Solução_Calculo_Stewart_6e

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F. TX.10 PRINCIPLES OF PROBLEM SOLVING 1. By using the area formula for a triangle, 1 (base)(height), in two ways, we see that 2 1 (4) (y) = 1 (h)(a),soa = 4y 2 2 h .Since42 + y 2 = h 2 , y = √ h 2 − 16,and a = 4√ h 2 − 16 . h 3. |2x − 1| = 2x − 1 if x ≥ 1 2 1 − 2x if x< 1 2 and |x +5| = x +5 −x − 5 if x ≥−5 if x
F. TX.10 40 ¤ CHAPTER 1 PRINCIPLES OF PROBLEM SOLVING 9. |x| + |y| ≤ 1. The boundary of the region has equation |x| + |y| =1. In quadrants I,II,III,andIV,thisbecomesthelinesx + y =1, −x + y =1, −x − y =1,and x − y =1respectively. 11. (log 2 3)(log 3 4)(log 4 5) ···(log 31 32) = ln 3 ln 2 ln 4 ln 3 ln 5 ··· ln 4 ln 32 ln 32 ln 25 = = ln 31 ln 2 ln 2 = 5ln2 ln 2 =5 13. ln x 2 − 2x − 2 ≤ 0 ⇒ x 2 − 2x − 2 ≤ e 0 =1 ⇒ x 2 − 2x − 3 ≤ 0 ⇒ (x − 3)(x +1)≤ 0 ⇒ x ∈ [−1, 3]. Since the argument must be positive, x 2 − 2x − 2 > 0 ⇒ x − 1 − √ 3 x − 1+ √ 3 > 0 ⇒ x ∈ −∞, 1 − √ 3 ∪ 1+ √ 3, ∞ . The intersection of these intervals is −1, 1 − √ 3 ∪ 1+ √ 3, 3 . 15. Let d be the distance traveled on each half of the trip. Let t 1 and t 2 be the times taken for the first and second halves of the trip. For the first half of the trip we have t 1 = d/30 and for the second half we have t 2 = d/60. Thus, the average speed for the entire trip is is 40 mi/h. total distance total time = 2d t 1 + t 2 = 2d d 30 + d 60 17. Let S n be the statement that 7 n − 1 is divisible by 6. • S 1 is true because 7 1 − 1=6is divisible by 6. · 60 60 = 120d 2d + d = 120d =40. The average speed for the entire trip 3d • Assume S k is true, that is, 7 k − 1 is divisible by 6. Inotherwords,7 k − 1=6m for some positive integer m. Then 7 k+1 − 1=7 k · 7 − 1=(6m +1)· 7 − 1=42m +6=6(7m +1), which is divisible by 6,soS k+1 is true. • Therefore, by mathematical induction, 7 n − 1 is divisible by 6 for every positive integer n. 19. f 0 (x) =x 2 and f n+1 (x) =f 0 (f n (x)) for n =0, 1, 2,.... f 1 (x) =f 0 (f 0 (x)) = f 0 x 2 = x 22 = x 4 , f 2 (x) =f 0 (f 1 (x)) = f 0 (x 4 )=(x 4 ) 2 = x 8 , f 3(x) =f 0(f 2(x)) = f 0(x 8 )=(x 8 ) 2 = x 16 , .... Thus, a general formula is f n(x) =x 2n+1 .
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Solução_Calculo_Stewart_6e

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