MA139 Final Review Sheet NEW 2024 PDF

MA139 Bentley University CICCARELLI Final Review Sheet The Final Exam is cumulative: you are responsible for all of the material covered in class (even if you w...

MA139 Bentley University CICCARELLI Final Review Sheet The Final Exam is cumulative: you are responsible for all of the material covered in class (even if you were not there!), and all of the homework from the entire semester. If you did not do certain homework problems, I suggest you do them. If you missed any classes, get notes from someone in the class. Write down the day, date and time of your final exam here: _________________________________ (It’s on the syllabus – don’t ask me!) The exam will be in our classroom. There will be NO MAKEUPS OR RESCHEDULING, so make your travel plans accordingly! ___________________________________________________________________________________________________________________________________ No notes, books, laptops, etc. are allowed on the final. You will need a scientific (or graphing) calculator for the test, so BRING ONE. As usual, partial credit will be given on the exam, BUT ONLY IF YOU SHOW YOUR WORK! Chapters 8 & 9 homework will be accepted only until our last class, so catch up if you need to. ___________________________________________________________________________________________________________________________________ This Final Review Sheet covers only Chapters 8 & 9, not the whole semester. (Otherwise it would be 80 problems long!) So use our previous quizzes, tests and review sheets to review older material – I gave you solutions to everything! Remember that if you didn’t do so well on one of the tests, it’ll only count as 10% of your course grade (hopefully!), but that means that this final will count as 35%. So get to work – I want you all to finish up strong. Make sure to try this Review Sheet by our last review class, and have lots of questions ready! Thanks for being such a good group – I’ve had a blast this semester. Best of luck with all of your finals! Your Fearless Leader, Pete  MA139 Practice Problems for Final Exam CICCARELLI Sections 8.4 – 9.5 Here are some extra practice problems related to the topics we’ve studied in class. These are not due; however, problems like these (possibly exactly like these) may be on the last quiz &/or final exam. Feel free to work together, bouncing ideas off of each other, but each person should be able to write up their own solutions, as you’ll need to on a quiz or test. I) DIFFERENTIAL EQUATIONS 1.) For the following relationships between some quantity and its rate of change, name some variables and formulate a differential equation to model the relationship. Do not attempt to solve it. Recall that ∝ is the symbol for proportional. Use a (multiplicative) constant of proportionality 𝑘 to turn the “∝” into an 𝑑𝑦 𝑑𝑦 “=”. For example, if 𝑑𝑡 ∝ 𝑦, then 𝑑𝑡 = 𝑘𝑦. a) The rate of change of atmospheric pressure with respect to altitude is proportional to pressure. b) The rate of change of an object’s temperature with respect to time is proportional to the temperature difference between the object and its ambient surroundings. c) The time rate of a rumor’s spread throughout a community of 𝑀 people is proportional to the product of the number of people who have heard the rumor & the number who have not yet heard it. 2.) Give the definition of a separable differential equation, and make up an example. (No need to solve it.) 3.) Solve the separable differential equations. If given an initial condition (pt. on the graph), find the specific solution; otherwise, find the general solution. Remember to look for any equilibrium solutions! 