Applied Derivatives and Lab Math Session 5 : 9/20 1
Human Population Dochy et al. (1995) (Acta Biotheoretica 43: 241 247, 1995) developed the following function to describe the increase in human population over time: x ( t ) = 1 C β t Where x(t) is the human population (in millions) at year t, and C and β are constants with values: C = 9.833x10 3 β = 4.849x10 6 Using this model, 1. Predict how rapidly the human population was increasing in 1450, and in 1995 2. Based on this model, at what time was the human population increasing at 2,000,000 people per year? Session 5 : 9/20 2
Graphical Solution Global Population Over Time 6000 Population (in millions of people) 5000 4000 3000 2000 1000 Green points represent actual historical data Blue line represents model data Year (A.D.) 500 1000 1500 1950 Pop. From Data (in millions) 206 254 460 2527 Pop. From Model (in millions) 134 200 391 2649 0 0 500 1000 1500 2000 2500 Year (A.D.) Historical Data From: http://www.census.gov/ipc/www/worldhis.html Session 5 : 9/20 3
Gravitational Force Between Objects Newton s Law of Universal Gravitation: r m m 2 1 F G m m = 1 2 r 2 F = Gravitational force between masses (in Newtons) G = Gravitational constant (6.67x10 11 N m 2 kg 2 m 1 = Mass of object 1 (kg) M 2 = Mass of object 2 (kg) Session 5 : 9/20 4
Completely Irrelevant Fact: The gravitational force between two people (150 lbs each) standing at 10m distance is 3.08x10 9 N, which is approximately equal to 6.92x10 10 pound force, which would be approximately the same force that would be exerted on your body if a fruit fly was cut into 49,000 pieces, and one of those pieces was put on your head. Relevant Fact: Fruit fly weight from: Lehman et al. (2001), J. Exp. Biology 204, 627 635. The distance between Earth and Mars has a greater range than any other planets, from 5.7x10 10 m to 4.02x10 11 m. Question: How does the force change with respect to distance when the planets are at their furthest distance? At Session 5 : 9/20 5 their closest distance?
Again Asking for the Slope = Derivative Gravitational Force Between Earth and Mars Gravitational Force (N) 1.4E+17 1.2E+17 1E+17 8E+16 6E+16 4E+16 2E+16 0 0.00E+00 5.00E+10 1.00E+11 1.50E+11 2.00E+11 2.50E+11 3.00E+11 3.50E+11 4.00E+11 4.50E+11 5.00E+11 Separation Distance (m) Session 5 : 9/20 6
Minimizing production cost Recently, researchers at UCSD have discovered that the efficiency of solar cells can be greatly increased by coating the cell surfaces with indium phosphate nanowires. Novotny et al. (2008), NanoLetters 8, 775 779 A company that produces photocells has developed the following predictive model of average cost per unit to produce x InP coated cells (per batch): InP nanowires grown on solar cell surface. 1500 C cell = + 4300 + 0. 003 x x Where C = cost per cell ($) x = number of cells produced Question: What is the optimum number of photocells that should be produced per batch to minimize the average cost per cell? If the optimum number are produced, what is the cost per cell? Session 5 : 9/20 7
Average Cost per Cell 4316 4314 Average Cost ($) 4312 4310 4308 4306 4304 4302 0 1000 2000 3000 4000 5000 Number of Cells Produced Session 5 : 9/20 8
Lab Math! Scientific Notation and Powers of 10 Metric System Prefixes Converting Units Concentration Moles, Molar Weight, Molarity Dilutions ph Session 5 : 9/20 9
Scientific Notation Write numbers taking advantage of powers of 10 10 3 =1000 10 2 =100 10 1 =10 10 0 =1 10 1 =0.1 10 2 =0.01 10 3 =0.001 Session 5 : 9/20 10
To Convert to Scientific Notation To put in scientific notation, typically convert original number to have only a single number to the left of the decimal**, then multiply by powers of 10 accordingly. Examples: 3125 = 3.125x10 3 415 = 4.15x10 2 0.002 = 2.0x10 3 **In some situation, e.g. comparing data, you will want to have all data expressed to the same power of 10, so the number of digits to the left of the decimal may vary. Session 5 : 9/20 11
