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4th week MondayWe also discuss several other topics like irc study guide
Practice with experience logs
Recall y = ln(x) is inverse to y = exIf you want to learn more check out What is the use of response variable?
Example:
 ln(e1) = 1
 ln(e2) = 2 ln(e)
We also discuss several other topics like What are two legislative branches collectively known as Congress?
Logs cheat order of operations
Because ex, ln(x) are inverses and e0 = 1 so flip inputs / outputs
No. because there is no input of ex which gives zero as a outputWe also discuss several other topics like What is a binomial prob dist?
For the same reason ln (negative number) is not defined
We also discuss several other topics like What is meant by gragedy of the commons?
Example: convert b(t) = 1.3t to exponential form
Bt = b0 eln(3t)
= b0eln(3)tIf you want to learn more check out Enumerate the types of system.
Example: The doubling time of a population P is 10 years; initially there are 2. Find the growth rate if P(t) = P0ecct models the population growth
Solution: P0 = 21 if we plug in t = 10, 4 should come out (it doubles)
So 4 = 2ea10
Ln4 = ln(2e10a)
No!
Ln4 = ln(2) ln(e10a)
Ln4 = ln(2) + 10a
Cc = ==
Notes  lung model
The goal is to find an updating fxn which models the following situation
Some concentration of chemical is in the air, and we want to know after each breath how much chemical enters someone’s lungs
Suppose initially there is 2 mmol/ L of chemical concentrated in the lung and there is 5 mmol/ L in the air also, the lung is 6 L large and 6 L
Volume of lung(L)
Total chemical in lung(mmol)
Concentration of mmol / L chemical in lung
Initially
6
12
2
Exhale
6
1.2
2
How much air is left
5,4
10,8
2
Inhale
6
3
5
Air in lung finally
6
13.8
1.8/6
1.8 Notes
 Radians  the distance along the perimeter of the circle of radius 1
 Basic identity  2rad = 3600
Conversion factor: 1 = =
1 = =
 Trigonometric functions describe simple oscillations like heartbeats and breathing
 Oscillations  process that repeat in cycles
 Sine and cosine fxns, take angles inputs and return numbers between 1 and 1 as outputs
 The Cartesian coordinates of the point on the unit circle an angle O measured counter  clockwise from (1,0) are (cos C0), sin(0))
 Both sine and cosine fxns repeat every 2 radians; the value 2 is called period of the oscillation; adding or subtracting multiples of 2 from the argument does not change the value for any value of 0 and any integer n,
Cos  = cos(0 + 2) = cos(0 + 4) = cos (0 + 2n)
cos(0) = cos(02) = cos(0  4) = cos(0.2n)
Radians
Degrees
Cos (0)
sin(0)
0
00
1
0
/6
300
½
/4
450
/3
600
½
/2
900
0
1
2/3
1200
½
3/4
1350

5/6
1500

½
1800
1
=
7
2100

½
4/3
2400
½

/2
3000
½