𝑑𝑦 𝑦2 a) = 𝑥 3 through pt. (1, 1) 𝑑𝑥 𝑑𝑦 b) = (𝑦 − 5)2 𝑑𝑥 𝑑𝑦 cos 𝑦 2 c) = ( sin 𝑥 ) 𝑑𝑥 𝑑𝑦 d) 𝑥 + 2𝑦 ∙ √1 + 𝑥 2 𝑑𝑥 = 0 through pt. (0, 1) 𝑑𝑦 2𝑥+sec2 𝑥 e) = through pt. (0, −5) 𝑑𝑥 2𝑦 𝑑𝑦 f) = 3𝑥 2 𝑦 through pt. (0, 2) 𝑑𝑥 𝑑𝑦 g) 𝑦 cos 𝑥 ∙ 𝑑𝑥 = 𝑒 −𝑦 sin(2𝑥) through pt. (0, 0) 𝑑𝑦 𝑦 h) 𝑥𝑦 3 𝑑𝑥 = 𝑥−1 4.) a) Give the definition of an autonomous differential equation, and make up an example. (No need to solve it.) b) Explain how you can tell if a slope field is that of an autonomous differential equation. c) Which (if any) of the differential equations in Problem 3 are autonomous? 5.) Give the definition of a first-order linear differential equation, and make up an example. (No need to solve it.) 6.) Find the specific solutions to the following first-order linear differential equations. 𝑑𝑦 a) (1 + 𝑥) 𝑑𝑥 + 𝑦 = 1 + 𝑥 where 𝑥 > 0 and 𝑦(0) = 2 2 𝑑𝑦 𝑒𝑥 b) (𝑥 + 1) 𝑑𝑥 − 2𝑥(𝑥 + 1)𝑦 = 𝑥+1 where 𝑥 > −1 and 𝑦(0) = 5 𝑑𝑦 𝜋 𝜋 c) + (tan 𝑥) ∙ 𝑦 = cos 2 𝑥 where 𝑥 is in (− 2 , 2 ) and 𝑦(0) = 4 𝑑𝑥 𝑑𝑦 d) + (cos 𝑥) ∙ 𝑦 = cos 𝑥 where 𝑦(0) = 2 𝑑𝑥 𝑑𝑦 e) + 𝑦 = 𝑥 + 𝑒 𝑥 where 𝑦(0) = 0 𝑑𝑥 𝑑𝑦 35 f) 𝑥 𝑑𝑥 + 2𝑦 = 𝑥 4 ln 𝑥 where 𝑦(1) = 36 7.) (Exponential Decay Model) Einsteinium-252 decays to 64.3% of its original amount in 300 days. a) How long until 30% of the original material is left? b) Find the half-life of Einsteinium-252. 8.) (Exponential Decay Model) Due to inflation, the value of a dollar “decays” to about 97% of its original value in a year. Assuming the trend continues indefinitely: a) Find the “half-life” of a dollar. b) How long until a 2020 dollar is worth 30¢ (30 cents, or $.30)? c) Should you put your money under a mattress for “safe-keeping”, as many people used to do years ago, because they didn’t trust banks? Why or why not? 9.) (Exponential Decay Model) Recall that for carbon-14 ( 14C), the half-life is 5730 years. The famous “Lucy” skeleton found in Hadar, Ethiopia in 1974 has been shown to be about 3.2 million years old. a) What percent of the original 14C would you expect to be left? b) Why do you think carbon dating is only used for artifacts under about 50,000 years old? 10.) (Exponential Growth Model) A bacteria colony starts with 500 bacteria and grows at a rate proportional to its size. After 3 hours, there are 8000 bacteria. When will the population reach 30,000? 11.) (Exponential Growth Model) How long will it take $1000 to double, if it is invested at 3% APR compounded 𝑚 𝑚𝑡 a) monthly? (𝑃(𝑡) = 𝑃0 (1 + 𝑚) $ after 𝑡 yrs., compounded 𝑚 times/yr., & 𝑟 = APR as a decimal) b) continuously? ( 𝑃(𝑡) = 𝑃0 𝑒 𝑟𝑡 $ after 𝑡 yrs.) 12.) (Logistic Model) Upon return from Spring Break, the Messed Style Virus spreads through a dormitory of 100 students. The spread of the disease (symptoms include purple mullets, plastic clogs and Yankees 100 hats) is modeled by 𝐼(𝑡) = 1+99𝑒 −.85𝑡 infected people, 𝑡 days later. a) How many people brought the disease back from Spring Break (were initially infected)? b) How many were infected 4 days later? c) Take a limit to see how many people get infected “in the long run”. Then graph 𝑦 = 𝐼(𝑡) and compare your limit to the graph’s behavior. d) When was the disease spreading most rapidly? “Eyeball it” from the graph, but then find the answer algebraically. 