FDWKCalcSM_ch2

# FDWKCalcSM_ch2 - Section 2.1 51 65(a N(t = 4 2t(b 4 days 4...

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Section 2.1 51 65. (a) N ( t ) = 4 2 t (b) 4 days: 4 2 4 = 64 guppies 1 week: 4 2 7 = 512 guppies (c) Nt t t t t () ln = ⋅= = = = 2000 4 2 2000 2 500 2 500 2 500 t =≈ . 500 2 8 9658 There will be 2000 guppies after 8.9658 days, or after nearly 9 days. (d) Because it suggests the number of guppies will continue to double indefinitely and become arbitrarily large, which is impossible due to the finite size of the tank and the oxygen supply in the water. 66. (a) y = 41.770 x + 414.342 (b) y =+ = −= 41 770 22 414 342 1333 1333 1432 .( ) . 99 The estimate is 99 less than the actual number. (c) y = mx + b m = 41.770 The slope represents the approximate annual increase in the number of doctorates earned by Hispanic Americans per year. 67. (a) y = (17467.361)(. ) 1 00398 x = (b) (17467.361) (. ) 1 00398 23 = 19,138 thousand or 19,138,000 19,138,000 19,190,000 = 52,000 The prediction is less than the actual by 52,000. (c) 17 558 19 138 23 0 0398 4 , (, ) .% = or 68. (a) m = 1 (b) y = x 1 (c) y = x + 3 (d) 2 69. (a) (2, ) x 2 > 0 (b) ( −∞ , ) all real numbers (c) fx l nx ln x ln x ex xe ( ) =− = 12 01 2 2 1 1 2 4 718 +≈ . (d) yl n x xl n y n y ey y x −=− 2 1 = + e x 1 2 (e) ( ) ( ( ) ) ( ) ( ff x f f x f e e x x ± −− == + + 11 1 1 2 21 1 )( ) ( ) ( ( ) =− − = = ln e xx ff x f f x x ± ) ( ( ) ) (() ) ( fl n x ee ln x ln x 1 2 22 2 ) =+ − = 70. (a) ( −∞ , ) all real numbers (b) [ 2, 4] 1 3 cos (2 x ) oscillates between 2 and 4 (c) ± (d) Even. cos θ = cos (e) x 2.526 Chapter 2 Limits and Continuity Section 2.1 Rates of Change and Limits (pp. 59–69) Quick Review 2.1 1. f ( ) () 2 5 2 40 32 =−+ = 2. f 2 42 5 24 11 12 2 3 = + = 3. f () s i n s i n 2 2 2 0 =⋅ ±± 4. f 2 1 1 3 2 = =

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52 Section 2.1 5. x x < −< < 4 44 6. xc cx c < −<< 2 22 7. x x x −< −< − < 23 32 3 15 8. xc d dx c d d cxd c −<−< −+<< + 2 9. xx x x 2 31 8 3 36 3 63 −− + = +− + =− () , 10. 2 21 21 1 1 1 2 2 2 x x x = −+ = + ( ) , Section 2.1 Exercises 1. Δ Δ = = y t 16 3 16 0 30 48 ft sec 2. Δ Δ = = y t 16 4 16 0 40 64 3. Δ Δ = = = + y t h h h 16 3 16 3 001 16 3 0 01 ( ) ,. (. say ) ( ) . ) . . 2 16 9 16 9 0601 16 9 144 9616 = = == 144 0 9616 96 16 . . . . Confirm Algebraically Δ Δ = = ++ − = + y t h h hh h h 16 3 16 3 16 9 6 144 96 2 ( ) 16 96 16 2 h h h =+ if h = 0, then Δ Δ = y t 96 4. Δ Δ = = + y t h h h 16 4 16 4 16 4 0 01 ( ) . ) ( ) . ) () . 16 4 16 16 0801 16 16 = = 257 2816 256 1 2816 128 16 . . . . . Confirm Algebraically Δ Δ = = = y t h h h 16 4 16 4 16 16 8 256 128 2 ( ) h h h h + 16 128 16 2 if h = 0, then Δ Δ = y t 128 5. lim ( ) x cc c =−+ 1 1 6. lim x + 43 2 1 9 = + c 2 1 9 7. ( ) / x →− −= − 12 2 2 3213 1 2 2 1 2 1 = 3 1 4 2 3 2 Graphical support: 8. lim ( ) ( ) ( ) x x →− += −= 4 1998 1998 1998 34 3 1 1 Graphical support: 9. l im( x xxx −=+ 1 3 2 3 2 17 1 3 1 2 1 17 13 2 17 15 −− = Graphical support:
Section 2.1 53 10. lim () y yy y ++ + = + == 2 22 56 2 25 26 20 4 5 Graphical support: 11. () () y y →− = −+−+ −− 3 2 2 2 2 43 3 34 33 0 6 0 Graphical support: 12. lim int int / x x 12 1 2 0 Note that substitution cannot always be used to find limits of the int function. Its use here can be justified by the Sandwich Theorem, using g ( x ) = h ( x ) = 0 on the interval (0, 1).

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FDWKCalcSM_ch2 - Section 2.1 51 65(a N(t = 4 2t(b 4 days 4...

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