1. Clicker Question 1 If a type of light bulb has a mean lifetime of 120 hours, what is the probability that one of them will last longer than 240 hours? – A. 13.5% – B. 86.5% – C. 50% – D. 25% – E. 16.7%

2. Clicker Question 2 For the same light bulbs (120 hour mean lifetime), what is the median lifetime? – A. 120 hours – B. 90 hours – C. 83 hours – D. 72 hours – E. 60 hours

3. Clicker Question 3 Suppose a variable is normally distributed with a mean of 80 and a standard deviation of 15. What is the probability that the variable will have a value greater than 110? – A. 5% – B. 95% – C. 2.5% – D. 16% – E. 32%

4. Differential Equations (10/27/10) A differential equation is an equation which contains derivatives within it. More specifically, it is an equation which may contain an independent variable x (or t) and/or a dependent variable y (or some other variable name), but definitely contains a derivative y ' = dy/dx (or dy/dt). It may also contain second derivatives y '', etc.

5. Examples of DE’s Every anti-derivative (i.e., indefinite integral) you have solved (or tried to solve) this semester is a differential equation! What is y if y ' = x 2 – 3x + 5 ? What is y if y ' = x / (x 2 + 4) What is y if dy/dt = e 0.67t Note that you also get a “constant of integration” in the solution.

6. New types of examples The following is a DE of a different type since it contains the dependent variable: y ' =.08y Say in words what this says! Note that we don’t see the independent variable at all – let’s call it t. What is a solution to this equation? And how can we find it?

7. The solutions to a DE A solution of a given differential equation is a function y which makes the equation work. Show that y = Ae 0.08t is a solution to the DE on the previous slide, where A is a constant. Note that we are using the old tried and true method for solving equations here called “guess and check”.

8. Examples of guess and check for DE’s Show that y = 100 – A e –t satisfies the DE y ' = 100 - y Show that y = sin(2t) satisfies the DE d 2 y / dt 2 = -4y Show that y = x ln(x) – x satisfies the DE y ' = ln(x) Of course one hopes for better methods to solve equations, but DE’s can be very hard.

9. Assignment for Friday Read over these slides (and try to solve the problems on them), and read Section 9.1. On page 571, do # 1 – 7 odd.