13.) (Logistic Model) A small town has 1000 inhabitants. The people of this community enjoy gossip immensely, almost as much as certain math professors enjoy making up inane Calculus problems. At noon, 50 people have heard the rumor, and by 4 PM, half the town knows it. a) Write the logistic model for the number of people who have heard the rumor, 𝑡 hours after noon. b) How long until 90% of the population has heard it? 𝑑𝑦 𝑦 14.) (Logistic Model) Give the logistic model that is the solution to 𝑑𝑡 =.8𝑦 (1 − 500) if 𝑦(0) = 50. 𝑑𝑦 𝑀 −𝑘𝑡 15.) (Gompertz Model) Show that the solution to the Gompertz model = 𝑘𝑦 ln ( 𝑦 ) is 𝑦(𝑡) = 𝑀𝑒 −𝐶𝑒. 𝑑𝑡 What is lim 𝑦(𝑡)? 𝑡→∞  𝑑𝑦 16.) (Limited Growth Model) Show that the specific solution to the Limited Growth Model = 𝑘(𝑀 − 𝑦) 𝑑𝑡 with initial condition 𝑦(0) = 𝑦0 is 𝑦(𝑡) = 𝑀 − 𝐶𝑒 −𝑘𝑡 , where 𝐶 = 𝑀 − 𝑦0. What is lim 𝑦(𝑡)? (This 𝑡→∞ model is often used for learning curves, where 𝑀 = max level of knowledge/performance, so usually the initial condition used is 𝑦(0) = 0.) 17.) (Extra Bonus Scary Differential Equation) We have mentioned that the exponential growth model 𝑑𝑦 = 𝑘𝑦 (where 𝑘 > 0, 𝑦0 > 0) is unrealistic, due to the fact that growth is uninhibited: the population 𝑑𝑡 𝑦(𝑡) = 𝑦0 𝑒 𝑘𝑡 → ∞ as time 𝑡 → ∞. Obviously, limited space on Earth and limited food supply would prevent this from occurring. 𝑑𝑦 𝒅𝒚 A scarier model than = 𝑘𝑦: = 𝒌𝒚𝟏+𝜺 , where 𝜀 > 0. It turns out that if the exponent is even 𝑑𝑡 𝒅𝒕 slightly larger than 1, we have something faster than exponential growth. This is called a DOOMSDAY EQUATION, because the population becomes infinite not as 𝑡 → ∞, but in a finite amount of time! (Hey, I’m not making this stuff up…) Here is an example: a) Solve the Doomsday Equation (find the general solution) if 𝑘 =.8 and 𝜀 =.01. b) Assume the population starts with 2 bunnies (so 𝑦(0) = 2). Use this initial condition to solve for the “𝐶” in the general solution, giving you the specific solution. c) If time 𝑡 is measured in months, how long is it until Doomsday (infinite bunnies)? 18.) (True/False With Reasons) Determine if the statement is true or false, and explain why. 𝑑𝑦 a) All solutions of the differential equation 𝑑𝑥 = −3 − 𝑦 4 are decreasing functions. ln 𝑥 𝑑𝑦 b) The function 𝑦 = is a solution of the differential equation 𝑥 2 𝑑𝑥 + 𝑥𝑦 = 1. 𝑥 𝑑𝑦 c) The equation 𝑑𝑥 = 2𝑥 − 𝑦 is separable. 𝑑𝑦 d) The equation 𝑑𝑥 = 3𝑦 − 2𝑥 + 6𝑥𝑦 − 1 is separable. 𝑑𝑦 e) The equation 𝑒 𝑥 𝑑𝑥 = 𝑦 is first-order linear. 𝑑𝑦 f) The equation 𝑑𝑥 + 𝑥𝑦 = 𝑒 𝑦 is first-order linear. 𝑑𝑦 g) The specific solution to the differential equation (2𝑥 − 𝑦) 𝑑𝑥 = 𝑥 + 2𝑦 that goes through pt. (3, 1) has slope 1 at that point. 𝑑𝑦 𝑦 h) If 𝑦(𝑡) is the solution to the differential equation = 2𝑦 (1 − 5) and 𝑦(0) = 1, then lim 𝑦(𝑡) = 5. 𝑑𝑡 𝑡→∞ II) PROBABILITY 19.) State the 2 properties a function 𝑓(𝑥) must have on [𝑎, 𝑏] in order to be a pdf on [𝑎, 𝑏]. 