Problem 1 - Study the continuity of the following function. Problem 2 - Calculate the following limits, if they exist, by using a combination of polar coordinates and L'Hopital rule.Problem 3 - Calculate the differential of the following functions. Problem 4 - Calculate the directional derivative at each assigned vector of the following functions. Problem 5 - Find the dw/dot of the following function by using an appropriate chain rule.Problem 6 - Classify the critical points of the following function (in other words say if they are maximum, minimum or saddle points). Problem 7 - Find, if they exist, the extrema of the function. Poblem A- Study the continuity of the followingfunctionf(x,y)(x.y)#(0;0)xエ+yProblem 2- kly Calculate the following limits,if they exist, bypolar coordinatesusing a combination ofand L'Hopital rule.1-cos (x2+y?)a) lim(xiy)>(0;0)b) lim(x,y)>(0;0)arctan (x²+y2)Calculate the differential ofProblem 3the following functions.a) f(x,4) = 3x +2x= cas (2x +y) followingat each assigned vectorfunctions.of thea) f(x.y)=x²-3y?uミ/Va Ja2.b) f (x y)= gnt 3yt(0, 1)Problem 5- Ite Find the dw ldt of thefollowing function lby using an appropriate.Chain rule.Wミxy-3yz+x2xミニーt、2=もProblem 6 - Classify the critical points of thefollowing function (in other words say if they aremaximum, minimumf(x,y)="2x3+ 6xy+ 342or saddle points).Problem 7 - Find, if they exist, the extrema ofthe function.f(x,y)=x²ry2-1Subject to the constraint x-y%=2, by usingthe Lagrange's multipliers method.

Answered: Poblem 4 - Study the continuity of the…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Problem 1 - Study the continuity of the following function. Problem 2 - Calculate the following limits, if they exist, by using a combination of polar coordinates and L'Hopital rule. Problem 3 - Calculate the differential of the following functions. Problem 4 - Calculate the directional derivative at each assigned vector of the following functions. Problem 5 - Find the dw/dot of the following function by using an appropriate chain rule. Problem 6 - Classify the critical points of the following function (in other words say if they are maximum, minimum or saddle points). Problem 7 - Find, if they exist, the extrema of the function.

Poblem A- Study the continuity of the following
function
f(x,y)
(x.y)#(0;0)
xエ+y
Problem 2- kly Calculate the following limits,
if they exist, by
polar coordinates
using a combination of
and L'Hopital rule.
1-cos (x2+y?)
a) lim
(xiy)>(0;0)
b) lim
(x,y)>(0;0)
arctan (x²+y2)
Calculate the differential of
Problem 3
the following functions.
a) f(x,4) = 3x +2x
= cas (2x +y)

following
at each assigned vector
functions.
of the
a) f(x.y)=x²-3y?
uミ/Va Ja
2.
b) f (x y)= gnt 3yt
(0, 1)
Problem 5- Ite Find the dw ldt of the
following function lby using an appropriate.
Chain rule.
Wミxy-3yz+x2
xミ
ニーt、2=も
Problem 6 - Classify the critical points of the
following function (in other words say if they are
maximum, minimum
f(x,y)="2x3+ 6xy+ 342
or saddle points).
Problem 7 - Find, if they exist, the extrema of
the function.
f(x,y)=x²ry2-1
Subject to the constraint x-y%=2, by using
the Lagrange's multipliers method.

Expert Solution

Step 1

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We need to examine the continuity of the given function .

A function $f (x, y)$ is continuous at a point $(a, b)$ if it satisfies the following conditions :

$(i) \lim_{(x, y) \to (a, b)} f (x, y)$ exist .

$(i i) f (a, b)$ exist .

$(i i i) \lim_{(x, y) \to (a, b)} f (x, y) = f (a, b)$

If any of the above condition is not satisfied then the given function will not be continuous .

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