Exploring the Physics of Baumgartner's Sound Barrier Jump
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American High**We aren't endorsed by this school
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MATH AP
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Mathematics
Date
Dec 11, 2024
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12
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Can a Human Break the Sound Barrier? 1.Baumgartner is in free fall for 4 minutes and 20 seconds (260 seconds) before he deploys his parachute at an elevation of 8,420 feet above sea level. a.What was the vertical distance of the freefall? b.What was his average velocity during the freefall? 2.His elevation (in feet) above sea-level, t seconds after stepping off the balloon can be approximated by !(#) = 127850 − 16#!for 0 ≤ # ≤ 50. a.Look at the graph of !(#)below. Label both axes. b.Was Baumgartner traveling at a constant velocity? How do you know? c.What time does it look like Baumgartner is traveling the fastest? How can you tell?3.Let’s see if we can estimate his velocity exactly 30 seconds after leaving the balloon. a.What is his average velocity between # = 20and # = 30? Show your work. Is this faster or slower than the velocity at exactly 30 seconds? Explain. b.What is his average velocity between # = 30and # = 40? Show your work. Is this faster or slower than the velocity at exactly 30 seconds? Explain. On October 14th, 2012, Austrian skydiver Felix Baumgartner broke a world record for a high-altitude dive when he ascended 127,850 feet in a helium balloon and then went into a free fall lasting more than 4 minutes. 127,850-8420=119,430ftinfreetall-119430fT=-459346+elevation(feet(M·_time(S)No,there'snoconstantslopet=so,that'swhentheslopeissteepest.f(30)f(20)/30-20=-800+4/sslowerSlope@t=20&t=30isflatherthanslope&+=30f(40)-f(30)=10,2250-113,450=-11287t/S40-3010fasterSlopeissleeperafter+=30.
4.Let’s take an interval even closer to 30. a.Find the average velocity between # = 29and # = 30. Show your work. b.Find the average velocity between # = 30and # = 31. Show your work. 5.Are the estimates in 4a and 4b better or worse than those in 3a and 3b? Why? 6.How could we get an even better estimate? 7.We’re going to find the average velocity between # = 30and # = 30 + ℎ. Let’s break it down into steps. a.Find !(30 + ℎ). Simplify. b.Find !(30 + ℎ) − !(30). c.Write the expression for "($%&'))"($%)'using what you found above. d.What value of h would represent his velocity at exactly # = 30? Explain. e.Show how you could determine this velocity. 8.The speed of sound is 1,125.3 feet per second. Did Baumgartner go supersonic? f(s0)-+(29)=113480-114394=944++/30-29f(31)f(30)=112474-113450=ab++3138Theestimatesarebetterin44bbecausetheyareclosertogether.Useasmallerinterval127850-16(30+n)2=113450-940h-16h2113450960n-16h2- 113450=-960n-16h2-960h-16h2nn=0benotimehaspassedlim960h-1642=um(-960-16h=-90nephNo
