1.2: Mathjax v3 Test
 Page ID
 10776
Basic
PageIndex Test $$\PageIndex{1}$$
 A continuous random variable \(X\) has a uniform distribution on the interval \([5,12]\). Sketch the graph of its density function.
 A continuous random variable \(X\) has a uniform distribution on the interval \([3,3]\). Sketch the graph of its density function.
 A continuous random variable \(X\) has a normal distribution with mean \(100\) and standard deviation \(10\). Sketch a qualitatively accurate graph of its density function.
 A continuous random variable \(X\) has a normal distribution with mean \(73\) and standard deviation \(2.5\). Sketch a qualitatively accurate graph of its density function.
 A continuous random variable \(X\) has a normal distribution with mean \(73\). The probability that \(X\) takes a value greater than \(80\) is \(0.212\). Use this information and the symmetry of the density function to find the probability that \(X\) takes a value less than \(66\). Sketch the density curve with relevant regions shaded to illustrate the computation.
 A continuous random variable \(X\) has a normal distribution with mean \(169\). The probability that \(X\) takes a value greater than \(180\) is \(0.17\). Use this information and the symmetry of the density function to find the probability that \(X\) takes a value less than \(158\). Sketch the density curve with relevant regions shaded to illustrate the computation.
 A continuous random variable \(X\) has a normal distribution with mean \(50.5\). The probability that \(X\) takes a value less than \(54\) is \(0.76\). Use this information and the symmetry of the density function to find the probability that \(X\) takes a value greater than \(47\). Sketch the density curve with relevant regions shaded to illustrate the computation.
 A continuous random variable \(X\) has a normal distribution with mean \(12.25\). The probability that \(X\) takes a value less than \(13\) is \(0.82\). Use this information and the symmetry of the density function to find the probability that \(X\) takes a value greater than \(11.50\). Sketch the density curve with relevant regions shaded to illustrate the computation.
 The figure provided shows the density curves of three normally distributed random variables \(X_A,\; X_B\; \text{and}\; X_C\). Their standard deviations (in no particular order) are \(15\), \(7\), and \(20\). Use the figure to identify the values of the means \(\mu _A,\: \mu _B,\; \text{and}\; \mu _C\) and standard deviations \(\sigma _A,\: \sigma _B,\; \text{and}\; \sigma _C\) of the three random variables.
 The figure provided shows the density curves of three normally distributed random variables \(X_A,\; X_B\; \text{and}\; X_C\). Their standard deviations (in no particular order) are \(20\), \(5\), and \(10\). Use the figure to identify the values of the means \(\mu _A,\: \mu _B,\; \text{and}\; \mu _C\) and standard deviations \(\sigma _A,\: \sigma _B,\; \text{and}\; \sigma _C\) of the three random variables.
Applications
 Dogberry's alarm clock is battery operated. The battery could fail with equal probability at any time of the day or night. Every day Dogberry sets his alarm for \(6:30\; a.m.\) and goes to bed at \(10:00\; p.m.\). Find the probability that when the clock battery finally dies, it will do so at the most inconvenient time, between \(10:00\; p.m.\) and \(6:30\; a.m.\).
 Buses running a bus line near Desdemona's house run every \(15\) minutes. Without paying attention to the schedule she walks to the nearest stop to take the bus to town. Find the probability that she waits more than \(10\) minutes.
 The amount \(X\) of orange juice in a randomly selected halfgallon container varies according to a normal distribution with mean \(64\) ounces and standard deviation \(0.25\) ounce.
 Sketch the graph of the density function for \(X\).
 What proportion of all containers contain less than a half gallon (\(64\) ounces)? Explain.
 What is the median amount of orange juice in such containers? Explain.
 The weight \(X\) of grass seed in bags marked \(50\) lb varies according to a normal distribution with mean \(50\) lb and standard deviation \(1\) ounce (\(0.0625\) lb).
 Sketch the graph of the density function for \(X\).
 What proportion of all bags weigh less than \(50\) pounds? Explain.
 What is the median weight of such bags? Explain.
Answers
 The graph is a horizontal line with height \(1/7\) from \(x = 5\) to \(x = 12\)
 The graph is a bellshaped curve centered at \(100\) and extending from about \(70\) to \(130\).
 \(0.212\)
 \(0.76\)
 \(\mu _A=100,\; \mu _B=200,\; \mu _C=300,\; \sigma _A=7,\; \sigma _B=20,\; \sigma _C=15\)
 \(0.3542\)

