(a) Find a formula for the single variable function f(0, y). f(0, y) = What is f(0, 0) for this function? ƒ(0,0) = Find its limit as y → 0: limf(0, y) = y→0° (b) Based on your work in (a), is the single variable function f(0, y) continuous? (c) Next, similarly consider f(x, 0). f(x,0) = f(0,0) = limf(x, 0) = x-0 (d) Based on this work in (a), is the single variable function f(x, 0) continuous? (e) Finally, consider f along rays emanating from the origin. Note that these are given by y = mx, for some (constant) value of m. Find and simplify f on the ray y = x: f(x,x) = = (Notice that this means that y = x is a contour off. Be sure you can explain why this is.) Find and simplify f on any ray y = mx. f(x, mx) = (Again, notice that this means that any ray y = mx a contour of f; be sure you can explain why.) (f) Is f(x, y) continuous at (0, 0)?
(a) Find a formula for the single variable function f(0, y). f(0, y) = What is f(0, 0) for this function? ƒ(0,0) = Find its limit as y → 0: limf(0, y) = y→0° (b) Based on your work in (a), is the single variable function f(0, y) continuous? (c) Next, similarly consider f(x, 0). f(x,0) = f(0,0) = limf(x, 0) = x-0 (d) Based on this work in (a), is the single variable function f(x, 0) continuous? (e) Finally, consider f along rays emanating from the origin. Note that these are given by y = mx, for some (constant) value of m. Find and simplify f on the ray y = x: f(x,x) = = (Notice that this means that y = x is a contour off. Be sure you can explain why this is.) Find and simplify f on any ray y = mx. f(x, mx) = (Again, notice that this means that any ray y = mx a contour of f; be sure you can explain why.) (f) Is f(x, y) continuous at (0, 0)?
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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Question
Transcribed Image Text:f(x, y)
=
N
xy
x²+y²
0,
{
(x, y) = (0, 0)
(x, y) = (0, 0).
Transcribed Image Text:(a) Find a formula for the single variable function f(0, y).
f(0, y) =
What is f(0, 0) for this function?
ƒ(0,0) =
Find its limit as y → 0:
lim f(0, y):
=
y→0°
(b) Based on your work in (a), is the single variable function f(0, y) continuous?
(c) Next, similarly consider f(x, 0).
f(x,0) =
f(0, 0) =
limf(x, 0) :
=
x→0
(d) Based on this work in (a), is the single variable function f(x, 0) continuous?
(e) Finally, consider f along rays emanating from the origin. Note that these are given by y = mx, for some (constant) value of m.
Find and simplify f on the ray y = x:
f(x,x) =
(Notice that this means that y = x is a contour off. Be sure you can explain why this is.)
Find and simplify f on any ray y = mx.
f(x, mx) =
(Again, notice that this means that any ray y = mx is a contour of f; be sure you can explain why.)
(f) Is f(x, y) continuous at (0, 0)?
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