Question 1. Consider the surface S given by the implicit equation 2° – xy + yz + y³ =1. (a) Find the tangent plane to S at (1,0, 1). Determine the direction of greatest increase for the function g(x, y, z) = z - y – x² – sin xyz at the point (1,0, 1). Among all directions tangent to S at (1,0, 1), which gives the greatest increase for g(x, y, z)? (b)

Q: 6. Consider the parametric curve x(t) = 1 + cost+cos 2t, y(t) = sint + sin 2t = sin t(1+2 cos t)…

A: Given that                Then            At  , we need to find the value of .

Q: 5. Find an equation plane to the surface (u, v) = (3 cos(u) sin(v), 2 sin(u) sin(v), cos(u)) at the…

A: First, we find the point on the surface where the tangent plane would be drawn. Next, we calculate…

Q: 8. For the curve given by r(t)=√√² +8 i + In(t² + 8)j + tk 2 find: a. r'(t)

A: Introduction: For a vector curve rt=ati+btj+ctk, the tangent line at the point of contact t=t0 is…

Q: Consider the following curves: r₁(t) = (sint, sint, √2 cost), r2(t)=(√2 cost, √2 sint,0), t = [0,…

A: As per the guidelines repost the other question.

Q: 3. Let f(x, y) = xy+ y². Find the equation of the tangent line in parametric form in the x-direction…

A: Solve the following

Q: al Let fix,y) = xey². a) Find the equation for the tangent plane to the graph off at (12) b. What…

A:

Q: Find the equation of the plane tangent to the graph of z + xyz = 33 at (1,2,3).

A:

Q: 2. Find a set of parametric equations for the tangent line to the curve of intersection of the…

A: Step 1:To find the parametric equations for the tangent line to the curve of intersection of the…

Q: Find an equation of the plane tangent to the following surface at the given points. z=ln(1+xy)​;…

A:

Q: 4. Consider the parametric curve x = cost, y = sin cost, te [-T, π]. (a) Show that (0,0) is a point…

A: Consider the parametric curve given by  and , where  belongs to .a) To show that  is a point of…

Q: Find the equation of the tangent plane to the surface z = cos(6x) cos(3y) at the point (-2π/2,…

A: