Question

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)

Answer #1

For f(x,y)=ln(x^2−y+3). -> Find the domain
and the range of the function z=f(x,y).
-> Sketch the domain, then
separately sketch three distinct level curves.
-> Find the linearization of
f(x,y) at the point
(x,y)=(4,18).
-> Use this linearization to determine the
approximate value of the function at the point (3.7,17.7).

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

Let f(x,y) = sqrt(22−2x^2−y^2). Find the linearization of the
function f at (1,2) and use it to approximate f(1.1,2.1).

6. Consider the function f defined by f (x, y) = ln(x
− y). (a) Determine the natural domain of f. (b) Sketch the level
curves of f for the values k = −2, 0, 2. (c) Find the gradient of f
at the point (2,1), that is ∇f(2,1). (d) In which unit vector
direction, at the point (2,1), is the directional derivative of f
the smallest and what is the directional derivative in that
direction?

Find the linearization of the function f(x,y) = √xy at the point
P(1,1) to approximate f(4/5, 11/10).

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

Determine the absolute maximum and minimum values of the
function f(x)= e^-x^2 over the interval [-2,1]

Determine the absolute maximum and minimum values of the
function f(x)= e^-x^2 over the interval [-2,1].

Find the linearization of the function
f(x,y)=√(22−1x2−3y2 )at the point (-1,
2).
L(x,y)=_______
Use the linear approximation to estimate the value of
f(−1.1,2.1)=_________

consider the 2 variable function f(x,y) = 4 - x^2 - y^2 - 2x -
2y + xy
a.) find the x,y location of all critical points of f(x,y)
b.) classify each of the critical points found in part a.)

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