 The graph is a bellshaped curve centered at \(64\) and extending from about \(63.25\) to \(64.75\).
 \(0.5\)
 \(64\)
Basic
 Use Figure 7.1.5: Cumulative Normal Probability to find the probability indicated.
 \(P(Z < 1.72)\)
 \(P(Z < 2.05)\)
 \(P(Z < 0)\)
 \(P(Z > 2.11)\)
 \(P(Z > 1.63)\)
 \(P(Z > 2.36)\)
 Use Figure 7.1.5: Cumulative Normal Probability to find the probability indicated.
 \(P(Z < 1.17)\)
 \(P(Z < 0.05)\)
 \(P(Z < 0.66)\)
 \(P(Z > 2.43)\)
 \(P(Z > 1.00)\)
 \(P(Z > 2.19)\)
 Use Figure 7.1.5: Cumulative Normal Probability to find the probability indicated.
 \(P(2.15 < Z < 1.09)\)
 \(P(0.93 < Z < 0.55)\)
 \(P(0.68 < Z < 2.11)\)
 Use Figure 7.1.5: Cumulative Normal Probability to find the probability indicated.
 \(P(1.99 < Z < 1.03)\)
 \(P(0.87 < Z < 1.58)\)
 \(P(0.33 < Z < 0.96)\)
 Use Figure 7.1.5: Cumulative Normal Probability to find the probability indicated.
 \(P(4.22 < Z < 1.39)\)
 \(P(1.37 < Z < 5.11)\)
 \(P(Z < 4.31)\)
 \(P(Z < 5.02)\)
 Use Figure 7.1.5: Cumulative Normal Probability to find the probability indicated.
 \(P(Z > 5.31)\)
 \(P(4.08 < Z < 0.58)\)
 \(P(Z < 6.16)\)
 \(P(0.51< Z < 5.63)\)
 Use Figure 7.1.5: Cumulative Normal Probability to find the probability listed. Find the second probability without referring to the table, but using the symmetry of the standard normal density curve instead. Sketch the density curve with relevant regions shaded to illustrate the computation.
 \(P(Z < 1.08),\; P(Z > 1.08)\)
 \(P(Z < 0.36),\; P(Z > 0.36)\)
 \(P(Z < 1.25),\; P(Z > 1.25)\)
 \(P(Z < 2.03),\; P(Z > 2.03)\)
 Use Figure 7.1.5: Cumulative Normal Probability to find the probability listed. Find the second probability without referring to the table, but using the symmetry of the standard normal density curve instead. Sketch the density curve with relevant regions shaded to illustrate the computation.
 \(P(Z < 2.11),\; P(Z > 2.11)\)
 \(P(Z < 0.88),\; P(Z > 0.88)\)
 \(P(Z < 2.44),\; P(Z > 2.44)\)
 \(P(Z < 3.07),\; P(Z > 3.07)\)
 The probability that a standard normal random variable \(Z\) takes a value in the union of intervals \((\infty ,\alpha ]\cup [\alpha ,\infty )\), which arises in applications, will be denoted \(P(Z \leq a\; or\; Z \geq a)\). Use Figure 7.1.5: Cumulative Normal Probability to find the following probabilities of this type. Sketch the density curve with relevant regions shaded to illustrate the computation. Because of the symmetry of the standard normal density curve you need to use Figure 7.1.5: Cumulative Normal Probability only one time for each part.
 \(P(Z < 1.29\; or\; Z > 1.29)\)
 \(P(Z < 2.33\; or\; Z > 2.33)\)
 \(P(Z < 1.96\; or\; Z > 1.96)\)
 \(P(Z < 3.09\; or\; Z > 3.09)\)
 The probability that a standard normal random variable \(Z\) takes a value in the union of intervals \((\infty ,\alpha ]\cup [\alpha ,\infty )\), which arises in applications, will be denoted \(P(Z \leq a\; or\; Z \geq a)\). Use Figure 7.1.5: Cumulative Normal Probability to find the following probabilities of this type. Sketch the density curve with relevant regions shaded to illustrate the computation. Because of the symmetry of the standard normal density curve you need to use Figure 7.1.5: Cumulative Normal Probability only one time for each part.
 \(P(Z < 2.58 \; or\; Z > 2.58 )\)
 \(P(Z < 2.81 \; or\; Z > 2.81 )\)
 \(P(Z < 1.65 \; or\; Z > 1.65 )\)
 \(P(Z < 2.43 \; or\; Z > 2.43 )\)
Answers

