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Math 1432 Solve these questions before class: 1) 2) x 1 y cos x , 2 x 4 ln x 2 dy ? dx dx ? 3) Give the equation for the tangent line to the graph of f x 2 3 s i n x at the point where x = 0. 1 Recall: d x a a x ln a dx d u a u' au ln a dx d 1 log a x dx x ln a d u' log a u dx u ln a ax a dx ln a C x and au a du ln a C u 2 Example: Find the derivatives: y 52 x y 2 sin x y log 2 5 x y log6 x 2 x 3 Example: x 2 x3 5 dx log x 3 Example: 2 dx x 4 Example: Find the solution to this differential equation subject to the given initial condition. y' 5y y 0 6 5 Exponential Growth and Decay Section 7.6 6 For example, exponential Growth and Decay models are used in: Population Growth/Decay Radioactive Decay Investments Mixing problems Newton’s Law of Cooling 7 Write and solve a differential equation for the rate of change of y with respect to time is proportional to y, given that y > 0. 8 Example: Find a function that satisfies y’ = –2y and y(0) = 3 9 Naming Conventions Population Growth: P(t) = population at time t P t P0 e k t P0= P(0) = initial population k = growth rate Radioactive Decay: A t A 0 e kt A(t) = amount at time t A0= A(0) = initial amount k = growth/decay rate Continuous Compound Interest: A t A 0 e rt A(t) = principle at time t A0= A(0) = initial investment r = annual interest rate 10 Note that in all cases we have a quantity that changes at a rate proportional to itself! Example: At what rate r of continuous compounding does a sum of money double in 10 years? 11 Doubling Time Half-Life 12 Example: In a bacteria growing experiment, a biologist observes that the number of bacteria in a certain culture triples every 4 hours. After 12 hours, it is estimated that there are 1 million bacteria in the culture. a. How many bacteria were present initially? b. What is the doubling time for the bacteria population? 13 Example: A 100-liter tank initially full of water develops a leak at the bottom. Given that 10% of the water leaks out in the first 5 minutes, find the amount of water left in the tank 15 minutes after the leak develops if the water drains off at a rate that is proportional to the amount of water present. 14 Next week, we will cover derivatives and integrals of inverse trig functions. Review inverse trig functions before class! Exercise: Suppose a culture of bacteria is growing in such a way that the change in the number of bacteria is proportional to the number present. The number of bacteria doubles every 200 minutes and there are currently 5000 bacteria in the culture. How many bacteria were present 2 hours ago? 20