 By Dumbrille J

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Extra resources for Built To Fade The Advent of the Biodegradable Brand

Example text

Sin (−1) is negative cos (−1) is positive tan (−1) is negative. 46 10. The period is 2π/3, because when t varies from 0 to 2π/3, the quantity 3t varies from 0 to 2π. The amplitude is 7, since the value of the function oscillates between −7 and 7. 11. The period is 2π/(1/4) = 8π, because when u varies from 0 to 8π, the quantity u/4 varies from 0 to 2π. The amplitude is 3, since the function oscillates between 2 and 8. 12. The period is 2π/2 = π, because as x varies from −π/2 to π/2, the quantity 2x + π varies from 0 to 2π.

18, so 18% leaves the body each hour. 04. 04 mg. We want to find the value of t when A = 1. 60 hours. 60 hours, the amount is 1 mg. 4 SOLUTIONS 33 40. 17t . 36. We estimate the half-life by estimating t when the caffeine is reduced by half (so A = 50); this occurs at approximately t = 4 hours. 077. 077 hours. 36. 41. Since y(0) = Ce0 = C we have that C = 2. Similarly, substituting x = 1 gives y(1) = 2eα so 2eα = 1. Rearranging gives eα = 1/2. 693. Finally, y(2) = 2e2(− ln 2) = 2e−2 ln 2 = 1 . 2 42.

As x → ±∞, the lower-degree terms of f (x) become insignificant, and f (x) becomes approximated by the highest degree term of the numerator and denominator. Thus, as x → ±∞, we see that x2 f (x) → 2 = 1. x There is a horizontal asymptote at y = 1. To find the vertical asymptotes, we set the denominator equal to zero. When x2 − 4 = 0, we have x = ±2 so there are vertical asymptotes at x = −2 and at x = 2. 38. To find the horizontal asymptote, we look at end behavior. As x → ±∞, the lower-degree terms of f (x) become insignificant, and f (x) becomes approximated by the highest degree term of the numerator and denominator.