Explanation Given: sin x ≤ x Formula used: If f and g are integrable and g ( x ) ≤ f ( x ) for x in [ a , b ] then ∫ a b g ( x ) d x ≤ ∫ a b f ( x ) d x Calculation: (a) By the comparison theorem, we have ∫ 0 x t d t ≥ ∫ 0 x sin t d t [ t 2 2 ] 0 x ≥ [ ( − cos t ) ] 0 x x 2 2 ≥ 1 − cos x 1 − x 2 2 ≤ cos x We know cos x ≤ 1 Hence , 1 − x 2 2 ≤ cos x ≤ 1 for x ≥ 0 1 − ( − x ) 2 2 ≤ cos x = cos ( − x ) ≤ 1 so , 1 − x 2 2 ≤ cos x ≤ 1 (b) Using comparison theorem, ∫ 0

3rd Edition

ISBN: 9781464125263

Author: Jon Rogawski, Colin Adams

Publisher: W.H. Freeman & Co

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Question

Chapter 5.4, Problem 52E

To determine

(a) We need to apply the comparison theorem to the inequality $\sin x \leq x$ valid for $x \geq 0$ to prove that $1 - \frac{x^{2}}{2} \leq \cos x \leq 1$ .

(b) We need to prove that $x - \frac{x^{3}}{6} \leq \sin x \leq x (for x \geq 0)$ .

(c) We need to verify the inequalities for $x = 0.3$ .

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Chapter 5.4, Problem 51E

Chapter 5.4, Problem 53E

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Prove that : (a) 2 ³ ₁ ( ² · ² ; ³ ; x ² ) = ↓, sin¯¹ x (b) 2F1 (12 3 2 +²) -tan-1 X X

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Let n and m be integers.

Chapter 5 Solutions

Calculus - Standalone book

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(a) We need to apply the comparison theorem to the inequality sin x ≤ x valid for x ≥ 0 to prove that 1 − x 2 2 ≤ cos x ≤ 1 . (b) We need to prove that x − x 3 6 ≤ sin x ≤ x ( for x ≥ 0 ) . (c) We need to verify the inequalities for x = 0.3 .

Solution Summary: The author explains the comparison theorem and the inequalities for x = 0.3.

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(a) We need to apply the comparison theorem to the inequality $\sin x \leq x$ valid for $x \geq 0$ to prove that $1 - \frac{x^{2}}{2} \leq \cos x \leq 1$ .

(b) We need to prove that $x - \frac{x^{3}}{6} \leq \sin x \leq x (for x \geq 0)$ .

(c) We need to verify the inequalities for $x = 0.3$ .

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Chapter 5 Solutions

(a) We need to apply the comparison theorem to the inequality sin x ≤ x valid for x ≥ 0 to prove that 1 − x 2 2 ≤ cos x ≤ 1 . (b) We need to prove that x − x 3 6 ≤ sin x ≤ x ( for x ≥ 0 ) . (c) We need to verify the inequalities for x = 0.3 .

Solution Summary: The author explains the comparison theorem and the inequalities for x = 0.3.

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(a) We need to apply the comparison theorem to the inequality sinx≤xvalid for x≥0to prove that 1−x22≤cosx≤1. (b) We need to prove that x−x36≤sinx≤x(for x≥0). (c) We need to verify the inequalities for x=0.3.

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Chapter 5 Solutions

(a) We need to apply the comparison theorem to the inequality $\sin x \leq x$ valid for $x \geq 0$ to prove that $1 - \frac{x^{2}}{2} \leq \cos x \leq 1$ .

(b) We need to prove that $x - \frac{x^{3}}{6} \leq \sin x \leq x (for x \geq 0)$ .

(c) We need to verify the inequalities for $x = 0.3$ .