Topic 2.1 – Instantaneous Rates of Change QuickNotes Check Your Understanding 1.Let !(4) = 4!− 44. a.Find the average rate of change on the interval [-1,5]. b.Find the instantaneous rate of change at 4 = 3. 2.Write, but do not evaluate, an expression that gives the instantaneous rate of change of 5(4) =)*$+at 4 = 2. 3.Which of the following gives the instantaneous rate of change of !(4)at 4 = −1. Choose all that apply: lim!→#$(−1 + ℎ) − $(−1)ℎlim$→%$(+) − $(−1)−1lim$→&'$(+) − $(−1)+ − 1lim!→($(−1 + ℎ) − $(−1)ℎlim!→($(+ + ℎ) − $(+)ℎlim$→&'$(+) − $(−1)+ − (−1)lim!→&'$(−1 + ℎ) − $(−1)ℎlim$→&'$(−1) − $(,)+ − (−1)+Isl-t=lim2 limnim-imX-2--
Find the instantaneous rate of change of ࠵?ሺ࠵?ሻ ൌ ࠵? െ ࠵?ଶat ࠵? ൌ െ1. lim→ሺାሻିሺሻlim௫→ሺ௫ሻିሺሻ௫ିIdentify the function we are working with. Then identify the x-value for the instantaneous rate of change (slope of the tangent line at a point). 1. lim→ହ ୪୬ቀమరశቁିହ ୪୬ቀభమቁFunction: ࠵?ሺ࠵?ሻ ൌ5 lnቀଶ௫ቁInstantaneous rate at ࠵? ൌ42. lim௫→ഏమୱ୧୬ ௫ିଵ௫ିഏమFunction: ࠵?ሺ࠵?ሻ ൌsinሺ࠵?ሻInstantaneous rate at ࠵? ൌగଶ2.1 Average and Instantaneous Rate of ChangeCalculusFind the average rate of change of each function on the given interval. Use appropriate units if necessary. 1. ࠵?ሺ࠵?ሻ ൌ ࠵?ଶെ2; ሾെ1, 3ሿ2. ࠵?ሺ࠵?ሻ ൌ2௧; ሾ2, 4ሿ࠵?represents years ࠵?represents dollars 3. ℎሺ࠵?ሻ ൌtanሺ࠵?ሻ 4; ቂగସ,ଷగସቃℎrepresents hair ࠵?represents months 4. ࠵?ሺ࠵?ሻ ൌln࠵?on the interval 2 ࠵? 7. 5. ࠵?ሺ࠵?ሻ ൌcos࠵?on the interval െ1 ࠵? 0. Practice Write your questions and thoughts here! In7-In27-2=025/cos(0)-cos(=0
Use the following table to find the average rate of change on the given interval. ࠵?(Minutes) 0341213࠵?ሺ࠵?ሻ(Feet) െ24െ75106. ሾ3, 13ሿ7. 0 ࠵? 128. ሾ3, 4ሿUse the following graph to find the average rate of change on the given interval. 9. െ5 ࠵? െ210. ሾെ1, 5ሿ11. െ4 ࠵? െ2The graphs of ࠵?and ࠵?are given below. For each function, find the average rate of change on the given interval. 12. ℎሺ࠵?ሻ ൌ ࠵?ሺ࠵?ሻ ࠵?ሺ࠵?ሻon ሾെ4,3ሿ13. ࠵?ሺ࠵?ሻ ൌ ࠵?൫࠵?ሺ࠵?ሻ൯on ሾെ4,0ሿ14. ࠵?ሺ࠵?ሻ ൌ ࠵?൫࠵?ሺ࠵?ሻ൯on ሾെ2,3ሿxyxyf(x)g(x)51-2)12-8F2++/min52=-S-1-12n(s-n=
Find the instantaneous rate of change of each function at the given x-value. Use the form ࠵?࠵?࠵?࠵?→࠵?࠵?ሺ࠵?ା࠵?ሻି࠵?ሺ࠵?ሻ࠵?. 15. ࠵?ሺ࠵?ሻ ൌ ࠵?ଶെ ࠵?at ࠵? ൌ െ116. ࠵?ሺ࠵?ሻ ൌ √࠵?at ࠵? ൌ517. ࠵?ሺ࠵?ሻ ൌଵ௫at ࠵? ൌ2Find the instantaneous rate of change of each function at the given x-value. Use the form ࠵?࠵?࠵?࠵?→࠵?࠵?ሺ࠵?ሻି࠵?ሺ࠵?ሻ࠵?ି࠵?. 18. ࠵?ሺ࠵?ሻ ൌ2࠵?ଶ3࠵? െ4at ࠵? ൌ െ319. ࠵?ሺ࠵?ሻ ൌ √3࠵?at ࠵? ൌ720. ࠵?ሺ࠵?