Math 102: A Log-jam. f(x+h) f(x) h. = 10 x ( 10 h 1. = 10x+h 10 x h. = 10x 10 h 10 x h. 2. The hyperbolic cosine function is defined by

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1 Mat 102: A Log-jam 1. If f(x) = 10 x, sow tat f(x+) f(x) ( 10 = 10 x ) 1 f(x+) f(x) = 10x+ 10 x = 10x x = 10 x ( 10 1 ) 2. Te yperbolic cosine function is defined by cos(x) = ex +e x 2 Te yperbolic sine function is defined by sin(x) = ex e x 2 Use tese definitions to prove te following identities. (a) cos( x) = cos(x) (b) sin( x) = sin(x) (c) [cos(x)] 2 [sin(x)] 2 = 1 (d) sin(x + y) = sin(x) cos(y) + cos(x) sin(y) 3. Wen a certain medical drug is administered to a patient, te number of milligrams remaining in te patient s bloodstream after t ours is modeled by D(t) = 50e 0.2t How many milligrams of te drug remain in te patient s bloodstream after 3 ours? D(3) = 50e 0.2(3) =

2 4. A radioactive substance decays in suc a way tat te amount of mass remaining after t days is given by te function m(t) = 13e 0.015t were m(t) is measured in kilograms. (a) Find te mass at time t = 0. m(0) = 13 (b) How muc of te mass remains after 45 days? m(45) = 13e 0.015(45) = (c) Wen will only one-fourt of te initial amount remain? t =

3 5. Radioactive iodine is used by doctors as a tracer in diagnosing certain tyroid gland disorders. Tis type of iodine decays in suc a way tat te mass remaining after t days is given by te function were m(t) is measured in grams. (a) Find te mass at time t = 0. m(t) = 6e 0.087t m(0) = 6 (b) How muc of te mass remains after 20 days? m(20) = (c) Find te alf-life of radioactive iodine. t = A skydiver jumps from a reasonable eigt above te ground. Te air resistance se experiences is proportional to er velocity, and te constant of proportionality is 0.2. It can be sown tat te downward velocity of te sky diver at time t is given by v(t) = 80(1 e 0.2t ) were t is measured in seconds and v(t) is measured in feet per second (ft/s). (a) Find te initial velocity of te sky diver. v(0) = 0 (b) Find te velocity after 5 s and after 10 s. v(5) = m/s, v(10) = m/s 7. Assume tat a population of rabbits beaves according to te logistic growt model 300 n(t) = 0.05+( 300 n 0 0.5)e 0.55t were n 0 is te initial rabbit population. (a) If te initial population is 50 rabbits, wat will te population be after 12 years? n(12) =

4 (b) Draw graps of te function n(t) for n 0 = 50,000, 2000, 8000, and (c) From te graps in part (ii) observe tat, regardless of te initial population, te rabbit population seems to approac a certain number as time goes on. Wat is tat number? Wat does tat number represent? Solution. After graping tese, te population appears to stabilize around 6057 rabbits. 4

5 8. If $10,000 is invested at an interest rate of 10% per year compounded semiannually, find te value of te investment after te given number of years. (a) 5 years: $ (b) 10 years $ (c) 20 years $ (d) Wen te te investment quadruple in value? 14.2 years 9. If $10,000 is invested at an interest rate of 10% per year compounded continuously, find te value of te investment after te given number of years. (a) 5 years $ (b) Wen will te investment double in value? 6.93 years 10. Suppose you are offered a job tat lasts one mont, and you are to be very well paid. Wic of te following metods of payment is more profitable for you? (a) One million dollars at te end of te mont, or (b) Two cents on te first day of te mont, 4 cents on te second day, 8 cents on te tird day, and, in general, 2 n cents on te n t day. Te doubling of 2 cents per day for one mont is 2 30 =$10,737, Te two pennies doubled eac day for one mont is te better salary. 11. Your matematics instructor asks you to sketc a grap of te exponential function f(x) = 2 x for x between 0 and 40, using a scale of 10 units to one inc. Wat are te dimensions of te seet of paper you will need to sketc tis grap? Te paper needs to be 4 inces wide and 1,735,340 miles long. (Note, one mile is equivalent to 5280 feet, and 12 inces equals one foot.) 12. Teageofanancientartifactcanbe determinedbyte amountofradioactive carbon-14 remaining in it. If D 0 is te original amount of carbon-14 and D is te amount remaining, ten te artifact s age A (in years) is given by ( ) D A = 8267ln D 0 Find te age of te object if te amount D of carbon-14 tat remains in te object is 73% of te original amount D 0. Te artifact is years old. 5

