Exam 1 Formula Sheet

# Exam 1 Formula Sheet - Math 127 Formula and Examples Sheet...

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Math 127 Formula and Examples Sheet Exam #1 Average Rate Δ Y f(b) f(a) of Change Δ X b a Slope Intercept Form y-y 0 = m(x-x 0 ) Half Life (t 1/2 ) = - ln(2) Doubling Time = ln(2) k k Periodic Function = P 0 a t a = e k k = ln(a) Continuous Function = P 0 e kt r = a-1 a = r + 1 Decay when rate is negative, Growth rate is positive. EXAMPLES: 1. Prozac has a half-life of 3 days. What percent remains after a week? P(t) = P 0 e -kt t 1/2 = ln(2) k = ln(2) k = -.23 k 3 P(7) = P 0 e -k(7) = P 0 (.198) ≈ P 0 (.2), only 20% of original substance remains after a week. 2. A substance loses 10% of its mass in 5 hours. Find the half-life of this substance. (Exponential Decay) P(5) = P 0 e -5k r = 1 - .1 = .9 (of original substance) .9P 0 = P 0 e -5k .9 = e -5k ln(.9) = ln(e -5k ) -5kln(e) = ln(.9) * NOTE: ln(e) = 1 k = ln(.9) = .021 t 1/2 = ln(2) = 32.9 hours 5 .021 3. A painting is supposedly by Vermeer (1632-1675) and it contains 99.5% of its carbon-14. Its half-life is 5,730 years. Is the painting fake? t 1/2 = ln(2) = 5730 k = ln(2) = 1.21x10 -4 k 5730 (.995)P 0 = P 0 e -.000121t .995 = e -.000121t t = ln(.995) = 41.43 -.000121 years The painting is a fake.
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