a) Find parametric equations for the tangent line to the curve of intersection of the cvlinders y -r2 and z - r2 at the point (1, -1,1) (b) Find an equation for the osculating plane of the curve ア(t) 〈cos 3t, 4t, sin 3t) at the point (-1.4T,0).
uestion 7[value16jp (a) Find parametric equations for the tangent line to the curve of intersection of the cvlinders y -r2 and z - r2 at the point (1, -1,1) (b)...
1. 2. Which vector function represents the curve of intersection of the surfaces x = y2...
1.
2.
Which vector function represents the curve of intersection of the surfaces x = y2 and y² + x2 25 ? = Find a vector that is parallel to both of the planes 2 – y + 2z = 2 and x + y + 3z = 13.
30 6 9 Compute the slope of the line tangent to the 36 Consider the upper...
30 6 9 Compute the slope of the line tangent to the 36 Consider the upper half of the ellipsoid f(x,y) = and the point P on the level curve f(x,y) - level curve at P, and verify that the tangent line is orthogonal to the gradient at that point. 245 A. The slope is 5 OB. The slope is undefined, so the tangent line is vertical Verify that the tangent line is orthogonal to the gradient at P Select...
6 (20 pts). Let F(x, y, z) = x2 + y2 + x2 - 6xyz. (1)...
6 (20 pts). Let F(x, y, z) = x2 + y2 + x2 - 6xyz. (1) Find the gradient vector of F(x, y, z); (2) Find the tangent plane of the level surface F(x, y, z) = x2 + y2 + x2 - 6xyz = 4 at (0, 0, 2); (3) The level surface F(x, y, z) = 4 defines a function z = f(x,y). Use linear approxi- mation to approximate z = = f(-0.002,0.003).
2. Given the paraboloid: z + x2 + y2 = 6. a) Find the symmetric equations of the normal line at the point (1, 2, 1). b) Find the equation of the tangent plane at the point (1, 2, 1). Simplify. c) Find the angle of inclination of the tangent plane to the xy-plane in degrees. Round to the tenths place.
2. We say that two curves intersect orthogonally if they intersect and their tangent lines are orthogonal at each point in the intersection. For example, the curve y = 0 intersects the curve x2 + y2-1 orthogonally at (-1,0) and (1,0). Let H be the set of curves y2b with b ER. (a) Prove that the tangent line of each curve in H at a point (r, y) with y / 0 has slope (b) Let y -f(x) be a...
2. We say that two curves intersect orthogonally if they intersect and their tangent lines are orthogonal at each point in the intersection. For example, the curve y = 0 intersects the curve x2 + y2-1 orthogonally at (-1,0) and (1,0). Let H be the set of curves y2-2.2-b with b є R. (a) Prove that the tangent line of each curve in H at a point (x, y) with y 0 has slope - (b) Let y-f(x) be a...
Question (2): (5 Marks) x-1-3-y x-1-6-y:+2 are (A) Determine intersecting or skew. If they intersect, find the point of intersection Given SI: x2-2y2 = 4z2-252 &s2: (0 Show that the tangent planes to the two surfaces at P(2,0,-8) are perpendicular. whether the lines parallel, 2-z & 12 Marks] 4x2 +9y2-24. (B) Find the points on Si at which the tangent plane is parallel to the plane x+y+32-5 3 Marks]
Question (2): (5 Marks) x-1-3-y x-1-6-y:+2 are (A) Determine intersecting or...
Let z = x2 + y2 be the surface, and x = -1+t, y= 2+t, z = 2t + 7 be the line. Find the incorrect answer in the following 4 Select one: The acute angle between tangent to the surface and the given line at the -1 point (0,3,9) is į – cos V6 37 The normal to the surface at the point (0,3, 9) is 6 j-k. The line is normal to the surface. The line intersects at...
2. Find the slope of the tangent line to f(x, y) 6-x2 + xy - y2 at (4, 2), toward the point (7, 1). Then find the maximum slope and its direction
2. Find the slope of the tangent line to f(x, y) 6-x2 + xy - y2 at (4, 2), toward the point (7, 1). Then find the maximum slope and its direction
uestion 7[value16jp (a) Find parametric equations for the tangent line to the curve of intersection of the cvlinders y -r2 and z - r2 at the point (1, -1,1) (b) Find an equation for the osculating plane of the curve ア(t) 〈cos 3t, 4t, sin 3t) at the point (-1.4T,0).
uestion 7[value16jp (a) Find parametric equations for the tangent line to the curve of intersection of the cvlinders y -r2 and z - r2 at the point (1, -1,1) (b)...
1.
2.
Which vector function represents the curve of intersection of the surfaces x = y2 and y² + x2 25 ? = Find a vector that is parallel to both of the planes 2 – y + 2z = 2 and x + y + 3z = 13.
30 6 9 Compute the slope of the line tangent to the 36 Consider the upper half of the ellipsoid f(x,y) = and the point P on the level curve f(x,y) - level curve at P, and verify that the tangent line is orthogonal to the gradient at that point. 245 A. The slope is 5 OB. The slope is undefined, so the tangent line is vertical Verify that the tangent line is orthogonal to the gradient at P Select...
6 (20 pts). Let F(x, y, z) = x2 + y2 + x2 - 6xyz. (1) Find the gradient vector of F(x, y, z); (2) Find the tangent plane of the level surface F(x, y, z) = x2 + y2 + x2 - 6xyz = 4 at (0, 0, 2); (3) The level surface F(x, y, z) = 4 defines a function z = f(x,y). Use linear approxi- mation to approximate z = = f(-0.002,0.003).
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