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chapter 2 2 - 66 CHAPTER 2 LlMlTS AND DERIVATIVES 4 For the...

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Unformatted text preview: 66 CHAPTER 2 LlMlTS AND DERIVATIVES 4. For the curve y = In a: and the point P(27 1n 2): (a) Q (b) The slope appears to be %. (1.5. 0.405465) 0.575364 (C) y—1n2 2 500—2) or (19, 0.641854) 0.512933 1 (1.99, 0.688135) 0.501254 y : 55 _ 1+ 1112 (1999.0.692647) 0.500125 (2.5.0.916291) 0.446287 (2.1. 0.741937) 0.487902 (2.01. 0.698135) 0.498754 (2.001. 0.693647) 0.499875 (d) y tline atx= 1.5 /SeCan 1 /<——" tangent line at x = 2 q: secant line at x : 2.5 y : In x l" 2 secant line at X = 2.5 \, 0.5 secant line \y : lnx atx=1.5 1— 1— 1— -> 0 1,5 2 2.5 x 5. (a) y : y(t) : 4075 4 16t2. At t 2 2, y : 40(2) — 16(2)2 : 16. The average velocity between times 2 and 2 + h a“ (2 + h) — 2 h h ('1) [22.5]: h : 0.5. um : —32 ft/s (ii) [22.1]: h = 0.1. um = -25.6 ft/s (iii) [2. 2.05]: h : 0.05. 22.... = —24.8 ft/s (iv) [2. 2.01]: h : 0.01.71ave : —24.16 ft/s : —244 16h. ifh 74 0. (b) The instantaneous velocity when t : 2 (h approaches 0) is #24 ft / s. 6. The average velocity between t and t + h seconds is 5816 h —0.83t ht 585—08312 1 . — . 2 ( + ) ( + ) ( ) 58h 1 66th 0 83h : 58 1.6615 0.83h ifh 7’: 0. h h (a) Here If : 1. so the average velocity is 58 — 1.66 - 0.83h = 56.34 — 0.83h. (i) [1.2]: h : 1, 55.51 m/s (ii) [1.15]; h : 0.5. 55.925 m/s (iii) [11.1]: h : 0.1. 56.257 m/s (iv) [1.1.01]: h : 0.01. 56.3317 m/s (v) [1.1.001]: h : 0.001. 56.33917 m/s (b) The instantaneous velocity after 1 second is 56.34 m/s. ...
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