 \(0.0427\)
 \(0.9798\)
 \(0.5\)
 \(0.9826\)
 \(0.0516\)
 \(0.0091\)

 \(0.1221\)
 \(0.5326\)
 \(0.2309\)

 \(0.0823\)
 \(0.9147\)
 \(0.0000\)
 \(1.0000\)

 \(0.1401,\; 0.1401\)
 \(0.3594,\; 0.3594\)
 \(0.8944,\; 0.8944\)
 \(0.9788,\; 0.9788\)

 \(0.1970\)
 \(0.01980\)
 \(0.0500\)
 \(0.0020\)
Basic
 \(X\) is a normally distributed random variable with mean \(57\) and standard deviation \(6\). Find the probability indicated.
 \(P(X < 59.5)\)
 \(P(X < 46.2)\)
 \(P(X > 52.2)\)
 \(P(X > 70)\)
 \(X\) is a normally distributed random variable with mean \(25\) and standard deviation \(4\). Find the probability indicated.
 \(P(X < 27.2)\)
 \(P(X < 14.8)\)
 \(P(X > 33.1)\)
 \(P(X > 16.5)\)
 \(X\) is a normally distributed random variable with mean \(112\) and standard deviation \(15\). Find the probability indicated.
 \(P(100<X<125)\)
 \(P(91<X<107)\)
 \(P(118<X<160)\)
 \(X\) is a normally distributed random variable with mean \(72\) and standard deviation \(22\). Find the probability indicated.
 \(P(78<X<127)\)
 \(P(60<X<90)\)
 \(P(49<X<71)\)
 \(X\) is a normally distributed random variable with mean \(500\) and standard deviation \(25\). Find the probability indicated.
 \(P(X < 400)\)
 \(P(466<X<625)\)
 \(X\) is a normally distributed random variable with mean \(0\) and standard deviation \(0.75\). Find the probability indicated.
 \(P(4.02 < X < 3.82)\)
 \(P(X > 4.11)\)
 \(X\) is a normally distributed random variable with mean \(15\) and standard deviation \(1\). Use Figure 7.1.5: Cumulative Normal Probability to find the first probability listed. Find the second probability using the symmetry of the density curve. Sketch the density curve with relevant regions shaded to illustrate the computation.
 \(P(X < 12),\; P(X > 18)\)
 \(P(X < 14),\; P(X > 16)\)
 \(P(X < 11.25),\; P(X > 18.75)\)
 \(P(X < 12.67),\; P(X > 17.33)\)
 \(X\) is a normally distributed random variable with mean \(100\) and standard deviation \(10\). Use Figure 7.1.5: Cumulative Normal Probability to find the first probability listed. Find the second probability using the symmetry of the density curve. Sketch the density curve with relevant regions shaded to illustrate the computation.
 \(P(X < 80),\; P(X > 120)\)
 \(P(X < 75),\; P(X > 125)\)
 \(P(X < 84.55),\; P(X > 115.45)\)
 \(P(X < 77.42),\; P(X > 122.58)\)
 \(X\) is a normally distributed random variable with mean \(67\) and standard deviation \(13\). The probability that \(X\) takes a value in the union of intervals \((\infty ,67a]\cup [67+a,\infty )\) will be denoted \(P(X\leq 67a\; or\; X\geq 67+a)\). Use Figure 7.1.5: Cumulative Normal Probability to find the following probabilities of this type. Sketch the density curve with relevant regions shaded to illustrate the computation. Because of the symmetry of the density curve you need to use Figure 7.1.5: Cumulative Normal Probability only one time for each part.
 \(P(X<57\; or\; X>77)\)
 \(P(X<47\; or\; X>87)\)
 \(P(X<49\; or\; X>85)\)
 \(P(X<37\; or\; X>97)\)
 \(X\) is a normally distributed random variable with mean \(288\) and standard deviation \(6\). The probability that \(X\) takes a value in the union of intervals \((\infty ,288a]\cup [288+a,\infty )\) will be denoted \(P(X\leq 288a\; or\; X\geq 288+a)\). Use Figure 7.1.5: Cumulative Normal Probability to find the following probabilities of this type. Sketch the density curve with relevant regions shaded to illustrate the computation. Because of the symmetry of the density curve you need to use Figure 7.1.5: Cumulative Normal Probability only one time for each part.
 \(P(X<278\; or\; X>298)\)
 \(P(X<268\; or\; X>308)\)
 \(P(X<273\; or\; X>303)\)
 \(P(X<280\; or\; X>296)\)
 The amount \(X\) of beverage in a can labeled \(12\) ounces is normally distributed with mean \(12.1\) ounces and standard deviation \(0.05\) ounce. A can is selected at random.