20.) Normalize the function 𝑔(𝑥) on the given interval [𝑎, 𝑏] to make a pdf 𝑓(𝑥) on [𝑎, 𝑏]: a) 𝑔(𝑥) = 5𝑥 − 𝑥 2 on [0, 5] b) 𝑔(𝑥) = √𝑥 on [0, 4] 6𝑥 2 c) 𝑔(𝑥) = 𝑥 3 +1 on [0, 1] d) 𝑔(𝑥) = |𝑥| on [−2, 2] 1 e) 𝑔(𝑥) = 𝑥 3 on [1, ∞) 1 f) 𝑔(𝑥) = on [1, ∞) √𝑥 21.) Find the mean 𝜇, variance Var(𝑋) and standard deviation 𝜎(𝑋) for the following pdfs: 3 a) 𝑓(𝑥) = 4 𝑥 2 (2 − 𝑥) on [0, 2] 3 b) 𝑓(𝑥) = 2 √𝑥 on [0, 1] 4 c) 𝑓(𝑥) = (1+𝑥)5 on [0, ∞) 22.) 1 a) Show 𝑓(𝑥) = 𝑥 2 is a pdf on [1, ∞). b) Try to calculate the mean and the variance – what happens? 23.) Explain the difference between a pdf 𝑓(𝑥) and its CDF 𝐹(𝑥) on [𝑎, 𝑏]. 24.) Given the following pdfs, find their CDFs: 1 a) 𝑓(𝑥) = 4 𝑥 on [1, 3] 2 b) 𝑓(𝑥) = (1+𝑥)2 on [0, 1] c) 𝑓(𝑥) = 3𝑒 −3𝑥 on [0, ∞) 25.) Find 𝑃(1 ≤ 𝑋 ≤ 4) first using the given CDF (on the given sample space) & then with the associated pdf: (HINT: How do you find the pdf from the CDF?) 1 1 a) 𝐹(𝑥) = 4 √𝑥 − 4 on [1, 25] 1 3⁄ b) 𝐹(𝑥) = 8 𝑥 2 on [0, 4] c) 𝐹(𝑥) = 1 − 𝑒 −2𝑥 on [0, ∞) 26.) Show that any CDF 𝐹(𝑥) on [𝑎, 𝑏] is an increasing function: That is, if 𝑥1 ≤ 𝑥2 in [𝑎, 𝑏], then we must have 𝐹(𝑥1 ) ≤ 𝐹(𝑥2 ). (HINT: Draw pictures of the areas 𝐹(𝑥1 ) & 𝐹(𝑥2 ).) 27.) For a UNIFORM random variable 𝑋 on [𝑎, 𝑏], give: a) situations where a uniform random variable might be a good model b) the pdf 𝑓(𝑥) and sketch its graph c) the CDF 𝐹(𝑥) d) the mean e) the variance f) the standard deviation 28.) The first time I get cut off in traffic is equally likely to occur any time in the first 10 minutes of my commute. So if 𝑋 = the first time I get cut off, then 𝑋 is uniformly distributed on [0, 10]. a) Find the pdf 𝑓(𝑥). b) Find the expected time it takes for me to first get cut off. c) Find and interpret 𝑃(4 ≤ 𝑋 ≤ 8). 29.) For an EXPONENTIAL random variable 𝑋 on [0, ∞) with parameter 𝑎 > 0, do parts a-f like in # 27. 1 1 30.) Suppose 𝑋 is an exponential random variable with pdf 𝑓(𝑥) = 3 𝑒 −3𝑥 on [0, ∞). a) Without doing any work, give the mean, variance and standard deviation. b) Find 𝑃(3 ≤ 𝑋 ≤ 6). c) Find 𝑃(𝑋 > 12). 31.) The number of minutes that a livery service is late is an exponential random variable with mean 5 minutes. a) Find the probability that a random fare waits less than 5 minutes. b) The company pays compensation if the limo driver is more than 15 minutes late. Find the probability that they pay compensation to a random client. 32.) For a NORMAL random variable 𝑋 on (−∞, ∞) with parameters 𝜎 > 0 and 𝜇, do parts a-f like in # 27. 33.) Suppose 𝑋 is a normal random variable with mean 7 and standard deviation 2. a) Give the pdf 𝑓(𝑥) and graph it, showing the mean and 1 standard deviation on either side of the mean (what happens to the graph in those 2 places?). b) Find 𝑃(𝑋 ≤ 4) by converting it to an equivalent area under the standard normal pdf using the “z-sub” 𝑥−𝜇 𝑧 = 𝜎 and the table of areas I gave you. (It’s also on Brightspace if you can’t find it.) c) Find 𝑃(6 ≤ 𝑋 ≤ 9). d) Find 𝑃(𝑋 ≥ 11). e) Find the value of 𝑥 marking the 90th percentile; that is, the value of 𝑥 where 𝑃(𝑋 ≤ 𝑥) =.90. (Do this by finding the 90th percentile for the standard normal curve from the table, and then using the conversion formula to find the corresponding 𝑥-value on our normal curve.) 34.) Lengths of human pregnancies are normally distributed with mean 268 days and standard deviation 15 days. What percentage of pregnancies last between 250 days and 280 days? 35.) The #𝑚 in a sample space [𝑎, 𝑏] is the MEDIAN of the random variable 𝑋 if 𝑃(𝑋 ≤ 𝑚) = ________. 