In Problems 27 through 31, a function y = g(x) is described by some geometric property of its graph. Write a differential equation of the form dy/dx = f(x, y) having the function g as its solution (or as one of its solutions). 27. The slope of the graph of g at the point (x, y) is the sum of x and y. 28. The line tangent to the graph of g at the point (x, y) inter- sects the x-axis at the point (x/2, 0). 29. Every straight line normal to the graph of g passes through the point (0, 1). Can you guess what the graph of such a function g might look like? 30. The graph of g is normal to every curve of the form y = x2 + k (k is a constant) where they meet. 31. The line tangent to the graph of g at (x, y) passes through the point (-y, x).

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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In Problems 27 through 31, a function y = g(x) is described
by some geometric property of its graph. Write a differential
equation of the form dy/dx = f(x, y) having the function g as
its solution (or as one of its solutions).
27. The slope of the graph of g at the point (x, y) is the sum
of x and y.
28. The line tangent to the graph of g at the point (x, y) inter-
sects the x-axis at the point (x/2, 0).
29. Every straight line normal to the graph of g passes through
the point (0, 1). Can you guess what the graph of such a
function g might look like?
30. The graph of g is normal to every curve of the form
y = x2 + k (k is a constant) where they meet.
31. The line tangent to the graph of g at (x, y) passes through
the point (-y, x).
Transcribed Image Text:In Problems 27 through 31, a function y = g(x) is described by some geometric property of its graph. Write a differential equation of the form dy/dx = f(x, y) having the function g as its solution (or as one of its solutions). 27. The slope of the graph of g at the point (x, y) is the sum of x and y. 28. The line tangent to the graph of g at the point (x, y) inter- sects the x-axis at the point (x/2, 0). 29. Every straight line normal to the graph of g passes through the point (0, 1). Can you guess what the graph of such a function g might look like? 30. The graph of g is normal to every curve of the form y = x2 + k (k is a constant) where they meet. 31. The line tangent to the graph of g at (x, y) passes through the point (-y, x).
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