Clicker Question 1 If a type of light bulb has a mean lifetime of 120 hours, what is the probability...
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![Page 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?](https://reader036.fdocuments.net/reader036/viewer/2022081811/5697c01d1a28abf838cd08f4/html5/thumbnails/1.jpg)
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%
![Page 2: 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?](https://reader036.fdocuments.net/reader036/viewer/2022081811/5697c01d1a28abf838cd08f4/html5/thumbnails/2.jpg)
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
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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%
![Page 4: 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?](https://reader036.fdocuments.net/reader036/viewer/2022081811/5697c01d1a28abf838cd08f4/html5/thumbnails/4.jpg)
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.
![Page 5: 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?](https://reader036.fdocuments.net/reader036/viewer/2022081811/5697c01d1a28abf838cd08f4/html5/thumbnails/5.jpg)
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 ' = x2 – 3x + 5 ? What is y if y ' = x / (x2 + 4) What is y if dy/dt = e0.67t
Note that you also get a “constant of integration” in the solution.
![Page 6: 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?](https://reader036.fdocuments.net/reader036/viewer/2022081811/5697c01d1a28abf838cd08f4/html5/thumbnails/6.jpg)
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?
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The solutions to a DE
A solution of a given differential equation is a function y which makes the equation work.
Show that y = Ae0.08t 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”.
![Page 8: 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?](https://reader036.fdocuments.net/reader036/viewer/2022081811/5697c01d1a28abf838cd08f4/html5/thumbnails/8.jpg)
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 DEd2y / 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.
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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.