Exponential Growth and Decay AP Calculus 5-8 Exponential Growth and Decay

𝑃 𝑡 = 𝑃 0 𝑒 𝑘𝑡 𝑦= 𝑃 0 𝑒 𝑘𝑡 Find 𝑦′ 𝑦 ′ =𝑘 𝑃 0 𝑒 𝑘𝑡 𝑦 ′ =𝑘𝑦

𝑃 0 = Initial Population 𝑘= Growth Constant 𝑡= Time 𝑃(𝑡)= 𝑃 0 𝑒 𝑘𝑡 𝑃 0 = Initial Population 𝑘= Growth Constant 𝑡= Time

In a laboratory, a number of Escherichia coli (E In a laboratory, a number of Escherichia coli (E. coli) bacteria grows exponentially with a growth constant of 𝑘=0.41 ℎ𝑜𝑢𝑟𝑠 −1 . Assume that 1000 bacteria are present at time 𝑡=0. Find the formula for the number of bacterial 𝑃(𝑡) at time 𝑡. How large is the population after 5 hours? When will the population reach 10,000?

𝑃 𝑡 =1000 𝑒 0.41𝑡 𝑃 5 =1000 𝑒 0.41(5) 𝑃 5 =7768 𝑏𝑎𝑐𝑡𝑒𝑟𝑖𝑎 10,000=1000 𝑒 0.41𝑡 10= 𝑒 0.41𝑡 ln 10 =0.41𝑡 𝑡=5.62 ℎ𝑜𝑢𝑟𝑠

𝑦 𝑡 =𝐶 𝑒 3𝑡 , where 𝐶 is the initial value 𝐶=𝑦 0 . Find all solutions to 𝑦 ′ =3𝑦. Which solution satisfies 𝑦 0 =9? 𝑦 𝑡 =𝐶 𝑒 3𝑡 , where 𝐶 is the initial value 𝐶=𝑦 0 . 𝑦 𝑡 =9 𝑒 3𝑡

Doubling Time/Half-life ln 2 𝑘 If 𝑘 is positive you are finding doubling time If 𝑘 is negative you are finding half-life

The Nazi Zombie Virus (NZV) is spreading quickly around the world The Nazi Zombie Virus (NZV) is spreading quickly around the world. The virus is doubling time of the virus is 11.55245301 days. What is the function for the virus? If the virus started in Norway, how long before everyone is infected in Norway? Norway’s Population is 5,100,000 people How long before the world is infected? 7 billion people

7,000,000,000= 𝑒 0.06𝑡 ln 7,000,000,000 0.06 =𝑡 𝑡=377.82 𝑑𝑎𝑦𝑠

ln 2 𝑘 =11.55245301 𝑘=0.06 𝑃 𝑡 =1 𝑒 0.06𝑡 5,100,000= 𝑒 0.06𝑡 ln 5,100,000 0.06 =𝑡 𝑡=257.41 𝑑𝑎𝑦𝑠

Dr. Strangelove has found an antidote that will cure all of those that are infected and prevent the virus from spreading. He finds that the decay rate of the virus is 0.09. Write the function for the virus What is the half-life of the virus? If after 300 days the antidote is starting to be distributed, how long before the virus is infected by only 10 people?

𝑃 𝑡 = 𝑃 0 𝑒 −0.09𝑡 ln 2 0.09 =7.7016 𝑑𝑎𝑦𝑠

𝑒 0.06(300) =65,659,969 65,659,969 𝑒 −0.09𝑡 =10 ln 10 65,659,969 −0.09 =𝑡 𝑡=174.416 𝑑𝑎𝑦𝑠

Problems 5-8 page 350 #1-21