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Let z = f(x, y) = x² + y² be the temperature at a point (x, y) = R². (a) Draw a sketch indicating the points in R2 where z = 0, where z = 1, where z = 2, and where z = 4. (b) Compute the directional derivative of f at (1, 1) in the eight directions (1,0), (1, 1), (0, 1), (−1, 1), (-1,0), (-1,-1), (0, -1), and (1,-1). (c) In which of these eight directions is the temperature (1) increasing most rapidly, (2) decreasing most rapidly, and (3) not changing?

Let z = f(x, y) = x² + y² be the temperature at a point (x, y) = R². (a) Draw a sketch indicating the points in R2 where z = 0, where z = 1, where z = 2, and where z = 4. (b) Compute the directional derivative of f at (1, 1) in the eight directions (1,0), (1, 1), (0, 1), (−1, 1), (-1,0), (-1,-1), (0, -1), and (1,-1). (c) In which of these eight directions is the temperature (1) increasing most rapidly, (2) decreasing most rapidly, and (3) not changing?

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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Let z = f(x, y) = x² + y² be the temperature at a point (x, y) = R². (a) Draw a sketch indicating the points in R2 where z = 0, where z = 1, where z = 2, and where z = 4. (b) Compute the directional derivative of f at (1, 1) in the eight directions (1,0), (1, 1), (0, 1), (-1, 1), (-1,0), (-1,-1), (0, -1), and (1, -1). (c) In which of these eight directions is the temperature (1) increasing most rapidly, (2) decreasing most rapidly, and (3) not changing? (d) How might one have deduced the result in preceding part from the sketch in part (a)?
Transcribed Image Text:Let z = f(x, y) = x² + y² be the temperature at a point (x, y) = R². (a) Draw a sketch indicating the points in R2 where z = 0, where z = 1, where z = 2, and where z = 4. (b) Compute the directional derivative of f at (1, 1) in the eight directions (1,0), (1, 1), (0, 1), (-1, 1), (-1,0), (-1,-1), (0, -1), and (1, -1). (c) In which of these eight directions is the temperature (1) increasing most rapidly, (2) decreasing most rapidly, and (3) not changing? (d) How might one have deduced the result in preceding part from the sketch in part (a)?
Expert Solution
Step 1

We know that the directional derivative of a function z=fx,y at a,b in a direction v→ is calculated as:

Dv→fa,b=fx,b,fya,b.v→v→.

We know that the function is increasing in a direction, if the directional is positive in that direction.

We know that the function is decreasing in a direction, if the directional is negative in that direction.

We know that the function is not changing in a direction, if the directional is zero in that direction.

Given that z=fx,y=x2+y2 be the temperature at a point x,y∈ℝ2.

We know that a level curve z=a is locus of all points in ℝ2, which satisfies fx,y=a.

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