79. Suppose that f(x) satisfies the following equation (an example of a differential equation): f"(x) = -f (x) (a) Show that f (x)2 + f'(x)2 = f (0)2 + f'(0)? for all x. Hint: Show that the function on the left has zero derivative. (b) Verify that sin x and cos x satisfy Eq. (1), and deduce that sin2 x+ cos? x = 1.

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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79. Suppose that f(x) satisfies the following equation (an example of a differential equation):
f"(x) = -f (x)
(a) Show that f (x)2 + f'(x)2 = f (0)2 + f'(0)? for all x. Hint: Show that the function on the left has zero derivative.
(b) Verify that sin x and cos x satisfy Eq. (1), and deduce that sin2 x+ cos? x = 1.
Transcribed Image Text:79. Suppose that f(x) satisfies the following equation (an example of a differential equation): f"(x) = -f (x) (a) Show that f (x)2 + f'(x)2 = f (0)2 + f'(0)? for all x. Hint: Show that the function on the left has zero derivative. (b) Verify that sin x and cos x satisfy Eq. (1), and deduce that sin2 x+ cos? x = 1.
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