 Find the probability that the can contains at least \(12\) ounces.
 Find the probability that the can contains between \(11.9\) and \(12.1\) ounces.
 The length of gestation for swine is normally distributed with mean \(114\) days and standard deviation \(0.75\) day. Find the probability that a litter will be born within one day of the mean of \(114\).
 The systolic blood pressure \(X\) of adults in a region is normally distributed with mean \(112\) mm Hg and standard deviation \(15\) mm Hg. A person is considered “prehypertensive” if his systolic blood pressure is between \(120\) and \(130\) mm Hg. Find the probability that the blood pressure of a randomly selected person is prehypertensive.
 Heights \(X\) of adult women are normally distributed with mean \(63.7\) inches and standard deviation \(2.71\) inches. Romeo, who is \(69.25\) inches tall, wishes to date only women who are shorter than he but within \(4\) inches of his height. Find the probability that the next woman he meets will have such a height.
 Heights \(X\) of adult men are normally distributed with mean \(69.1\) inches and standard deviation \(2.92\) inches. Juliet, who is \(63.25\) inches tall, wishes to date only men who are taller than she but within 6 inches of her height. Find the probability that the next man she meets will have such a height.
 A regulation hockey puck must weigh between \(5.5\) and \(6\) ounces. The weights \(X\) of pucks made by a particular process are normally distributed with mean \(5.75\) ounces and standard deviation \(0.11\) ounce. Find the probability that a puck made by this process will meet the weight standard.
 A regulation golf ball may not weigh more than \(1.620\) ounces. The weights \(X\) of golf balls made by a particular process are normally distributed with mean \(1.361\) ounces and standard deviation \(0.09\) ounce. Find the probability that a golf ball made by this process will meet the weight standard.
 The length of time that the battery in Hippolyta's cell phone will hold enough charge to operate acceptably is normally distributed with mean \(25.6\) hours and standard deviation \(0.32\) hour. Hippolyta forgot to charge her phone yesterday, so that at the moment she first wishes to use it today it has been \(26\) hours \(18\) minutes since the phone was last fully charged. Find the probability that the phone will operate properly.
 The amount of nonmortgage debt per household for households in a particular income bracket in one part of the country is normally distributed with mean \(\$28,350\) and standard deviation \(\$3,425\). Find the probability that a randomly selected such household has between \(\$20,000\) and \(\$30,000\) in nonmortgage debt.
 Birth weights of fullterm babies in a certain region are normally distributed with mean \(7.125\) lb and standard deviation \(1.290\) lb. Find the probability that a randomly selected newborn will weigh less than \(5.5\) lb, the historic definition of prematurity.
 The distance from the seat back to the front of the knees of seated adult males is normally distributed with mean \(23.8\) inches and standard deviation \(1.22\) inches. The distance from the seat back to the back of the next seat forward in all seats on aircraft flown by a budget airline is \(26\) inches. Find the proportion of adult men flying with this airline whose knees will touch the back of the seat in front of them.
 The distance from the seat to the top of the head of seated adult males is normally distributed with mean \(36.5\) inches and standard deviation \(1.39\) inches. The distance from the seat to the roof of a particular make and model car is \(40.5\) inches. Find the proportion of adult men who when sitting in this car will have at least one inch of headroom (distance from the top of the head to the roof).
