(a) Consider the contour plot for f(x, y) shown below. Use the contour plot to determine the sign Y P km 1 k=2 k=3 of each of the following derivatives of f at P. For each, write one sentence explaining your answer. (Assume the x and y axes have the same scale.) (i) fx (P) (ii) fy(P), (iv) Duf(P) where u = (1,1). (iii) Duf (P) where u = (-2,-1) k=4 k=5 k=6
(a) Consider the contour plot for f(x, y) shown below. Use the contour plot to determine the sign Y P km 1 k=2 k=3 of each of the following derivatives of f at P. For each, write one sentence explaining your answer. (Assume the x and y axes have the same scale.) (i) fx (P) (ii) fy(P), (iv) Duf(P) where u = (1,1). (iii) Duf (P) where u = (-2,-1) k=4 k=5 k=6
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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Transcribed Image Text:1. Provide a brief explanation or calculation to justify your answer:
(a) Consider the contour plot for f(x, y) shown below. Use the contour plot to determine the sign
P
●
x
k=1
2
k=3
of each of the following derivatives of f at P. For each, write one sentence explaining your
answer. (Assume the x and y axes have the same scale.)
(i) fx (P)
(ii) fy(P),
(iii) Duf (P) where u = (-2,-1)
(iv) Duf(P) where u = (1, 1).
(b) The function f(x, y) =
xxx + y² and its partial derivatives are continuous in the region defined
by x > 0 and y € R. Calculate fxxxxxy. (Note: x** = x(xª), not (x²)ª.)
(c) Let g(x, y) be a function of x and y with the following data at different points (x, y). Suppose
further that x and y are both functions of u and v with the following data at different points
(u, v). Calculated at the point (u, v) = (1, 2).
g(1,2) g(5,3) 9x (1,2) 9y (1,2)
2 -1
3
9x (5,3) 9y (5,3)
-2
4
2
x(1,2)
5
xu (1,2) xv(1,2) xu (5,3)
-2
-1
-1
xv(5,3)
3
y(1,2) Yu (1, 2) Yv(1,2) Yu (5,3)
3
Yv(5,3)
3
2
0
1
k=4
k=5
k=6
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