Make your data uniform! It is difficult to compare data when the order or expression are not uniform. Convert to express all data in the same units and to the same power. Example: BAD! Good! Year U.S.A. Population (# of people) Year U.S.A. Population (# of people) 1900 76,000,000 1900 7.6x10 7 1930 12.3x10 7 Convert to same power 1930 12.3x10 7 1974 213,853,928 1998 270.3x10 6 1974 21.4x10 7 1998 27.0x10 7 http://www.npg.org/facts/us_historical_pops.htm Session 5 : 9/20 12
Metric System Prefixes Factor 1,000,000,000 1,000,000 1000 100 10 0.1 0.01 0.001 0.000001 0.000000001 In Words Prefix Billion Giga Million Mega Thousand Kilo Hundred Hecto Ten Deca Tenth Deci Hundredth Centi Thousandth Milli Millionth Micro Billionth Nano symbol G M k h da d c m μ Session 5 : 9/20 13 n
Examples: 3 millionths of a meter = 0.000003 meter = 3x10 6 m = 3 micrometers = 3μm 4,500g = forty five thousand grams = 4.5x10 3 g = 4.5kg 0.0001 millimeters = 0.0001mm = 0.1μm = 100nm Session 5 : 9/20 14
Useful Conversion Constants Common Conversions: Length: Area: 1 foot = 3.048x10 1 m 1 acre = 4.047x10 3 m 2 1 inch = 2.54x10 2 m 1 foot 2 = 9,29x10 2 m 2 1 mile = 1.609x10 3 m 1 mile 2 = 2.59x10 6 m 2 1 yard = 9.144x10 1 m 1 yard 2 = 8.36x10 1 m 2 Mass: Volume: 1 ounce = 2.835x10 2 kg 1 foot 3 = 2.831 m 3 1 pound = 4.536x10 1 kg 1 gallon = 3.785 m 3 Time: 1 minute = 60 seconds 1 hour = 3.6x10 3 seconds 1 day = 8.64x10 4 seconds 1 year = 3.1536x10 7 seconds Session 5 : 9/20 15
Examples of Unit Conversion 1) 3.4 acres is how many m 2? 2) Convert 4.3 lbs to kg 3) How many ounces are in one kg? Session 5 : 9/20 16
Concentration Many different ways of expressing concentration. Most common: Mass/Volume Mass/Mass Parts per million (ppm) Parts per billion (ppb) Molarity (M) Session 5 : 9/20 17
Mass/Volume: kg/m 3, g/cm 3, mg/ml, g/l Mass/Mass: ngdna/g Dry Soil Parts per billion (ppb): one part x substance per one billion parts of the substance it is in so long as the two components are in the same units! Parts per million (ppm): just like above, but one part x substance per one million parts of what it is in Molarity: moles per liter Session 5 : 9/20 18
Moles A mole of a substance consists of Avogadro s Number (6.022x10 23 ) of atoms or molecules There are 6.022x10 23 carbon atoms in one mole of carbon There are 6.022x10 23 water molecules in one mole of water The molecular weight (mw) is the mass in grams of one mole of a substance (grams/mole) Molecular weight of a compound can be determined by adding the molecular weights of its elemental components carbon 6 C Element Atomic Number (# protons in nucleus) Element Symbol 12.01 Session 5 : 9/20 19 Molecular Weight (g/mol)
Example: Find the molecular weight of salt, NaCl MW Na =14.99 g/mol MW Cl = 35.45 g/mol MW NaCl = MW Na + MW Cl = 50.44g/mol Session 5 : 9/20 20
Molarity Molarity (M) =moles/l 25 moles of potassium chloride (KCl) is dissolved in 2 L of water. What is the molarity? Session 5 : 9/20 21
Converting from Mass to Moles Convert from mass to moles using molecular weight Mass = # moles * MW Example: What is the mass of 9 moles of glucose, C 6 H 12 O 6? Session 5 : 9/20 22
Converting mass concentration to molarity Example: What is the molarity of a 8g/cm 3 MgCl 2 solution? Session 5 : 9/20 23
Dilutions C 1 V 1 =C 2 V 2 Example: If you start with a 15 g/ml solution of NaCl, what volume of the solution would you have to mix with what volume of water to make 20mL of a 6g/mL NaCl solution? Session 5 : 9/20 24
10x Dilutions: To make 10x dilutions from a starting stock solution: Stock Suspension + 900 μl fresh media + 900 μl fresh media + 900 μl fresh media + 900 μl fresh media Session 5 : 9/20 25
ph ph is a measure of how acidic or basic (alkaline) a substance is, and is determined by the abundance of hydrogen ions. More Acidic More Basic ACIDIC Average ph rainwater Human Blood Session 5 : 9/20 26