Please do g and h Activity 10.6.5. Consider the function f defined byf(x, y) = -x + 2xy — Y.a. Find the gradient ▼ƒ(1, 2) and sketch it on Figure 10.6.5.32+1Y23Figure 10.6.5. A plot for the gradient V ƒ(1, 2).X41b. 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 signof the directional derivative tell you?d. Consider the vector v = (2, -1) and sketch v on Figure 10.6.5 with itstail 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 thisdirectional derivative? Now verify your observation by computingDwf(1,2).e. In which direction (that is, for what unit vector u) is D„ƒ(1, 2) thegreatest? What is the slope of the graph in this direction?f. Corresponding, in which direction is Däƒ(1, 2) least? What is the slope ofthe graph in this direction? 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 signof the directional derivative tell you?d. Consider the vector v = (2, -1) and sketch v on Figure 10.6.5 with itstail 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 thisdirectional derivative? Now verify your observation by computingDwf(1,2).e. In which direction (that is, for what unit vector u) is D.ƒ(1, 2) thegreatest? What is the slope of the graph in this direction?f. Corresponding, in which direction is Däƒ(1, 2) least? What is the slope ofthe graph in this direction?g. Sketch two unit vectors u for which Duf(1, 2) = 0 and then findcomponent representations of these vectors.h. Suppose you are standing at the point (3, 3). In which direction shouldyou move to cause f to increase as rapidly as possible? At what rate doesf increase in this direction?

To find unit vectors for which , we need to find vectors such that the directional derivative of…

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

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