g. Sketch two unit vectors u for which Duf(1, 2) = 0 and then find component representations of these vectors. h. Suppose you are standing at the point (3,3). In which direction should you move to cause f to increase as rapidly as possible? At what rate does f increase in this direction?
g. Sketch two unit vectors u for which Duf(1, 2) = 0 and then find component representations of these vectors. h. Suppose you are standing at the point (3,3). In which direction should you move to cause f to increase as rapidly as possible? At what rate does f increase in this direction?
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
Please do g and h
Transcribed Image Text:Activity 10.6.5. Consider the function f defined by
f(x, y) = -x + 2xy — Y.
a. Find the gradient ▼ƒ(1, 2) and sketch it on Figure 10.6.5.
3
2+
1
Y
2
3
Figure 10.6.5. A plot for the gradient V ƒ(1, 2).
X
4
1
b. Sketch the unit vector z = (-1/2) -√2) on Figure 10.6.5 with its tail at
√2
(1,2). Now find the directional derivative D₂ƒ(1,2).
c. What is the slope of the graph of f in the direction z? What does the sign
of the directional derivative tell you?
d. Consider the vector v = (2, -1) and sketch v on Figure 10.6.5 with its
tail at (1, 2). Find a unit vector w pointing in the same direction of v.
Without computing Dwƒ(1, 2), what do you know about the sign of this
directional derivative? Now verify your observation by computing
Dwf(1,2).
e. In which direction (that is, for what unit vector u) is D„ƒ(1, 2) the
greatest? What is the slope of the graph in this direction?
f. Corresponding, in which direction is Däƒ(1, 2) least? What is the slope of
the graph in this direction?
Transcribed Image Text:Figure 10.6.5. A plot for the gradient Vƒ(1, 2).
b. Sketch the unit vector z = ( - 1/2 - 1/12) on Figure 10.6.5 with its tail at
√2' √2
(1, 2). Now find the directional derivative D₂ƒ(1,2).
c. What is the slope of the graph of f in the direction z? What does the sign
of the directional derivative tell you?
d. Consider the vector v = (2, -1) and sketch v on Figure 10.6.5 with its
tail at (1, 2). Find a unit vector w pointing in the same direction of v.
Without computing Dwƒ(1, 2), what do you know about the sign of this
directional derivative? Now verify your observation by computing
Dwf(1,2).
e. In which direction (that is, for what unit vector u) is D.ƒ(1, 2) the
greatest? What is the slope of the graph in this direction?
f. Corresponding, in which direction is Däƒ(1, 2) least? What is the slope of
the graph in this direction?
g. Sketch two unit vectors u for which Duf(1, 2) = 0 and then find
component representations of these vectors.
h. Suppose you are standing at the point (3, 3). In which direction should
you move to cause f to increase as rapidly as possible? At what rate does
f increase in this direction?
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
This is a popular solution!
Trending now
This is a popular solution!
Step by step
Solved in 3 steps with 16 images
Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
