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. Graph and discuss the continuity of the function sin xy xy f(x, y) = if xy # 0 if xy = 0

. Graph and discuss the continuity of the function sin xy xy f(x, y) = if xy # 0 if xy = 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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ху + lim 21. yz² + xz² (x, y, z) (0, 0, 0) x² + y² + zª 1922x9 76 21 to anley 10,0 x² y ² z ² lim 22. (x, y, z)→(0, 0, 0) x² + y² + z² abimed bits sure, mes to pouzit limit does not exist. 23-24 Use a computer graph of the function to explain why the lim (x, y)→(0, 0) 23. 2x² + 3xy + 4y² 3x² + 5y² Su is continuous. 25-26 Find h(x, y) = g(f(x, y)) and the set of points at which h 26. g(t) = t + ln t, f(x, y) 25. g(t) = 1² + √t, f(x, y) = 2x + 3y - 6 1 - xy 1 + x²y² = 27-28 Graph the function and observe where it is discontinu- ous. Then use the formula to explain what you have observed. 27. f(x, y) = e¹/(x-y)2 xobi 28. f(x, y) = nom TRUM 1 ungu 1 - x² - y² weddines (1) mar (0 (1) xy 1 + ex-y 24. lim = 29-38 Determine the set of points at which the function is continuous. 29. F(x, y) xy³ (x, y) (0,0) x² + yo x²y³ 37. f(x, y) = 2x² + y² aten 1 + x² + y² 31. F(x, y) = 1- x² - y² - bascles vd 33. G(x, y) = √√x + √√1 - x² - y² broqu 30. F(x, y) = cos √/1 + x = y 34. G(x, y) = ln(1 + x - y) (0) 35. f(x, y, z) = arcsin(x + y + z) 36. f(x, y, z)=√√y - x² ln z 32. H(x, y) 14.3 Partial Derivatives -How notonun aid et + ex ety - 1 sulav orb gnizu (03 = (oP)g partial if (x, y) = (0, 0) if (x, y) = (0, 0) pritisch voz mno 101998 and svitelst or base ai ownerqines gaboys 101 (be) 38. f(x, y) wor bills 39. 41. 1 39-41 Use polar coordinates to find the limit. [If (r, 0) are polar coordinates of the point (x, y) with r = 0, note that r→ 0* as (x, y) → (0, 0).] lim (x,y) → (0,0) 40. lim (x,y) → (0,0) da yd zusin (zshni isen SECTION 14.3 Partial Derivatives ху x² + xy + y² 0 x³ + y³ x² + y² (x² + y²) In(x² + y²) e-x²-y² - 1 lim (x, y) (0,0) x² + y² 44. Let 42. At the beginning of this section we considered the function sin(x² + y²) x² + y² (29) B f(x, y) - →>>> and guessed on the basis of numerical evidence that (0, 0). Use polar coordinates to f(x, y) → 1 as (x, y) confirm the value of the limit. Then graph the function. 43. Graph and discuss the continuity of the function boxd f(x, y) f(x, y): -{ = if (x, y) = (0, 0) if (x, y) = (0, 0) = = 1 sin xy xy 0 if y ≤ 0 0 911 if xy # 0 if xy = 0 or or y = x¹ if 0 <y< xª 1 (a) Show that f(x, y) →0 as (x, y) → (0, 0) along any path mx with 0 < a < 4. through (0, 0) of the form y (b) Despite part (a), show that f is discontinuous at (0, 0). (c) Show that f is discontinuous on two entire curves. 45. Show that the function f given by f(x) = |x|is continuous on R". [Hint: Consider |x-a/2 = (x - a) (x-a).] 46. If c E Vn, show that the function f given by f(x) = cx is continuous on R". On a hot day, extreme humidity makes us think the temperature is higher than it really is, whereas in very dry air we perceive the temperature to be lower than the thermom- eter indicates. The National Weather Service has devised the heat index (also called the temperature-humidity index, or humidex, in some countries) to describe the combined
