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chapter 2 20

# chapter 2 20 - 84 I/4:n — 3|< 0.5 4 2.5< x/4x 1<...

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Unformatted text preview: 84 . I./4:n + — 3| < 0.5 4:) 2.5 < x/4x + 1 < 3.5. We plot the CHAPTER 2 LIMITS AND DERIVATIVES . On the left side ofx = 2. we need Ix 1 2| < 1—70 — 2| = %. On the right side. we need Ix — 2| < Il—3O — 2| : g. For both of these conditions to be satisﬁed at once. we need the more restrictive of the two to hold. that is. Ix — 2| < 3. So we can choose 6 = %. or any smaller positive number. . On the left side. we need Ix — 5| < I4 — 5|: 1. On the right side. we need Ix — 5| < I5.7 — 5| 2 0.7. For both conditions to be satisﬁed at once. we need the more restrictive condition to hold; that is. Ix — 5| < 0.7. So we can choose 6 = 0.7. or any smaller positive number. The leftmost question mark is the solution of ﬁ: : 1.6 and the rightmost. ﬂ : 2.4. So the values are 1.62 : 2.56 and 2.42 : 5.76. On the left side. we need Ix — 4| < I256 — 4| 2 1.44. On the right side. we need Ix — 4| < I576 i 4| : 1.76. To satisfy both conditions. we need the more restrictive condition to hold——namely. Ix — 4| < 1.44. Thus. we can Choose 6 = 1.44. or any smaller positive number. The left—hand uestion mark is the ositive solution of x2 : 1. that is. x : i. and the ri ht—hand uestion mark is the positive solution of \$2 : % that is. x = ﬂ On the left side. we need Ix 1 1I < If — 1| m 0.292 (rounding down to be safe). On the right side. we need Ix ~ 1I < |\/§ 4 1| m 0.224. The more restrictive of these two conditions must apply. so we Choose 5 : 0.224 (or any smaller positive number). three parts of this inequality on the same screen and identify the x-coordinates of the points of intersection using the cursor. It appears that the inequality holds for 1.3125 g x 3 2.8125. Since I2 - 1.3125I : 0.6875 and I2 — 2.8125I : 0.8125. we choose 0 < 6 < min {0687508125} : 0.6875. 8. Isin x — %| < 0.1 <=> 0.4 < sinx < 0.6. From the graph. we see that for this inequality to hold. we need 0.42 S x g 0.64. So since I05 1 0.42I : 0.08 and I05 , 0.64I : 0.14. we choose 0 < 6 g min {0.08.014} : 0.08. 0.7 ...
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