 The useful life of a particular make and type of automotive tire is normally distributed with mean \(57,500\) miles and standard deviation \(950\) miles.
 Find the probability that such a tire will have a useful life of between \(57,000\) and \(58,000\) miles.
 Hamlet buys four such tires. Assuming that their lifetimes are independent, find the probability that all four will last between \(57,000\) and \(58,000\) miles. (If so, the best tire will have no more than \(1,000\) miles left on it when the first tire fails.) Hint: There is a binomial random variable here, whose value of \(p\) comes from part (a).
 A machine produces large fasteners whose length must be within \(0.5\) inch of \(22\) inches. The lengths are normally distributed with mean \(22.0\) inches and standard deviation \(0.17\) inch.
 Find the probability that a randomly selected fastener produced by the machine will have an acceptable length.
 The machine produces \(20\) fasteners per hour. The length of each one is inspected. Assuming lengths of fasteners are independent, find the probability that all \(20\) will have acceptable length. Hint: There is a binomial random variable here, whose value of \(p\) comes from part (a).
 The lengths of time taken by students on an algebra proficiency exam (if not forced to stop before completing it) are normally distributed with mean \(28\) minutes and standard deviation \(1.5\) minutes.
 Find the proportion of students who will finish the exam if a \(30\)minute time limit is set.
 Six students are taking the exam today. Find the probability that all six will finish the exam within the \(30\)minute limit, assuming that times taken by students are independent. Hint: There is a binomial random variable here, whose value of \(p\) comes from part (a).
 Heights of adult men between \(18\) and \(34\) years of age are normally distributed with mean \(69.1\) inches and standard deviation \(2.92\) inches. One requirement for enlistment in the military is that men must stand between \(60\) and \(80\) inches tall.
 Find the probability that a randomly elected man meets the height requirement for military service.
 Twentythree men independently contact a recruiter this week. Find the probability that all of them meet the height requirement. Hint: There is a binomial random variable here, whose value of \(p\) comes from part (a).
 A regulation hockey puck must weigh between \(5.5\) and \(6\) ounces. In an alternative manufacturing process the mean weight of pucks produced is \(5.75\) ounce. The weights of pucks have a normal distribution whose standard deviation can be decreased by increasingly stringent (and expensive) controls on the manufacturing process. Find the maximum allowable standard deviation so that at most \(0.005\) of all pucks will fail to meet the weight standard. (Hint: The distribution is symmetric and is centered at the middle of the interval of acceptable weights.)
 The amount of gasoline \(X\) delivered by a metered pump when it registers \(5\) gallons is a normally distributed random variable. The standard deviation \(\sigma\) of \(X\)measures the precision of the pump; the smaller \(\sigma\) is the smaller the variation from delivery to delivery. A typical standard for pumps is that when they show that \(5\) gallons of fuel has been delivered the actual amount must be between \(4.97\) and \(5.03\) gallons (which corresponds to being off by at most about half a cup). Supposing that the mean of \(X\) is \(5\), find the largest that \(\sigma\) can be so that \(P(4.97 < X < 5.03)\) is \(1.0000\) to four decimal places when computed using Figure 7.1.5: Cumulative Normal Probability which means that the pump is sufficiently accurate. (Hint: The \(z\)score of \(5.03\) will be the smallest value of \(Z\) so that Figure 7.1.5: Cumulative Normal Probability gives \(P(Z<z)=1.0000\)).
Answers

