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- Does the sphere x2+y2+z2=100 have symmetry with respect to the a x-axis? b xy-plane?Use sine and cosine to parametrize the intersection of the cylinders x² + y = 81 and x? + z? = 81. Find a vector parametrization r(t) = (x(1), y(t), z(1)). (Use symbolic notation and fractions where needed. Use cosine for parametrization x variable.) x(t) = y(t) = + z(t) =2. Find the directional derivative of the function f(x,y,z) = xy? – 4x²y + z? at (1, -1, 2) in the direction of the vector ü = 6î + 2ĵ + 3k.
- Find the vector equation that represents the curve of intersection of the cylinder z + y = 16 and the surface z = ze". Write the equation so the r(t) term contains a cos(t) term. z(t) %3D y(t) z(t) =Consider the following function. T: R? - R?, T(x, y) = (3x², 3xy, y²) Find the following images for vectors u = (u,, u2) and v = (v,, v2) in R? and the scalar c. (Give all answers in terms of u,, uz, Va. and c.) %3D T(u) = T(v) = 3u + 31. T(u) + T(v) = T(u + v) - CT(u) = T(cu) = Determine whether the function is a linear transformation. linear transformation not a linear transformation1. Obtain the directional derivative of: a.. f(x,y) = x²-4x³y² at the point (1,-2) in the direction of a unit vector whose angle with the semi-axis x is 1, u = cos i + sen j b. f(x,y) = x²-xy + 3y² at the point (-1,-2) in the direction of a unit vector whose angle with the semi-axis x is u = cos 0 i+sen j c. f(x,y) = x²sin y at the point (1.1) in the direction of a vector v = 3i-4j
- Find the vector equation that represents the curve of intersection of the cylinder x2 + y? = 9 and the surface z = x* + y. %3D Write the equation so the (t) term contains a cos(t) term. 2(t) = 3 cos (t) y(t) = 3 sin(t) %3D z(t) = 9 cos(t) + sin (t)3. Find a vector function that represents the curve of intersection of the paraboloid z = r+y and the cylinder r+y = 16.Find the vector equation that represents the curve of intersection of the cylinder x² + y² 4 and the surface z = x+4y. = Write the equation so the x(t) term contains a cos(t) term. x(t) = y(t) = z(t) =
- The vector equation r (u, v) = u cos vi + u sin vj + vk, 0 < v < 5x, 0 < u < 1, describes a helicoid (spiral ramp). What is the surface area? 10.25piFind a parametrization for the cylinder x2 + y2 = 1.Consider the paraboloid defined by x=4y2+4z2 . (a) Which of the following vector-valued functions gives a parametrization of this paraboloid? r(u, v) = (u, v, 4u? + 4v²) r(u, v) = (4u? + 4u², u, v) r(u, v) = (4v cos(u), 4v sin(u), 16v²) o r(u, v) = (4v cos(u), 16v², 4v sin(u)) %3D (b) Let S be the portion of the paraboloid with 1s ys3 and 0s zs 9, and let f(x,y,z)=3yz2. Set up, but do not evaluate, an iterated double integral in the dudv order equal to f(x,y,z)dS du dv.