36.) Find the median for each of the following pdfs: a) 𝑓(𝑥) = 2𝑥 on [0, 1] 1 b) 𝑓(𝑥) = 6 on [1, 16] √𝑥 c) 𝑓(𝑥) = 3𝑒 −3𝑥 on [0, ∞) 37.) Fill in the blanks and draw a picture to illustrate each of the following: a) The #𝑥𝛼 in [𝑎, 𝑏] gives the 𝒂 − QUANTILE if 𝑃(𝑋 ≤ 𝑥𝛼 ) = __________. b) Thus, the median has 𝛼 = ______. c) The value 𝑥.25 gives the ________________________. d) The value 𝑥.75 gives the ________________________. e) The quantity 𝑥.75 − 𝑥.25 gives the ________________________. f) The value 𝑥.95 is called the ________________________ (when the pdf gives the probability of financial loss in a portfolio). 1 38.) Let 𝑓(𝑥) = 8 be a pdf on [1, 25]. √𝑥 a) Find the formula for the 𝛼 −quantiles. You can either do this with the pdf by solving for 𝑥𝛼 in the 𝑥 1 equation ∫1 𝛼 8 𝑥 𝑑𝑥 = 𝛼 or by finding the CDF 𝐹(𝑥) and solving for 𝑥𝛼 in the equation 𝐹(𝑥𝛼 ) = 𝛼. √ b) Use your formula from part a to compute the median. c) Compute the Interquartile Range. d) Compute the Value at Risk. 3 39.) Do the same for 𝑓(𝑥) = 16 √𝑥 on [0, 4]. 40.) Do the same for 𝑓(𝑥) = 2𝑒 −2𝑥 on [0, ∞). 41.) Recall that for 𝑐 in [𝑎, 𝑏], the CONDITIONAL EXPECTATION OF 𝒇, GIVEN 𝑿 ≥ 𝒄 is given by: 𝑏 𝑓(𝑥) 𝐸(𝑋|𝑋 ≥ 𝑐) = ∫𝑐 𝑥 ∙ 𝑓̂(𝑥) 𝑑𝑥, where 𝑓̂(𝑥) = 𝑏 is 𝑓 “renormalized” to be a pdf on [𝑐, 𝑏]. ∫𝑐 𝑓(𝑥)𝑑𝑥 1 a) Let 𝑓(𝑥) = 8 be a pdf on [1, 25]. Renormalize 𝑓 on [16, 25] to get 𝑓̂ and compute 𝐸(𝑋|𝑋 ≥ 16). √𝑥 b) A special case of a conditional expectation is the EXPECTED TAIL LOSS 𝐸(𝑋|𝑋 ≥ 𝑥.95 ), where 1 𝑥.95 is the Value at Risk (95th percentile). For our pdf 𝑓(𝑥) = 8 𝑥 on [1, 25], you already found 𝑥.95 √ in Problem 38d. Use it to renormalize 𝑓 to a pdf 𝑓̂ on [𝑥.95 , 25] and compute the expected tail loss. 42.) (True or False With Reasons) Determine if the statement is true or false, and explain why. 1 a) To normalize 𝑔(𝑥) = − 𝑥 + 2 and make it into a pdf on [0, 6], simply divide by the area 2 6 1 1 ∫0 (− 2 𝑥 + 2) 𝑑𝑥 = 3, so 𝑓(𝑥) = 3 𝑔(𝑥) is a pdf on [0, 6]. 3 1 1 b) Since ∫0 (𝑥 2 − 2𝑥 + 3) 𝑑𝑥 = 1, 𝑓(𝑥) = 𝑥 2 − 2𝑥 + 3 is already a pdf on [0, 3]. c) The mean 𝜇 of a random variable 𝑋 on [𝑎, 𝑏] can be negative. d) The variance Var(𝑋) of a random variable 𝑋 on [𝑎, 𝑏] can be negative. e) The standard deviation and the variance both tell us about the outcomes’ dispersal away from the mean, and have the same units. f) The mean and the median both lie along the 𝑥-axis, and are in the sample space 𝑆 = [𝑎, 𝑏]. g) The variance must lie in the sample space [𝑎, 𝑏]. h) If the mean of a random variable 𝑋 exists, then the variance and standard deviation exist. i) If the sample space 𝑆 = [𝑎, 𝑏], then the interquartile range IQR > 𝑏 − 𝑎. j) The Value at Risk is a quantile. k) The Expected Tail Loss must be greater than or equal to the Value at Risk. III) ECONOMICS – CONSUMERS’ & PRODUCERS’ SURPLUS 43.) Given demand 𝑝(𝑥) = −𝑥 2 − 4𝑥 + 280 $/unit and supply 𝑝𝑠 (𝑥) = 𝑥 2 + 4𝑥 + 160 $/unit, find: a) the equilibrium pt. (𝑥 ∗ , 𝑝∗ ) b) the consumers’ surplus (at the equilibrium pt.) c) the producers’ surplus (at the equilibrium pt.) d) the total social gain e) Sketch 𝑝(𝑥) & 𝑝𝑠 (𝑥) on the same set of axes, showing all of the above information. Label the axes’ units and give units for the areas. 12 44.) Repeat #43 if 𝑝(𝑥) = 𝑥+1 $/unit and 𝑝𝑠 (𝑥) =.2𝑥 + 1 $/unit.