 \(0.6628\)
 \(0.0359\)
 \(0.7881\)
 \(0.0150\)

 \(0.5959\)
 \(0.2899\)
 \(0.3439\)

 \(0.0000\)
 \(0.9131\)

 \(0.0013,\; 0.0013\)
 \(0.1587,\; 0.1587\)
 \(0.0001,\; 0.0001\)
 \(0.0099,\; 0.0099\)

 \(0.4412\)
 \(0.1236\)
 \(0.1676\)
 \(0.0208\)

 \(0.9772\)
 \(0.5000\)
 \(0.1830\)
 \(0.4971\)
 \(0.9980\)
 \(0.6771\)
 \(0.0359\)

 \(0.4038\)
 \(0.0266\)

 \(0.9082\)
 \(0.5612\)
 \(0.089\)
Basic
 Find the value of \(z\ast\) that yields the probability shown.
 \(P(Z<z*)=0.0075\)
 \(P(Z<z*)=0.9850\)
 \(P(Z>z*)=0.8997\)
 \(P(Z>z*)=0.0110\)
 Find the value of \(z\ast\) that yields the probability shown.
 \(P(Z<z*)=0.3300\)
 \(P(Z<z*)=0.9901\)
 \(P(Z>z*)=0.0055\)
 \(P(Z>z*)=0.7995\)
 Find the value of \(z\ast\) that yields the probability shown.
 \(P(Z<z*)=0.1500\)
 \(P(Z<z*)=0.7500\)
 \(P(Z>z*)=0.3333\)
 \(P(Z>z*)=0.8000\)
 Find the value of \(z\ast\) that yields the probability shown.
 \(P(Z<z*)=0.2200\)
 \(P(Z<z*)=0.6000\)
 \(P(Z>z*)=0.0750\)
 \(P(Z>z*)=0.8200\)
 Find the indicated value of \(Z\). (It is easier to find \(z_c\)and negate it.)
 \(Z_{0.025}\)
 \(Z_{0.20}\)
 Find the indicated value of \(Z\). (It is easier to find \(z_c\)and negate it.)
 \(Z_{0.002}\)
 \(Z_{0.02}\)
 Find the value of \(x\ast\) that yields the probability shown, where \(X\) is a normally distributed random variable \(X\) with mean \(83\) and standard deviation \(4\).
 \(P(X<x*)=0.8700\)
 P(X>x*)=0.0500P(X>x*)=0.0500\(P(X>x*)=0.0500\)
 Find the value of \(x\ast\) that yields the probability shown, where \(X\) is a normally distributed random variable \(X\) with mean \(54\) and standard deviation \(12\).
 \(P(X<x*)=0.0900\)
 P(X>x*)=0.0500P(X>x*)=0.0500\(P(X>x*)=0.6500\)
 \(X\) is a normally distributed random variable \(X\) with mean \(15\) and standard deviation \(0.25\). Find the values \(X_L\) and \(X_R\) of \(X\) that are symmetrically located with respect to the mean of \(X\) and satisfy \(P(X_L < X < X_R) = 0.80\). (Hint. First solve the corresponding problem for \(Z\)).
 \(X\) is a normally distributed random variable \(X\) with mean \(28\) and standard deviation \(3.7\). Find the values \(X_L\) and \(X_R\) of \(X\) that are symmetrically located with respect to the mean of \(X\) and satisfy \(P(X_L < X < X_R) = 0.65\). (Hint. First solve the corresponding problem for \(Z\)).
Applications
 Scores on a national exam are normally distributed with mean \(382\) and standard deviation \(26\).
 Find the score that is the \(50^{th}\) percentile.
 Find the score that is the \(90^{th}\) percentile.
 Heights of women are normally distributed with mean \(63.7\) inches and standard deviation \(2.47\) inches.
 Find the height that is the \(10^{th}\) percentile.
 Find the height that is the \(80^{th}\) percentile.
 The monthly amount of water used per household in a small community is normally distributed with mean \(7,069\) gallons and standard deviation \(58\) gallons. Find the three quartiles for the amount of water used.
 The quantity of gasoline purchased in a single sale at a chain of filling stations in a certain region is normally distributed with mean \(11.6\) gallons and standard deviation \(2.78\) gallons. Find the three quartiles for the quantity of gasoline purchased in a single sale.
 Scores on the common final exam given in a large enrollment multiple section course were normally distributed with mean \(69.35\) and standard deviation \(12.93\). The department has the rule that in order to receive an \(A\) in the course his score must be in the top \(10\%\) of all exam scores. Find the minimum exam score that meets this requirement.
