chapter_01 - January 27, 2005 11:42 L24-ch01 Sheet number 1...

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January 27, 2005 11:42 L24-ch01 Sheet number 1 Page number 1 black 1 CHAPTER 1 Functions EXERCISE SET 1.1 1. (a) 2 . 9 , 2 . 0 , 2 . 35 , 2 . 9 (b) none (c) y =0 (d) 1 . 75 x 2 . 15 (e) y max =2 . 8 at x = 2 . 6; y min = 2 . 2at x =1 . 2 2. (a) x = 1 , 4 (b) none (c) y = 1 (d) x , 3 , 5 (e) y max =9at x =6; y min = x 3. (a) yes (b) yes (c) no (vertical line test fails) (d) no (vertical line test fails) 4. (a) The natural domain of f is x 6 = 1, and for g it is the set of all x . Whenever x 6 = 1, f ( x )= g ( x ), but they have diFerent domains. (b) The domain of f is the set of all x 0; the domain of g is the same. 5. (a) around 1943 (b) 1960; 4200 (c) no; you need the year’s population (d) war; marketing techniques (e) news of health risk; social pressure, antismoking campaigns, increased taxation 6. (a) around 1983 (b) 1966 (c) the former (d) no, it appears to be levelling out 7. (a) 1999, $34,400 (b) 1985, $37,000 (c) second year; graph has a larger (negative) slope 8. (a) In thousands, approximately 43 . 2 37 . 8 6 = 5 . 4 6 per yr, or $900/yr (b) The median income during 1993 increased from $37.8K to $38K (K for ’kilodollars’; all ±gures approximate). During 1996 it increased from $40K to $42K, and during 1999 it decreased slightly from $43.2K to $43.1K. Thus the average rate of change measured on January 1 was (40 - 37.8)/3 for the ±rst three-yr period and (43.2 - 40)/3 for the second-year period, and hence the median income as measured on January 1 increased more rapidly in the second three-year period. Measured on December 31, however, the numbers are (42 - 38)/3 and (43.1 - 42)/3, and the former is the greater number. Thus the answer to the question depends on where in the year the median income is measured. (c) 1993 9. (a) f (0) = 3(0) 2 2= 2; f (2) = 3(2) 2 2 = 10; f ( 2)=3( 2) 2 2 = 10; f (3) = 3(3) 2 2 = 25; f ( 2) 2 2=4; f (3 t ) = 3(3 t ) 2 2=27 t 2 2 (b) f (0) = 2(0) = 0; f (2) = 2(2) = 4; f ( 2 )=2 ( 2) = 4; f (3) = 2(3) = 6; f ( 2 2; f (3 t )=1 / 3 t for t> 1 and f (3 t )=6 t for t 1. 10. (a) g (3) = 3+1 3 1 ; g ( 1) = 1+1 1 1 ; g ( π π +1 π 1 ; g ( 1 . 1) = 1 . 1 . 1 1 = 0 . 1 2 . 1 = 1 21 ; g ( t 2 1) = t 2 t 2 1 1 = t 2 t 2 2 (b) g (3) = 3+1 = 2; g ( 1 )=3 ; g ( π π +1; g ( 1 . 1 ; g ( t 2 1 )=3i f t 2 < 2 and g ( t 2 1) = t 2 1+1= | t | if t 2 2.
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January 27, 2005 11:42 L24-ch01 Sheet number 2 Page number 2 black 2 Chapter 1 11. (a) x 6 =3 (b) x ≤− 3or x 3 (c) x 2 2 x + 5 = 0 has no real solutions so x 2 2 x + 5 is always positive or always negative. If x = 0, then x 2 2 x +5=5 > 0; domain: ( −∞ , + ). (d) x 6 =0 (e) sin x 6 =1,so x 6 =(2 n + 1 2 ) π , n , ± 1 , ± 2 ,... 12. (a) x 6 = 7 5 (b) x 3 x 2 must be nonnegative; y = x 3 x 2 is a parabola that crosses the x -axis at x , 1 3 and opens downward, thus 0 x 1 3 (c) x 2 4 x 4 > 0, so x 2 4 > 0 and x 4 > 0, thus x> 4; or x 2 4 < 0 and x 4 < 0, thus 2 <x< 2 (d) x 6 = 1 (e) cos x 1 < 2, 2 cos 0, all x 13. (a) x 3 (b) 2 x 2 (c) x 0 (d) all x (e) all x 14. (a) x 2 3 (b) 3 2 x 3 2 (c) x 0 (d) x 6 (e) x 0 15. (a) Breaks could be caused by war, pestilence, flood, earthquakes, for example.
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chapter_01 - January 27, 2005 11:42 L24-ch01 Sheet number 1...

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