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Option c, e and f are correct.
Let Select all that apply Let z =f(x,y)= arctan(3x In(6) Select all that apply Your answer: The...
(b) Let C be the closed curve formed by intersecting the cylinder x2 +y= 1 with the plane x z= 2. Let the tangent to the curve from above. point in the anti-clockwise direction when viewed Calculate the line integral (e (e sin y+ 4) dy+(e(cos z+ sin z)+ay) dz. cos x2yz) dx + (b) Let C be the closed curve formed by intersecting the cylinder x2 +y= 1 with the plane x z= 2. Let the tangent to the...
5 and 6 please 5) Given the surface f(x, y, z) = 0 or z = f(x,y), find the tangent plane at P. a) z2 – 2x2 – 2y2 = 12 @ P=(1,-1,4) b) f(x,y) = 2x - 3xy3 @ 12,-1) c) f(x,y) = sin(x) @ (3,5) 6) Find an equation of the tangent plane and the equation of the normal line to surface f(x..zb=0 @P x2 + y2 + z2 = 9 P = (2,2,1)
0 intersect only at (0,0) g(r)at z arctan(3z) Show that the graph y f(x) and its tangent line y po Consider the ftunction f(x) Intermediate steps: 1) The lIne tangent to y f(x)atz -0isy g(x) where g(r) 9(a)- 2Let H(x) f(x) - 9(x) The derivative ot H (x)s H'(z) = which is zero only when x = Rolle's theorem to H (x) on the interval [ri, 0]. Get a contradiction. 4) Now assume that we have zp O where f(2)-9(T2)...
Consider the vector field F(x, y, z) = (z arctan(y2), 22 In(22 +1), 32) Let the surface S be the part of the sphere x2 + y2 + x2 = 4 that lies above the plane 2=1 and be oriented downwards. (a) Find the divergence of F. (b) Compute the flux integral SS. F . ñ ds.
find an equation of the tangent plane and parametric equations of the normal line to the surface at the given point z=-9+4x-6y-x^2-y^2 (2,-3,4) Find the relative extrema. A) f(x, y) = x3-3xyザ B) f(x, y)=xy +-+- Find the relative extrema. A) f(x, y) = x3-3xyザ B) f(x, y)=xy +-+-
Find the area of the lateral surface over the curve C in 6. the xy-plane and under the surface z - f(x,y) f(x,y)-h, C:y-1 -x2 from (1,0) to (0,1) Surface: Lateral surface area - f(x, y) ds z =f(x, y) Lateral surface xy) As C: Curve in xy-plane Find the area of the lateral surface over the curve C in 6. the xy-plane and under the surface z - f(x,y) f(x,y)-h, C:y-1 -x2 from (1,0) to (0,1) Surface: Lateral surface...
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).
5. [12 Marks) Consider the level surface of the function f(x, y, z) defined by f(x, y, z) = x2 + y2 + x2 = 2a?, (1) where a is a fixed real positive constant, and the point u = (0,a,a) on the surface f(x, y, z) = 2a. a) Find the gradient of f(x, y, z) at the point u. b) Calculate the normal derivative of f(x, y, 2) at u. c) Find the equation of the tangent plane...
Let f(x) = z² and g(x) = (1 - 5)² + 10. There is one line with positive slope that is tangent to both of the parabolas y = f(x) and y = g(x) simultaneously. ye9bx)/ y=f/ Find the equation of the line. y= On a separate piece of paper, sketch the graph of the parabola y = 2? + 6. On the same graph, plot the point(0, – 3). Note that there are two tangent lines of y =...
Consider the following function 6 f(x, y,z)=z - x? cos(my) + xy? (i) Find the gradient of the function f(x, y, z) at the point P,(2,-1,-7). (ii) Find the directional derivative of f(x, y, z) at P,(2,-1,-7) along the direction of the vector ū = 2î+j+2k. (iii) Find the equation of the tangent plane to the surface given below at the point P,(2,-1, -7). 6 :- xcos(ty) + = 0 xy