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25. Use the definition of differentiability to prove that the following function is differentiable at (0,0). You must produce functions &, and ez with the required properties. f(x.y) =x+y

25. Use the definition of differentiability to prove that the following function is differentiable at (0,0). You must produce functions &, and ez with the required properties. f(x.y) =x+y

College Algebra (MindTap Course List)
College Algebra (MindTap Course List)
12th Edition
ISBN:9781305652231
Author:R. David Gustafson, Jeff Hughes
Publisher:R. David Gustafson, Jeff Hughes
Chapter3: Functions
Section3.3: More On Functions; Piecewise-defined Functions
Problem 99E: Determine if the statemment is true or false. If the statement is false, then correct it and make it...
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25. Use the definition of differentiability to prove that the following function is differentiable at (0,0). You must produce functions &, and e2 with the required properties. f(x,y) =x+y What is the first step in proving differentiablility? OA. Show that f, (0,0) f, (0,0) =f(x.y). *B. Show that f, (0,0) and f,(0,0) exist. XC. Show that ,(0,0) f(0,0) = 1. O D. Show that fy (0,0) = fy (0,0). Find f, (0,0). (0,0) =. Find fy(0,0). fy(0,0) = What is the next step? O A. Show Az = f(0 + Ax,0 + Ay) – f(a,b) = f, (0,0)Ax + fy(0,0)Ay + &1 Ax+ ɛ2Ay, where e, and e2 are functions that depend only on Ax and Ay, with (E1,82)→0 as (Ax,Ay)→0. O B. Show Az = f(0 - Ax,0 - Ay) + f(a,b) = f, (0,0)Ax + fy(0,0)Ay + &1 Ax + ɛ2Ay, where &, and 82 are functions that depend only on Ax and Ay, with (e,,82)→0 as (Ax,Ay)→0. OC. Show Az = f(0- Ax,0 - Ay)- f(a,b) = -f(0,0)Ax-f,(0,0)Ay-e, Ax- 82Ay, where ɛ and 82 are functions that depend only on Ax and Ay, with (81,82)→0 as (Ax,Ay)→0. O D. Show Az = f(0+ Ax,0 + Ay)- f(a,b) = fy (0,0)Ax-fy(0,0)Ay-(8,4x + 82Ay), where ɛ, and ɛ2 are functions that depend only on Ax and Ay, with (E1,82)→0 as (Ax,Ay)→0. Evaluate the middle expression from the equation in the previous step. Az = Find the right-most expression found for Az. Choose the correct answer below. O A. Az = - Ax-Ay-eAx- E2Ay O B. Az=Ax+ Ay+81 Ax + €2Ay OC. Az=e,Ax+ €2Ay O D. Az= -&,Ax-82Ay O E. Az = Ax-Ay-8, Ax-82Ay Set the two resulting equations equal to each other to find &, and e2. Select all that apply. O A. e, = Ay and e2 = 0 B. e, =0 and e2 = Ax
Transcribed Image Text:25. Use the definition of differentiability to prove that the following function is differentiable at (0,0). You must produce functions &, and e2 with the required properties. f(x,y) =x+y What is the first step in proving differentiablility? OA. Show that f, (0,0) f, (0,0) =f(x.y). *B. Show that f, (0,0) and f,(0,0) exist. XC. Show that ,(0,0) f(0,0) = 1. O D. Show that fy (0,0) = fy (0,0). Find f, (0,0). (0,0) =. Find fy(0,0). fy(0,0) = What is the next step? O A. Show Az = f(0 + Ax,0 + Ay) – f(a,b) = f, (0,0)Ax + fy(0,0)Ay + &1 Ax+ ɛ2Ay, where e, and e2 are functions that depend only on Ax and Ay, with (E1,82)→0 as (Ax,Ay)→0. O B. Show Az = f(0 - Ax,0 - Ay) + f(a,b) = f, (0,0)Ax + fy(0,0)Ay + &1 Ax + ɛ2Ay, where &, and 82 are functions that depend only on Ax and Ay, with (e,,82)→0 as (Ax,Ay)→0. OC. Show Az = f(0- Ax,0 - Ay)- f(a,b) = -f(0,0)Ax-f,(0,0)Ay-e, Ax- 82Ay, where ɛ and 82 are functions that depend only on Ax and Ay, with (81,82)→0 as (Ax,Ay)→0. O D. Show Az = f(0+ Ax,0 + Ay)- f(a,b) = fy (0,0)Ax-fy(0,0)Ay-(8,4x + 82Ay), where ɛ, and ɛ2 are functions that depend only on Ax and Ay, with (E1,82)→0 as (Ax,Ay)→0. Evaluate the middle expression from the equation in the previous step. Az = Find the right-most expression found for Az. Choose the correct answer below. O A. Az = - Ax-Ay-eAx- E2Ay O B. Az=Ax+ Ay+81 Ax + €2Ay OC. Az=e,Ax+ €2Ay O D. Az= -&,Ax-82Ay O E. Az = Ax-Ay-8, Ax-82Ay Set the two resulting equations equal to each other to find &, and e2. Select all that apply. O A. e, = Ay and e2 = 0 B. e, =0 and e2 = Ax
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