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Consider the function whose graph is shown below. This function is given by (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 (b) Based on your work in (a), is the single variable function f(0, y) continuous? yes (c) Next, similarly consider f(x, 0). f(x,0) = f(0,0) = lim f(x,0) = x→0 (d) Based on this work in (a), is the single variable function f(x, 0) continuous? yes ✓ (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 f on any ray y = mx. f(x, mx) = 2xy f(x, y) = ²² 0, (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)? no V (x, y) = (0,0) (x, y) = (0,0).

Consider the function whose graph is shown below. This function is given by (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 (b) Based on your work in (a), is the single variable function f(0, y) continuous? yes (c) Next, similarly consider f(x, 0). f(x,0) = f(0,0) = lim f(x,0) = x→0 (d) Based on this work in (a), is the single variable function f(x, 0) continuous? yes ✓ (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 f on any ray y = mx. f(x, mx) = 2xy f(x, y) = ²² 0, (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)? no V (x, y) = (0,0) (x, y) = (0,0).

Advanced Engineering Mathematics
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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Consider the function f whose graph is shown below. This function is given by (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 (b) Based on your work in (a), is the single variable function f(0, y) continuous? yes (c) Next, similarly consider f(x, 0). f(x,0) = f(0,0) = lim f(x,0) = I→0 (d) Based on this work in (a), is the single variable function f(x, 0) continuous? yes ✓ (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 f on any ray y = mx. f(x, mx) = (Again, notice that this means that any ray y (f) Is f(x, y) continuous at (0, 0)? no V ma is a contour of f; be sure you can explain why.) f(x, y) = - {** 0, 2² (2,y) (0,0) (x, y) = (0,0).
Transcribed Image Text:Consider the function f whose graph is shown below. This function is given by (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 (b) Based on your work in (a), is the single variable function f(0, y) continuous? yes (c) Next, similarly consider f(x, 0). f(x,0) = f(0,0) = lim f(x,0) = I→0 (d) Based on this work in (a), is the single variable function f(x, 0) continuous? yes ✓ (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 f on any ray y = mx. f(x, mx) = (Again, notice that this means that any ray y (f) Is f(x, y) continuous at (0, 0)? no V ma is a contour of f; be sure you can explain why.) f(x, y) = - {** 0, 2² (2,y) (0,0) (x, y) = (0,0).
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