6 13. A certain strain of bacteria divides every tree ours. If a colony is started wit 50bacteria, ten te time t (in ours) required for te colonyto grow to N bacteria is given by t = 3 log(n/50) log2 Find te time required for te colony to grow to a million bacteria. t = years 14. Te rate at wic a battery carges is slower te closer te battery is to its maximum carge C 0. Te time (in ours) required to carge a fully discarged battery to a carge C is given by ) t = kln (1 CC0 were k is a positive constant tat depends on te battery. For a certain battery, k = If tis battery is fully discarged, ow long will it take to carge to 90% of its maximum carge C 0? t =.58 ours 15. A googol is , and a googolplex is 10 googol. Find log(log(log(googolplex))) and log(log(googol)) Solution. Bot expressions simplify to Witout using your calculator, wic is larger, log 4 17 or log Explain your reasoning. Solution. Te first expression log 4 17 is just over 2, for 4 2 = 16. Te second expression, log 5 24 < 2. Tus, log 4 17 is te larger value. 6

7 17. Use te Cange of Base Formula to sow tat loge = 1 ln10 Tis can be demonstrated by working from one side of te equation and ending at te oter side. I will sow tis identity by beginning on te rigt side and ending on te left side. 1 ln10 = (ln10) 1 ( log10 = loge = loge ) Simplify: (log 2 5)(log 5 7). Give an exact answer. (log 2 5)(log 5 7) = log5 log7 log2log5 = log7 log2 = log 2 7 7

8 19. Sow tat ln(x x 2 1) = ln(x+ x 2 1). 20. Determine weter eac equation is true or false for all values, ignore values of te variables for wic any term is undefined. (a) log( x y ) = logx log y False (b) log 2 (x y) = log 2 x log 2 y False (c) log 5 ( a b 2 ) = log 5 a 2log 5 b True (d) log2 z = zlog2 True (e) (logp)(logq) = logp +logq False (f) loga logb = loga logb False (g) (log 2 7) x = xlog 2 7 False () log a a a = a True (i) log(x y) = logx log y False (j) ln( 1 A ) = lna True 21. A sum of $1000 is invested at an interest rate of 4% per year. Find te time required for te amount to grow to $4000 if interest is compounded (a) Annually t = 35.3 years (b) Quarterly t = 34.8 years (c) Continuously t = 34.7 years 22. A sample of 15 g of radioactive iodine decays in suc a way tat te mass remaining after t days is given by m(t) = 15e 0.087t were t is measured in grams. After ow many days does only 5 g remain? Solution. t = days. 23. A small lake is stocked wit a certain species of fis. Te fis population is modeled by te function P = e.8t were P is te number of fis in tousands and t is measured in years since te lake was stocked. (a) Find te fis population after 3 years fis (b) After ow many years will te fis population reac 5000 fis? 1.73 years 8

9 24. Solve te equation x 1 log x = 5 Provide support for your answers. No solution. 25. Solve te following. (a) (x 1) log(x 1) = 100(x 1) Take te log of eac side. x = 99, 1.1 (b) log 2 x+log 4 x+log 8 x = 11 Cange all logs to base 2. x = 64 (c) 4 x 2 (x+1) = 3 Write as a quadratic in 2 x. x = ln3 ln2 26. An infections strain of bacteria increase in number at a relative growt rate of 200% per our. Wen a certain critical number of bacteria are present in te bloodstream, a person become ill. If a single bacterium infects a person, te critical level is reaced in 24 ours. How long will it take for te critical level to be reaced if te same person is infected wit 10 bacteria? ours 27. Radium-221 as a alf-life of 30s. How long will it take for 95% of a sample to decay? 129 seconds 28. A wooden artifact from an ancient tomb contains 65% of te carbon-14 tat is present in living trees. How long ago was te artifact made? (Te alf-life of carbon-14 is 5730 years.) 3561 years old 29. Te burial clot of an Egyptian mummy is estimated to contain 59% of te carbon-14 it contained originally. How long ago was te mummy buried? (Te alf-life of carbon-14 is 5730 years.) 4361 years old 9