ሻ ൌଵହ௫at ࠵? ൌ െ2Each limit represents the instantaneous rate of change of a function. Identify the original function, and the x-value of the instantaneous rate of change. 21. lim௫→భඥೣమషమೣିభ√యఱ௫ିFunction: ࠵?ሺ࠵?ሻ ൌInstantaneous rate at ࠵? ൌ22. lim௫→ିଶ൫ଷ௫ିଽ௫మ൯ାሺସଶሻ௫ାଶFunction: ࠵?ሺ࠵?ሻ ൌInstantaneous rate at ࠵? ൌ23. lim→ଷ ୪୬ሺଶାሻିଷ ୪୬ ଶFunction: ࠵?ሺ࠵?ሻ ൌInstantaneous rate at ࠵? ൌ24. lim→ଷሺଵାሻమିሺଵାሻାଵାሺଷሻFunction: ࠵?ሺ࠵?ሻ ൌInstantaneous rate at ࠵? ൌ25. lim௫→ഏమ௫మୱ୧୬ ௫ିయഏమమ௫ିഏమFunction: ࠵?ሺ࠵?ሻ ൌInstantaneous rate at ࠵? ൌ26. lim→୪୭ሺଶିସሺିହሻሻି୪୭ሺଶଶሻFunction: ࠵?ሺ࠵?ሻ ൌInstantaneous rate at ࠵? ൌ27. lim௫→ହభ√యೣିభ√భఱ௫ିହFunction: ࠵?ሺ࠵?ሻ ൌInstantaneous rate at ࠵? ൌ28. lim→లሺయశሻశభିభవFunction: ࠵?ሺ࠵?ሻ ൌInstantaneous rate at ࠵? ൌmm
29. Let ࠵?be the function defined by ࠵?ሺ࠵?ሻ ൌln 7࠵?. The average rate of change of ࠵?over the interval ሾ2,࠵?ሿis 41, where ࠵? 2. Which of the following is an equation that could be used to find the value of ࠵?? (A) ࠵?ሺ࠵?ሻ ൌ41(B) ࠵?ሺ࠵?ሻ െ ࠵?ሺ2ሻ ൌ41(C) ሺሻିሺଶሻିଶൌ41(D) ሺሻାሺଶሻଶൌ4130. Find the average rate of change of ࠵?ሺ࠵?ሻ ൌsin࠵?ln࠵?on the interval 1 ࠵? ࠵?. 31. Today’s school lunch was inappropriately thrown over the school fence by Mr. Kelly. For 0 ࠵? 90, the amount of food remaining (assuming no animals eat it) is modeled by ࠵?ሺ࠵?ሻ ൌ544.311ሺ0.907ሻ௧, where ࠵?ሺ࠵?ሻis measured in grams and ࠵?is measured in days. Find the average rate of change of ࠵?ሺ࠵?ሻover the interval 0࠵? 90. Indicate units of measure. 32. A continuous function ࠵?is shown above and defined on the closed interval െ5 ࠵? 4. For how many values of ࠵?, െ5൏ ࠵? ൏4, is the average rate of change of ࠵?on the interval ሾ࠵?, 1ሿequal to 0? Give a reason for your answer. xyTest Prep 2.1 Average and Instantaneous Rate of Change2.Aug=0meansaslopeofzero(horizontalrivel
2.2 Defining the DerivativeCalculusFind the derivative using limits. If the equation is given as ࠵? ൌ, use Leibniz Notation: ࠵?࠵?࠵?࠵?. If the equation is given as ࠵?ሺ࠵?ሻ ൌ, use Lagrange Notation: ࠵?ᇱሺ࠵?ሻ. WRITE SMALL!!1. ࠵?ሺ࠵?ሻ ൌ7െ6࠵?2. ࠵? ൌ5࠵?ଶെ ࠵?3. ࠵? ൌ √5࠵? 24. ࠵?ሺ࠵?ሻ ൌଵ௫ିଶFor each problem, use the information given to identify the meaning of the two equations in the context of the problem. Write in full sentences! 5. ࠵?is the number of championships Sully has won while coaching basketball. ࠵?is the number of years since 2002 for the function ࠵?ሺ࠵?ሻ. ࠵?ሺ12ሻ ൌ3and ࠵?ᇱሺ12ሻ ൌ0.46. ࠵?is the distance (in miles) from home when you walk to school. ℎis the number of hours since 7:00 a.m. for the function ࠵?ሺℎሻ. ࠵?ሺ0.5ሻ ൌ1.2and ࠵?ᇱሺ0.5ሻ ൌ െ11Practice f'(X)=lim<b(x+h)(-6x=-6HimS(x+-(h) -IInn+0=10X-1=im=5/222014:Sullywon3championshipsand0.47:30:12milesfromhomeagoingbackchampionshipsperyr.home&11mph
