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11:31 Aa » Q A QuamaiVC TOMULION adout a SolutiOIl y differential equation can often be obtained from the equation itself. Before working Problems 59–62, recall the geometric significance of the derivatives dy/dx and d²y/dx². 59. Consider the differential equation dy/dx = e *. (a) Explain why a solution of the DE must be an increasing function on any interval of the x-axis. (b) What are lim dy/dx and lim dy/dx? What does this suggest about a solution curve as x → ±x? (c) Determine an interval over which a solution curve is concave down and an interval over which the curve is concave up. (d) Sketch the graph of a solution y = $(x) of the differential equation whose shape is suggested by parts (a)–(c). 60. Consider the differential equation dy/dx = 5 y. (a) Either by inspection or by the method suggested in Problems 37–40, find a constant solution of the DE. (b) Using only the differential equation, find intervals on the y-axis on which a nonconstant solution y = p(x) is increasing. Find intervals on the y-axis on which y = 4(x) is decreasing. 61. Consider the differential equation dy/dx = y(a – by), where a and b are positive constants. (a) Either by inspection or by the method suggested in Problems 37–40, find two constant solutions of the DE. (b) Using only the differential equation, find intervals on the y-axis on which a nonconstant solution y = $(x) is increas- ing. Find intervals on which y = (x) is decreasing. (c) Using only the differential equation, explain why 14 Reader Notebook Bookmarks Flashcards Contents IK ШО

11:31 Aa » Q A QuamaiVC TOMULION adout a SolutiOIl y differential equation can often be obtained from the equation itself. Before working Problems 59–62, recall the geometric significance of the derivatives dy/dx and d²y/dx². 59. Consider the differential equation dy/dx = e *. (a) Explain why a solution of the DE must be an increasing function on any interval of the x-axis. (b) What are lim dy/dx and lim dy/dx? What does this suggest about a solution curve as x → ±x? (c) Determine an interval over which a solution curve is concave down and an interval over which the curve is concave up. (d) Sketch the graph of a solution y = $(x) of the differential equation whose shape is suggested by parts (a)–(c). 60. Consider the differential equation dy/dx = 5 y. (a) Either by inspection or by the method suggested in Problems 37–40, find a constant solution of the DE. (b) Using only the differential equation, find intervals on the y-axis on which a nonconstant solution y = p(x) is increasing. Find intervals on the y-axis on which y = 4(x) is decreasing. 61. Consider the differential equation dy/dx = y(a – by), where a and b are positive constants. (a) Either by inspection or by the method suggested in Problems 37–40, find two constant solutions of the DE. (b) Using only the differential equation, find intervals on the y-axis on which a nonconstant solution y = $(x) is increas- ing. Find intervals on which y = (x) is decreasing. (c) Using only the differential equation, explain why 14 Reader Notebook Bookmarks Flashcards Contents IK ШО

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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11:31 Aa » Q A QuamaiVC TOMULION adout a SolutiOIl y differential equation can often be obtained from the equation itself. Before working Problems 59–62, recall the geometric significance of the derivatives dy/dx and d²y/dx². 59. Consider the differential equation dy/dx = e *. (a) Explain why a solution of the DE must be an increasing function on any interval of the x-axis. (b) What are lim dy/dx and lim dy/dx? What does this suggest about a solution curve as x → ±x? (c) Determine an interval over which a solution curve is concave down and an interval over which the curve is concave up. (d) Sketch the graph of a solution y = $(x) of the differential equation whose shape is suggested by parts (a)–(c). 60. Consider the differential equation dy/dx = 5 y. (a) Either by inspection or by the method suggested in Problems 37–40, find a constant solution of the DE. (b) Using only the differential equation, find intervals on the y-axis on which a nonconstant solution y = p(x) is increasing. Find intervals on the y-axis on which y = 4(x) is decreasing. 61. Consider the differential equation dy/dx = y(a – by), where a and b are positive constants. (a) Either by inspection or by the method suggested in Problems 37–40, find two constant solutions of the DE. (b) Using only the differential equation, find intervals on the y-axis on which a nonconstant solution y = $(x) is increas- ing. Find intervals on which y = (x) is decreasing. (c) Using only the differential equation, explain why 14 Reader Notebook Bookmarks Flashcards Contents IK ШО
Transcribed Image Text:11:31 Aa » Q A QuamaiVC TOMULION adout a SolutiOIl y differential equation can often be obtained from the equation itself. Before working Problems 59–62, recall the geometric significance of the derivatives dy/dx and d²y/dx². 59. Consider the differential equation dy/dx = e *. (a) Explain why a solution of the DE must be an increasing function on any interval of the x-axis. (b) What are lim dy/dx and lim dy/dx? What does this suggest about a solution curve as x → ±x? (c) Determine an interval over which a solution curve is concave down and an interval over which the curve is concave up. (d) Sketch the graph of a solution y = $(x) of the differential equation whose shape is suggested by parts (a)–(c). 60. Consider the differential equation dy/dx = 5 y. (a) Either by inspection or by the method suggested in Problems 37–40, find a constant solution of the DE. (b) Using only the differential equation, find intervals on the y-axis on which a nonconstant solution y = p(x) is increasing. Find intervals on the y-axis on which y = 4(x) is decreasing. 61. Consider the differential equation dy/dx = y(a – by), where a and b are positive constants. (a) Either by inspection or by the method suggested in Problems 37–40, find two constant solutions of the DE. (b) Using only the differential equation, find intervals on the y-axis on which a nonconstant solution y = $(x) is increas- ing. Find intervals on which y = (x) is decreasing. (c) Using only the differential equation, explain why 14 Reader Notebook Bookmarks Flashcards Contents IK ШО
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