Transcribed Image Text:ху + lim 21. yz² + xz² (x, y, z) (0, 0, 0) x² + y² + zª 1922x9 76 21 to anley 10,0 x² y ² z ² lim 22. (x, y, z)→(0, 0, 0) x² + y² + z² abimed bits sure, mes to pouzit limit does not exist. 23-24 Use a computer graph of the function to explain why the lim (x, y)→(0, 0) 23. 2x² + 3xy + 4y² 3x² + 5y² Su is continuous. 25-26 Find h(x, y) = g(f(x, y)) and the set of points at which h 26. g(t) = t + ln t, f(x, y) 25. g(t) = 1² + √t, f(x, y) = 2x + 3y - 6 1 - xy 1 + x²y² = 27-28 Graph the function and observe where it is discontinu- ous. Then use the formula to explain what you have observed. 27. f(x, y) = e¹/(x-y)2 xobi 28. f(x, y) = nom TRUM 1 ungu 1 - x² - y² weddines (1) mar (0 (1) xy 1 + ex-y 24. lim = 29-38 Determine the set of points at which the function is continuous. 29. F(x, y) xy³ (x, y) (0,0) x² + yo x²y³ 37. f(x, y) = 2x² + y² aten 1 + x² + y² 31. F(x, y) = 1- x² - y² - bascles vd 33. G(x, y) = √√x + √√1 - x² - y² broqu 30. F(x, y) = cos √/1 + x = y 34. G(x, y) = ln(1 + x - y) (0) 35. f(x, y, z) = arcsin(x + y + z) 36. f(x, y, z)=√√y - x² ln z 32. H(x, y) 14.3 Partial Derivatives -How notonun aid et + ex ety - 1 sulav orb gnizu (03 = (oP)g partial if (x, y) = (0, 0) if (x, y) = (0, 0) pritisch voz mno 101998 and svitelst or base ai ownerqines gaboys 101 (be) 38. f(x, y) wor bills 39. 41. 1 39-41 Use polar coordinates to find the limit. [If (r, 0) are polar coordinates of the point (x, y) with r = 0, note that r→ 0* as (x, y) → (0, 0).] lim (x,y) → (0,0) 40. lim (x,y) → (0,0) da yd zusin (zshni isen SECTION 14.3 Partial Derivatives ху x² + xy + y² 0 x³ + y³ x² + y² (x² + y²) In(x² + y²) e-x²-y² - 1 lim (x, y) (0,0) x² + y² 44. Let 42. At the beginning of this section we considered the function sin(x² + y²) x² + y² (29) B f(x, y) - →>>> and guessed on the basis of numerical evidence that (0, 0). Use polar coordinates to f(x, y) → 1 as (x, y) confirm the value of the limit. Then graph the function. 43. Graph and discuss the continuity of the function boxd f(x, y) f(x, y): -{ = if (x, y) = (0, 0) if (x, y) = (0, 0) = = 1 sin xy xy 0 if y ≤ 0 0 911 if xy # 0 if xy = 0 or or y = x¹ if 0 <y< xª 1 (a) Show that f(x, y) →0 as (x, y) → (0, 0) along any path mx with 0 < a < 4. through (0, 0) of the form y (b) Despite part (a), show that f is discontinuous at (0, 0). (c) Show that f is discontinuous on two entire curves. 45. Show that the function f given by f(x) = |x|is continuous on R". [Hint: Consider |x-a/2 = (x - a) (x-a).] 46. If c E Vn, show that the function f given by f(x) = cx is continuous on R". On a hot day, extreme humidity makes us think the temperature is higher than it really is, whereas in very dry air we perceive the temperature to be lower than the thermom- eter indicates. The National Weather Service has devised the heat index (also called the temperature-humidity index, or humidex, in some countries) to describe the combined
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