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d. In what direction, relative to Vf(zo.to), does / decrease most rapidly at the point (ro. 30)?

d. In what direction, relative to Vf(zo.to), does / decrease most rapidly at the point (ro. 30)?

Calculus For The Life Sciences
Calculus For The Life Sciences
2nd Edition
ISBN:9780321964038
Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Publisher:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Chapter6: Applications Of The Derivative
Section6.CR: Chapter 6 Review
Problem 48CR
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Bo Res Acrobat File Edit View E-Sign Window Help Home Tools Bookmarks ActiveCalcMVand... x X 世届凤 > Multivariable and Vector Functions Derivatives of Multivariable Functions Multiple Integrals Vector Calculus d 84 (88 of 244) Ď FEB 43 7 104 ActiveCalcMVand Vector-workbook-1.pdf + 75% Activity 10.6.4 In this activity we investigate how the gradient is related to the directions of greatest increase and decrease of a function. Let ƒ be a differentiable function and u a unit vector. as large as possible. € a. Let be the angle between Vf(xo, yo) and u. Use the relationship be- tween the dot product and the angle between two vectors to explain why Duf(xo, yo) = |(fx (xo, Yo), fy (xo, yo))| cos(0). (10.6.2) سلام b. At the point (xo, yo), the only quantity in Equation (10.6.2) that can change is (which determines the direction u of travel). Explain why 0 0 makes the quantity |(fx (xo, yo), fy (xo, yo))| cos(0) 1 c. When 0 = 0, in what direction does the unit vector u point relative to Vf(xo, yo)? Why? What does this tell us about the direction of greatest increase of f at the point (xo, yo)? 80,490 d. In what direction, relative to Vf(xo, yo), does f decrease most rapidly at the point (xo, yo)? O e. State the unit vectors u and v (in terms of Vƒ (xo, yo)) that provide the directions of greatest increase and decrease for the function f at the point (xo, yo). What important assumption must we make regarding Vf(xo, yo) in order for these vectors to exist? IJ 4 BEN CON GREAT -257 ០១ Tue Feb 7 8:04 AM Cue ↑
Transcribed Image Text:Bo Res Acrobat File Edit View E-Sign Window Help Home Tools Bookmarks ActiveCalcMVand... x X 世届凤 > Multivariable and Vector Functions Derivatives of Multivariable Functions Multiple Integrals Vector Calculus d 84 (88 of 244) Ď FEB 43 7 104 ActiveCalcMVand Vector-workbook-1.pdf + 75% Activity 10.6.4 In this activity we investigate how the gradient is related to the directions of greatest increase and decrease of a function. Let ƒ be a differentiable function and u a unit vector. as large as possible. € a. Let be the angle between Vf(xo, yo) and u. Use the relationship be- tween the dot product and the angle between two vectors to explain why Duf(xo, yo) = |(fx (xo, Yo), fy (xo, yo))| cos(0). (10.6.2) سلام b. At the point (xo, yo), the only quantity in Equation (10.6.2) that can change is (which determines the direction u of travel). Explain why 0 0 makes the quantity |(fx (xo, yo), fy (xo, yo))| cos(0) 1 c. When 0 = 0, in what direction does the unit vector u point relative to Vf(xo, yo)? Why? What does this tell us about the direction of greatest increase of f at the point (xo, yo)? 80,490 d. In what direction, relative to Vf(xo, yo), does f decrease most rapidly at the point (xo, yo)? O e. State the unit vectors u and v (in terms of Vƒ (xo, yo)) that provide the directions of greatest increase and decrease for the function f at the point (xo, yo). What important assumption must we make regarding Vf(xo, yo) in order for these vectors to exist? IJ 4 BEN CON GREAT -257 ០១ Tue Feb 7 8:04 AM Cue ↑
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Given that f be a differentiable function and u a unit vector

 

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