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(Answers to Exercises for Chapter 4: Applications of Derivatives) A.4.1 CHAPTER 4: APPLICATIONS OF DERIVATIVES SECTION 4.1: EXTREMA 1) a) A.Max Value: 23, A.Max Point: 2, 23 () ; A.Min Value: 5; A.Min Point: ± 1, 5 . b) A.Max Value: 10, A.Max Point: 0,10 ; A.Min Value: ± 34 3 ; A.Min Point: 2, ± 34 3 ² ³ ´ µ · . c) A.Max Value: 20, A.Max Point: ± 4, 20 ; A.Min Value: 12; A.Min Point: ± 2, 12 . 2) a) Dom f = ±² , ² ; CNs: ± 2 , 3 16 , and 2. b) Dom g = , ± 6 ( ³ ´ µ 6, ² · ) ; CNs: ± 6 and 6. c) Dom h = 1 4 , ± ² ³ ´ µ · ; CN: 1 4 . d) Dom p = , ² ; CNs: x ± ± x = ² 2 + n ,o r x = 6 + 2 n r x = 5 6 + 2 nn integer ³ ´ µ · ¸ ¹ ; Hint: After differentiating, use a Double-Angle ID. e) Dom q = x ± ± x ² ³ integer {} ; CNs: NONE.

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(Answers to Exercises for Chapter 4: Applications of Derivatives) A.4.2 SECTION 4.2: MEAN VALUE THEOREM (MVT) FOR DERIVATIVES 1) a) f satisfies the hypotheses on 1, 5 ± ² ³ ´ ; c = 3 . b) f does not satisfy the hypotheses on 3, 7 ± ² ³ ´ , because f 3 () ± f 7 . c) f satisfies the hypotheses on ± 6, ± 1 ² ³ ´ µ ; c = ± 5 , or c = ± 7 2 = ± 3.5 , or c = ± 2 . d) f does not satisfy the hypotheses on ± 4, 4 ² ³ ´ µ , because f is not differentiable at 0, and 0 ±² 4, 4 ; therefore, f is not differentiable on ± 4, 4 . 2) a) f satisfies the hypotheses on 1, 4 ± ² ³ ´ ; c = 2 . Note 1: ± 2 ² ³ ´ µ . Note 2: Rolle’s Theorem also applies! b) f satisfies the hypotheses on ± 2, 3 ² ³ ´ µ ; c = ± 5 + 139 6 = 139 ± 5 6 ² 1.1316 . Note: ± 5 ± 139 6 = ± 5 + 139 6 ²± 2.7983 , so ± 5 ± 139 6 2, 3 ³ ´ µ . c) f does not satisfy the hypotheses on ± 8, 8 ² ³ ´ µ , because f is not differentiable at 0, and 0 ; therefore, f is not differentiable on ± . d) f satisfies the hypotheses on 0, 2 ± ² ³ ´ ; all real values in 0, 2 satisfy the theorem. (Can you see graphically why this is true?)
(Answers to Exercises for Chapter 4: Applications of Derivatives) A.4.3 SECTION 4.3: FIRST DERIVATIVE TEST 1) a) Dom f () = ±² , ² . f is neither even nor odd. y -intercept: ± 5 , or 0, ± 5 . Holes: None. VAs: None. HAs: None. SAs: None. Points at critical numbers: ± 5, ± 105 , a local minimum point; ± 7 2 , ± 1599 16 ² ³ ´ µ · , or ± 3.5, ± 99.9375 , a local maximum point; ± 2, ± 105 , a local minimum point. f is increasing on ± ± 7 2 ² ³ ´ µ · , ± 2, ¸ ² ³ ) ; or ± ± 3.5 ² ³ ´ µ , ± 2, ² ³ ) . f is decreasing on , ± 5 ( ³ ´ , ± 7 2 , ± 2 µ · ³ ´ ¸ , or , ± 5 ( ³ ´ , ± 3.5, ± 2 µ ³ ´ .

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