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(d) Your next task is to find a curve which passes through (3, 5) in such a way that the gradient Vf(x, y) is tangent to the curve at every point on the curve, not just at (3, 5). Notice that your vector in part (b) NOT part (c) - has a vertical component, let's call that the "rise", and a horizontal component, which may as well be the "run". Now you know the slope of the tangent line to your dy curve at the point (3,5). From Calc 1, you know the slope is the derivative You now have da a (first-order) differential equation in x and y. Can you solve this equation? If not, that's okay. WolframAlpha will do that for you. What is the solution to this differential equation? (e) Your solution should have an arbitrary constant, C. Now, here's where the point (x, y) = (3,5) comes in. Use that (as your initial condition) to find your value of C.

(d) Your next task is to find a curve which passes through (3, 5) in such a way that the gradient Vf(x, y) is tangent to the curve at every point on the curve, not just at (3, 5). Notice that your vector in part (b) NOT part (c) - has a vertical component, let's call that the "rise", and a horizontal component, which may as well be the "run". Now you know the slope of the tangent line to your dy curve at the point (3,5). From Calc 1, you know the slope is the derivative You now have da a (first-order) differential equation in x and y. Can you solve this equation? If not, that's okay. WolframAlpha will do that for you. What is the solution to this differential equation? (e) Your solution should have an arbitrary constant, C. Now, here's where the point (x, y) = (3,5) comes in. Use that (as your initial condition) to find your value of C.

Calculus: Early Transcendentals
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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using the quadratic function: 3x^2+2xy+4y^2+5x+6y+7

(d) Your next task is to find a curve which passes through (3, 5) in such a way that the gradient Vf(x, y) is tangent to the curve at every point on the curve, not just at (3, 5). Notice that your vector in part (b) NOT part (c) - has a vertical component, let's call that the "rise", and a horizontal component, which may as well be the "run". Now you know the slope of the tangent line to your dy curve at the point (3,5). From Calc 1, you know the slope is the derivative You now have da a (first-order) differential equation in x and y. Can you solve this equation? If not, that's okay. WolframAlpha will do that for you. What is the solution to this differential equation? (e) Your solution should have an arbitrary constant, C. Now, here's where the point (x, y) = (3,5) comes in. Use that (as your initial condition) to find your value of C.
Transcribed Image Text:(d) Your next task is to find a curve which passes through (3, 5) in such a way that the gradient Vf(x, y) is tangent to the curve at every point on the curve, not just at (3, 5). Notice that your vector in part (b) NOT part (c) - has a vertical component, let's call that the "rise", and a horizontal component, which may as well be the "run". Now you know the slope of the tangent line to your dy curve at the point (3,5). From Calc 1, you know the slope is the derivative You now have da a (first-order) differential equation in x and y. Can you solve this equation? If not, that's okay. WolframAlpha will do that for you. What is the solution to this differential equation? (e) Your solution should have an arbitrary constant, C. Now, here's where the point (x, y) = (3,5) comes in. Use that (as your initial condition) to find your value of C.
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