22Example: Graph and Determine the horizontal asymptotes 𝑓 𝑥 = 7 1+3∗ .6 𝑥
23AnswerHorizontal asymptotes at y=0 and y=7Y-intercept at (0,7/4)
24Group Work: Graph and determine the horizontal asymptotes 𝐺(𝑥)= 𝑒 −4𝑥
25AnswerHorizontal asymptotes y=0 and y=26Y-intercept at (0,26/3)
26Word Problems: Year 2000 782,248 people Year 2010 923,135 people Use this information to determine when the population will surpass 1 million people? (hint use exponential function)
27Group Work Year 1990 156,530 people Year 2000 531,365 people Use this information and determine when the population will surpass 1.5 million people?
28Word Problem The population of New York State can be modeled by 𝑓 𝑡 = 𝑒 − 𝑡𝑓 𝑖𝑠 𝑡ℎ𝑒 𝑝𝑜𝑝𝑢𝑙𝑎𝑡𝑖𝑜𝑛 𝑖𝑛 𝑚𝑖𝑙𝑙𝑖𝑜𝑛𝑠 𝑎𝑛𝑑 𝑡 𝑖𝑠 𝑡ℎ𝑒 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑦𝑒𝑎𝑟𝑠 𝑠𝑖𝑛𝑐𝑒 1800A) What’s the population in 1850?B) What’s the population in 2010?C) What’s the maximum sustainable population?
30Group WorkIn chemistry, you are given half-life formulas 𝑃 𝑡 = 𝑃 0 𝑏 𝑡 𝑟 𝑟=ℎ𝑎𝑙𝑓 𝑙𝑖𝑓𝑒, 𝑡=𝑡𝑖𝑚𝑒, 𝑃 0 =𝑖𝑛𝑖𝑡𝑖𝑎𝑙 𝑎𝑚𝑜𝑢𝑛𝑡 If you are given a certain chemical have a half-life of 56.3 minutes. If you are given 80 g first, when will it become 16 g?
50Key Word Maximum sustainable population What does this mean? What function deals with this?
51Maximum Sustainable Population Exponential growth is unrestricted, but population growth often is not. For many populations, the growth begins exponentially, but eventually slows and approaches a limit to growth called the maximum sustainable population.
52Homework Practice (Do in class also) P 296 #1-44 eoo
72What you’ll learn about Properties of LogarithmsChange of BaseGraphs of Logarithmic Functions with Base bRe-expressing Data… and whyThe applications of logarithms are based on their manyspecial properties, so learn them well.
106Orders of MagnitudeThe common logarithm of a positive quantity is its order ofmagnitude.Orders of magnitude can be used to compare any like quantities:A kilometer is 3 orders of magnitude longer than a meter.A dollar is 2 orders of magnitude greater than a penny.New York City with 8 million people is 6 orders of magnitude bigger than Earmuff Junction with a population of 8.
107Note:In regular cases, how you determine the magnitude is by how many decimal places they differIn term of Richter scale and pH level, since the number is the power or the exponent, you just take the difference of them.
108Example:What’s the difference of the magnitude between kilometer and meter?It is 3 orders of magnitude longer than a meter
109Example:The order of magnitude between an earthquake rated 7 and Richter scale rated 5.5.The difference of magnitude is 1.5
110Group Work Find the order of magnitude: Between A dollar and a penny A horse weighing 500 kg and a horse weighing 50g8 million people vs population of 8
111Answer 2 orders of magnitude 4 orders of magnitude
112Group Work Find the difference of the magnitude: Sour vinegar a pH of 2.4 and baking soda pH of 8.4Earthquake in India 7.9 and Athens 5.9
113Answer6 orders of magnitude2 orders of magnitude
115Example:How many times more severe was the 2001 earthquake in Gujarat, India ( 𝑅 1 =7.9) than the 1999 earthquake in Athens, Greece ( 𝑅 2 =5.9)
116Group Work: Show workHow many times more severs was the earthquake in SF ( 𝑅 1 =6.5) than the earthquake in PS ( 𝑅 2 =3.6)?
117pHIn chemistry, the acidity of a water-based solution is measured by the concentration of hydrogen ions in the solution (in moles per liter). The hydrogen-ion concentration is written [H+]. The measure of acidity used is pH, the opposite of the common log of the hydrogen-ion concentration:pH=-log [H+]More acidic solutions have higher hydrogen-ion concentrations and lower pH values.
118Example:Sour vinegar has pH of 2.4 and a box of Leg and Sickle baking soda has a pH of 8.4.A) what are their hydrogen-ion concentration?B) How many more times greater is the hydrogen-ion concentration of the vinegar than of the baking soda?
119Group Work A substance with pH of 3.4 and another with pH of 8.1 A) what are their hydrogen-ion concentration?B) How many more times greater is the hydrogen-ion concentration?
136Example Compounding Continuously Suppose Paul invests $400 at 8% annual interest compounded continuously. Find the value of his investment after 5 years.
137Example Compounding Continuously Suppose Paul invests $400 at 8% annual interest compounded continuously. Find the value of his investment after 5 years.
138Group WorkSuppose you have $10000, you invest in a company where they give you 12% interest compounded continuously. Find the value of your investment after 40 years.
139Annual Percentage Yield A common basis for comparing investments is the annual percentage yield (APY) – the percentage rate that, compounded annually, would yield the same return as the given interest rate with the given compounding period.
140Example Computing Annual Percentage Yield Meredith invests $3000 with Frederick Bank at 4.65% annual interest compounded quarterly. What is the equivalent APY?
141Example Computing Annual Percentage Yield Meredith invests $3000 with Frederick Bank at 4.65% annual interest compounded quarterly. What is the equivalent APY?
143Future Value of an Annuity At the end of each quarter year, Emily makes a $500 payment into the Lanaghan Mutual Fund. If her investments earn 7.88% annual interest compounded quarterly, what will be the value of Emily’s annuity in 20 years?Remember i=r/k
144Group WorkYou are currently 18 and you want to retire at age 65. You decide to invest in your future. You are putting in $35 month. If your investment earn 12% annual interest compounded monthly, what will the value of your annuity when you retire?