Question

Let ?(?, ?) = ??^{2} + ln(??).

1. Write the equation of the tangent plane to the surface at (1, 1).

2. If ?(?,?)=?^(2?+?) and ?(?,?)=?+?,find **??/?u**
and **??/?v** if ?=1 and ?=−2.

3. In what direction would ??⃑ (1, 1) be steepest? What is that value?

Answer #1

(a) Find an equation of the plane tangent to the surface xy ln x
− y^2 + z^2 + 5 = 0 at the point (1, −3, 2)
(b) Find the directional derivative of f(x, y, z) = xy ln x −
y^2 + z^2 + 5 at the point (1, −3, 2) in the direction of the
vector < 1, 0, −1 >. (Hint: Use the results of partial
derivatives from part(a))

Let f(x, y) = x tan(xy^2) + ln(2y). Find the equation of the
tangent plane at (π, 1⁄2).

Find an equation of the tangent plane to the surface given
parametrically by x = u^2, y = v^2, z = u+4v at the point (1, 4,
9).

Find the equation of the tangent plane to the surface given by z =
ln (2tan x - tan y) at (pi/4, pi/4, 0).

Find the equation of the tangent plane and normal line
to the curve 2z−5=ln( x 2 y 3 )− 6y x 2z−5=ln(x2y3)−6yx at
(1,e)

Find the equation of the tangent plane (in terms of x, y and z)
to the surface given by x = u, y = v and z = uv at the point (3, 2,
6).

Find the equation of the tangent plane to the surface
z=e^(−2x/17)ln(3y) at the point (−2,4,3.1441).

Find an equation of the tangent plane to the parametric surface
r=(u,v)=ucosv I +usinv j +vk at u=1, v=pi/3
Find the surface area of the parametric surface r(u,v)=5sinucosv
I + 5sinusinv j+ 5cosu k, for 0 ,<= u <=pi and o<=v<=
2pi

Find an equation of the tangent plane to the surface x y 2 + 3 x
− z 2 = 4 at the point ( 2 , 1 , − 2 ) An equation of the tangent
plane is

let S be the surface defined by x^4-2x^2y^2+3z^2=12, Find the
equation of the tangent plane to the surface S at (0,1,2).

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