Question

Consider the function f(x,y)=y+sin(x/y)

a) Find the equation of the tangent plane to the graph offat the point(1,3)

b) Find the linearization of the function f at the point(1;3)and use it to approximate f(0:9;3:1).

c) Explain why f is differentiable at the point(1;3)

d)Find the differential of f

e) If (x,y) changes from (1,3) to (0.9,3.1), compare the values of ‘change in f’ and df

Answer #1

8).
a) Find an equation of the tangent plane to the surface z = x at
(−4, 2, −1).
b) Explain why f(x, y) = x2ey is differentiable at (1, 0). Then
find the linearization L(x, y) of the function at that point.

1. Find the equation of the tangent plane to the function f ( x,
y ) = x^2 + y^2 at the point (1,1).
2. Find a different solution to Laplace's equation.

Find the linearization of the function f (x y) =
arctan (y / x) at point (1,1) and the tangent plane at that
point.

Consider the function f(x,y)= (x+2*y)*e^(x-2y) for all real
values (x,y).
Determine the linearization to f at the point (2,1)
Use the linearization to approximate f(2.1,1.1)

Let f(x,y)=3xy−5x2−2y2
Then an equation for the tangent plane to the graph of ff at the
point (3,3)(3,3) is

Consider the curve given by the equation
sin(?−?)=2?+?sin(x−y)=2x+y. The equation defines ?y as a function
?(?)y(x) near (0,0)(0,0).
(a) Find the equation of the tangent line to this curve at the
point (0,0)(0,0).
(b) Find ?″(0)y″(0).

1)Find an equation of the tangent plane to the surface given by
the equation xy + e^2xz +3yz = −5, at the point, (0, −1, 2)
2)Find the local maximum and minimum values and saddle points
for the following function: f(x, y) = x − y+ 1 xy .
3)Use Lagrange multipliers to find the maximum and minimum
values of the function, f(x, y) = x^2 − y^2 subject to, x^2 + y 4 =
16.

Find the equation of the tangent line to the graph of the
function f(x)=cot2x at the point (π/4,0).

Consider the function f(x,y) = (e^{2x})lny whose domain is
{(x,y): y>0}. What is the equation of the plane
tangent to the surface z=f(x,y) at (x,y) = (3,5)?

f(x,y)=x^2+4xy-y^2 find an equation for the tangent
plane at the surface point (2,1,11)

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