a) Find an equation of the plane tangent to the graph of f at the given point P. Write your answer in the form ax + by +cz= d, where a, b, c, and d are integers with no common factor, and a is greater or equal to 0. f(x,y)= 2x3y + 4x-y, P(1,3,7)
b) Use a multivariable chain rule to find a formula for the given derivative or partial derivative. w= f(x,y), x=g(u,v), y=h(u,v); ∂w/ ∂v
1a.Find the equation of the tangent plane to the surface √ x +
√y + √ z = 4 at P(1, 1, 4).
1b.Let f be a function of x and y such that fx = 3x − 5y and fy
= 2y − 5x, which of the following is always TRUE?
a. (0, 0) is not a critical point of f.
b. f has a local minimum at (0, 0)
c. f has a local maximum at (0, 0)...
Find the equation of the tangent plane to the surface,
(4x^2)(y^3) + (5yz) + (2xz^3) = 7
at the point P(-1,1,1). Also nd the parametric equation of the
normal
line to that surface at that point . Sketch a picture that
illustrates what this
is all about.
2.
(a) Find an equation of the tangent plane to the surface
x4 +y4 +z4 = 18 at (2, 1, 1). Find a
derivative in direction (2,2,1) at point (2,1,1). (b) Use Lagrange
multipliers to find the minimum and maximum values of f(x,y,z) = 8x
+ y + z on the surface x4 + y4 + z4 = 18.
a. consider the plane with equation -x+y-z=2, and let p be the
point (3,2,1)in R^3. find the distance from P to the plane.
b. let P be the plane with normal vector n (1,-3,2) which passes
through the point(1,1,1). find the point in the plane which is
closest to (2,2,3)