**Consider the function** \( F(x,y) = e^{-x^2/2 - y^2/2} \) **and the point** \( P(2, -2) \).**a.** Find the unit vectors that give the direction of steepest ascent and steepest descent at \( P \).**b.** Find a vector that points in a direction of no change in the function at \( P \).---**a.** The direction of steepest ascent is \( \left( \, \Box, \Box \, \right) \).The direction of steepest descent is \( \left( \, \Box, \Box \, \right) \).**b.** Which of the following vectors points in a direction of no change of the function at \( P(2, -2) \)?- \( \text{A. } (-1, 1) \)- \( \text{B. } (0, 1) \)- \( \text{C. } (1, 0) \)- \( \text{D. } (1, 1) \)

Answered: Image /qna-images/answer/1a77d0f4-16c6-4d2b-9229-a0287901b4c1.jpg

Answered: Consider the function F(x,y) =…

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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70 P is a point on a straight line with position vector r = a + tb. Show that 7=a+2a-bt+ B² By completing the square, show that is a minimum for the point P for which t-a-b/B. Show that at this point OP is perpendicular to the line r= a + tb. (This proves the well-known result that the shortest distance from a point to a line is the length of the perpendicular from that point to the line.)
Let L₁ be the line passing through the points Q₁(-4, 5, −5) and Q₂(−1, 6, −2) and let L2 be the line passing through the point P₁ (9, 11, 3) with direction vector d = [4, 2, 2] T. Determine whether L₁ and L₂ intersect. If so, find the point of intersection.
-x²/3-²/3 Consider the function F(x,y) = e and the point P(3,3). a. Find the unit vectors that give the direction of steepest ascent and steepest descent at P. b. Find a vector that points in a direction of no change in the function at P. a. The direction of steepest ascent is
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You are on a rollercoaster, and the path of your body is modeled by a vector function r(t), where t is in seconds, the units of distance are in feet, and t = 0 represents the start of the ride. Assume the axes represent the standard cardinal directions and elevation (x is E/W, y is N/S, z is height). Explain what the following would represent physically, being as specific as possible. These are all common roller coaster shapes/behaviors and can be explained in specific language with regard to units: a. r(0)=r(120) b. For 0 ≤ t ≤ 30, N(t) = 0 c. r'(30) = 120 d. For 60 ≤ t ≤ 64, k(t) = 40 and z is constant. e. For 100 ≤ t ≤ 102, your B begins by pointing toward positive z, and does one full rotation in the normal (NB) plane while your T remains constant.
70 P is a point on a straight line with position vector r = a + tb. Show that 7=a+2a-bt+ B² By completing the square, show that is a minimum for the point P for which t-a-b/B. Show that at this point OP is perpendicular to the line r= a + tb. (This proves the well-known result that the shortest distance from a point to a line is the length of the perpendicular from that point to the line.)

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