7. ࠵?is the number of cartoon shows Mr. Kelly watches every week. ࠵?is the number of children Mr. Kelly has for the function ࠵?ሺ࠵?ሻ. ࠵?ሺ7ሻ ൌ25and ࠵?ᇱሺ7ሻ ൌ38.࠵?is the number of gray hairs on Mr. Brust’s head. ࠵?is the number of students in his 4thperiod. ࠵?ሺ26ሻ ൌ501and ࠵?ᇱሺ15ሻ ൌ130For each problem, create an equation of the tangent line of ࠵?at the given point. Leave in point-slope. 9. ࠵?ሺ7ሻ ൌ5and ࠵?ᇱሺ7ሻ ൌ െ210. ࠵?ሺെ2ሻ ൌ3and ࠵?ᇱሺെ2ሻ ൌ411. ࠵?ሺ࠵?ሻ ൌ3࠵?ଶ2࠵?; ࠵?ᇱሺ࠵?ሻ ൌ6࠵? 2;࠵? ൌ െ212. ࠵?ሺ࠵?ሻ ൌ10√6࠵? 1; ࠵?ᇱሺ࠵?ሻ ൌଷ√௫ାଵ;࠵? ൌ413. ࠵?ሺ࠵?ሻ ൌcos 2࠵?; ࠵?ᇱሺ࠵?ሻ ൌ െ2 sin 2࠵?;࠵? ൌగସ14. ࠵?ሺ࠵?ሻ ൌtan࠵?; ࠵?ᇱሺ࠵?ሻ ൌsecଶ࠵?;࠵? ൌగଷ15. Let ࠵?ᇱሺ࠵?ሻ ൌlim→ሺ௫ାሻమି௫మ. For what value of ࠵?does ࠵?ሺ࠵?ሻ ൌ4? (A) െ4(B) െ1(C) 1(D) 2(E) 4Test Prep 2.2 Defining the Derivative26kids-solgrayhairsif<kids,Iscartoonsperweekrateincreases3perweek.gaining130graysperkid.y-S=-z(x-7)y3=4(x+2)y-8=10(x+2)y=-z(x-4/4)O
16. The graph of the function ࠵?, along with a table of values, are shown below. Approximate the value of ࠵?ᇱሺ5.5ሻusing data from the table. Show computations that lead to your answer. ࠵?4.555.566.57࠵?ሺ࠵?ሻ2.1692.3212.45954.5417. The figure below shows the graph of the line tangent to the graph of ࠵?at ࠵? ൌ0. Of the following, which must be true? (A) ࠵?ᇱሺ0ሻ ൌ െ࠵?ሺ0ሻ(B) ࠵?ᇱሺ0ሻ ൌ ࠵?ሺ0ሻ(C) ࠵?ᇱሺ0ሻ ࠵?ሺ0ሻ(D) ࠵?ᇱሺ0ሻ ൏ ࠵?ሺ0ሻxyxy2459-232=0.2O
2.3 Estimating DerivativesCalculus Estimate the derivative at the given point by using a calculator.1. ࠵?(࠵?) = ࠵?√2 − ࠵?; find ࠵?ᇱ(−10). 2. ࠵?(࠵?) = sec(5࠵?); find ࠵?ᇱ(2). 3. ࠵?(࠵?) = ln൫√࠵?൯; find ࠵?ᇱ(1). 4. ࠵?(࠵?) = ࠵?ೣయ; find ࠵?ᇱ(4). 5. ࠵?(࠵?) = tan(sin ࠵?); find ࠵?ᇱ(−3). 6. ࠵?(࠵?) = 2୪୬(௫); find ࠵?ᇱ(2). 7. The model ࠵?(࠵?) =௫ୡ୭ୱ ௫measures the height of bird in meters where ࠵?is seconds. Find ࠵?ᇱ(2). 8. The model ࠵?(࠵?) = sinଶ(࠵?)measures the depth of a submarine measured in feet where ࠵?is minutes. Find ࠵?ᇱ(12.5). 9. The model ࠵?(࠵?) = √࠵?−ଵ௫ିଵmeasures the number of stocks sold where ࠵?is days. Find ࠵?ᇱ(12)For each function, write the equation of the tangent line at the given value of ࠵?. 10. ࠵?(࠵?) =୪୬ ଶ௫ସ௫at ࠵? = 1. 11. ࠵?(࠵?) = cos(tan(࠵?))at ࠵? = 2. 12. ࠵?(࠵?) =௫ర√௫at ࠵? = 3. 13. ࠵?(࠵?) = ࠵?ଶsin ቀଵ௫ቁat ࠵? = 7. Use the tables to estimate the value of the derivative at the given point. Indicate units of measures. 14. ࠵?Hours 13479࠵?(࠵?)visitors 120476595807902a. ࠵?ᇱ(8)b. ࠵?ᇱ(3.5)Practice 4.907-3864058098m/s-0132++/miny0173=0.0767(x-1y+0576=47188(x2)902-807/9-7=475Visitors/urbas-476/4-3=119visitors(nr
15. ࠵?cm 1123263245࠵?(࠵?)℃7151403610a. ࠵?ᇱ(17)b. ࠵?ᇱ(24.5)16. ࠵?years 0371520࠵?(࠵?)Students per year 5207−2−4a. ࠵?ᇱ(1.5)b. ࠵?ᇱ(11)17. ࠵?Days 513455070࠵?(࠵?)Pages per day 5120213658a. ࠵?ᇱ(47.5)b. ࠵?ᇱ(9)18. ࠵?seconds 10304565100࠵?(࠵?)Gallons per second 1005790786434209a. ࠵?ᇱ(20)b. ࠵?ᇱ(82.5)19. ࠵?Carries 312152130࠵?(࠵?)yards 1510798150272a. ࠵?ᇱ(25.5)b. ࠵?ᇱ(13.5)51-71/23-11=1.667·Cpercom40-51/26-23=3667·Cperum