(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 of f. Be sure you can explain why this is.) Find and simplify fon 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)? ? ✪
(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 of f. Be sure you can explain why this is.) Find and simplify fon 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)? ? ✪
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
answer only e and f
Transcribed Image Text:Consider the function f whose graph is shown below.
This function is given by
=
f(0,0)=
lim f(x, 0) =
x-0
=
f(x, y) =
{
Z
3ry
x² + y²
0,
(a) Find a formula for the single variable function f(0, y).
f(0, y) =
What is f(0, 0) for this function?
f(0,0) =
Find its limit as y → 0:
lim f(0, y)
y+0
f(x, mx) =
X
(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) =
(x, y) = (0,0)
(x, y) = (0,0).
(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 fon the ray y = x:
f(x, x) =
(Notice that this means that y = x is a contour of f. Be sure you can explain why this is.)
Find and simplify f on any ray y = 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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