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99. You are standing at the point (1, 1, 3) on the hill whose equation is given by z = 5y – x² – y². af nocant relat |(a) If you choose to climb in the direction of steepest ascent, what is your iniri -l - the horizontal distance? 146 / 237 (b) If you decide to go straight northwest, will you be ascending or descen (c) If you decide to maintain your altitude, in what directions can you go? Answer •.. ...

99. You are standing at the point (1, 1, 3) on the hill whose equation is given by z = 5y – x² – y². af nocant relat |(a) If you choose to climb in the direction of steepest ascent, what is your iniri -l - the horizontal distance? 146 / 237 (b) If you decide to go straight northwest, will you be ascending or descen (c) If you decide to maintain your altitude, in what directions can you go? Answer •.. ...

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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Please solve question no 99

Solution I In Problems 90–93, check that the point (2, 3) lies on the curve. Then, viewing the curve as a contour of f(x, y), use grad f(2, 3) to find a vector normal to the curve at (2, 3) and an equation for the tangent line to the curve at (2, 3). 90. x² + y² = 13 91. ху %3D6 Answer 92. y = x2 – 1 93. (у — х)2 + 2 %3 ху — 3 > Answer > Solution 94. The temperature H in °Fahrenheit y miles north of the Canadian border t hours after midnight is given by H = 30 – 0.05y – 5t. A moose runs north at a speed of 20 mph. At what rate does the moose perceive the temperature to be changing? 95. At a certain point on a heated plate, the greatest rate of temperature increase, 5°C per meter, is toward the northeast. If an object at this point moves directly north, at what rate is the temperature increasing? O Answer 96. An ant is at the point (1, 1, 3) on the surface of a bowl with equation z = x² + 2y², where x and y are in cm. In what two horizontal directions can the ant move away from the point (1, 1, 3) so that its initial rate of ascent is 2 vertical cm for each horizontal cm moved? Give your answers as vectors in the plane. 97. Let T = f(x, y) = 100e-(="/2)–v represent the temperature, in °C, at the point (x, y) with x and y in meters. (a) Describe the contours of f, and explain their meaning in the context of this problem. (b) Find the rate at which the temperature changes as you move away from the point (1, 1) toward the point (2, 3). Give units in your answer. (c) In what direction would you move away from (1, 1) for the temperature to increase as fast as possible? > Answer > Solution 98. You are climbing a mountain by the steepest route at a slope of 20° when you come upon a trail branching off at a 30° angle from yours. What is the angle of ascent of the branch trail? 99. You are standing at the point (1, 1, 3) on the hill whose equation is given by z = 5y – x² - y². (a) If you choose to climb in the direction of steepest ascent, what is your inisi- the horizontal distance? -nta af naeant relat 146 / 237 (b) If you decide to go straight northwest, will you be ascending or descen (c) If you decide to maintain your altitude, in what directions can you go? (> Answer Strengthen Your Understanding 1 In Problems 100–102, explain what is wrong with the statement. 100. A function f has a directional derivative given by f:(0,0) = 31 +43. 101. A function f has gradient grad f(0, 0) = 7. Answer > Solution 102. The gradient vector grad f(x, y) is perpendicular to the contours of f, and the closer together the contours for equally spaced values of f, the shorter the gradient vector. 1 In Problems 103–104, give an example of: 103. A unit vector i such that fa(0,0) < 0, given that f,(0, 0) = 2 and f,(0, 0) = 3. > Answer 104. A contour diagram of a function with two points in the domain where the gradients are parallel but different lengths. I Are the statements in Problems 105–116 true or false? Give reasons for your answer. 105. If the point (a, b) is on the contour f(x, y) = k, then the slope of the line tangent to this contour at (a, :::
Transcribed Image Text:Solution I In Problems 90–93, check that the point (2, 3) lies on the curve. Then, viewing the curve as a contour of f(x, y), use grad f(2, 3) to find a vector normal to the curve at (2, 3) and an equation for the tangent line to the curve at (2, 3). 90. x² + y² = 13 91. ху %3D6 Answer 92. y = x2 – 1 93. (у — х)2 + 2 %3 ху — 3 > Answer > Solution 94. The temperature H in °Fahrenheit y miles north of the Canadian border t hours after midnight is given by H = 30 – 0.05y – 5t. A moose runs north at a speed of 20 mph. At what rate does the moose perceive the temperature to be changing? 95. At a certain point on a heated plate, the greatest rate of temperature increase, 5°C per meter, is toward the northeast. If an object at this point moves directly north, at what rate is the temperature increasing? O Answer 96. An ant is at the point (1, 1, 3) on the surface of a bowl with equation z = x² + 2y², where x and y are in cm. In what two horizontal directions can the ant move away from the point (1, 1, 3) so that its initial rate of ascent is 2 vertical cm for each horizontal cm moved? Give your answers as vectors in the plane. 97. Let T = f(x, y) = 100e-(="/2)–v represent the temperature, in °C, at the point (x, y) with x and y in meters. (a) Describe the contours of f, and explain their meaning in the context of this problem. (b) Find the rate at which the temperature changes as you move away from the point (1, 1) toward the point (2, 3). Give units in your answer. (c) In what direction would you move away from (1, 1) for the temperature to increase as fast as possible? > Answer > Solution 98. You are climbing a mountain by the steepest route at a slope of 20° when you come upon a trail branching off at a 30° angle from yours. What is the angle of ascent of the branch trail? 99. You are standing at the point (1, 1, 3) on the hill whose equation is given by z = 5y – x² - y². (a) If you choose to climb in the direction of steepest ascent, what is your inisi- the horizontal distance? -nta af naeant relat 146 / 237 (b) If you decide to go straight northwest, will you be ascending or descen (c) If you decide to maintain your altitude, in what directions can you go? (> Answer Strengthen Your Understanding 1 In Problems 100–102, explain what is wrong with the statement. 100. A function f has a directional derivative given by f:(0,0) = 31 +43. 101. A function f has gradient grad f(0, 0) = 7. Answer > Solution 102. The gradient vector grad f(x, y) is perpendicular to the contours of f, and the closer together the contours for equally spaced values of f, the shorter the gradient vector. 1 In Problems 103–104, give an example of: 103. A unit vector i such that fa(0,0) < 0, given that f,(0, 0) = 2 and f,(0, 0) = 3. > Answer 104. A contour diagram of a function with two points in the domain where the gradients are parallel but different lengths. I Are the statements in Problems 105–116 true or false? Give reasons for your answer. 105. If the point (a, b) is on the contour f(x, y) = k, then the slope of the line tangent to this contour at (a, :::
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