 The average finishing time among all high school boys in a particular track event in a certain state is \(5\) minutes \(17\) seconds. Times are normally distributed with standard deviation \(12\) seconds.
 The qualifying time in this event for participation in the state meet is to be set so that only the fastest \(5\%\) of all runners qualify. Find the qualifying time. (Hint: Convert seconds to minutes.)
 In the western region of the state the times of all boys running in this event are normally distributed with standard deviation \(12\) seconds, but with mean \(5\) minutes \(22\) seconds. Find the proportion of boys from this region who qualify to run in this event in the state meet.
 Tests of a new tire developed by a tire manufacturer led to an estimated mean tread life of \(67,350\) miles and standard deviation of \(1,120\) miles. The manufacturer will advertise the lifetime of the tire (for example, a “\(50,000\) mile tire”) using the largest value for which it is expected that \(98\%\) of the tires will last at least that long. Assuming tire life is normally distributed, find that advertised value.
 Tests of a new light led to an estimated mean life of \(1,321\) hours and standard deviation of \(106\) hours. The manufacturer will advertise the lifetime of the bulb using the largest value for which it is expected that \(90\%\) of the bulbs will last at least that long. Assuming bulb life is normally distributed, find that advertised value.
 The weights \(X\) of eggs produced at a particular farm are normally distributed with mean \(1.72\) ounces and standard deviation \(0.12\) ounce. Eggs whose weights lie in the middle \(75\%\) of the distribution of weights of all eggs are classified as “medium.” Find the maximum and minimum weights of such eggs. (These weights are endpoints of an interval that is symmetric about the mean and in which the weights of \(75\%\) of the eggs produced at this farm lie.)
 The lengths \(X\) of hardwood flooring strips are normally distributed with mean \(28.9\) inches and standard deviation \(6.12\) inches. Strips whose lengths lie in the middle 80% of the distribution of lengths of all strips are classified as “averagelength strips.” Find the maximum and minimum lengths of such strips. (These lengths are endpoints of an interval that is symmetric about the mean and in which the lengths of \(80\%\) of the hardwood strips lie.)
 All students in a large enrollment multiple section course take common inclass exams and a common final, and submit common homework assignments. Course grades are assigned based on students' final overall scores, which are approximately normally distributed. The department assigns a \(C\) to students whose scores constitute the middle \(2/3\) of all scores. If scores this semester had mean \(72.5\) and standard deviation \(6.14\), find the interval of scores that will be assigned a \(C\).
 Researchers wish to investigate the overall health of individuals with abnormally high or low levels of glucose in the blood stream. Suppose glucose levels are normally distributed with mean \(96\) and standard deviation \(8.5\; mg/dl\), and that “normal” is defined as the middle \(90\%\) of the population. Find the interval of normal glucose levels, that is, the interval centered at \(96\) that contains \(90\%\) of all glucose levels in the population.
Additional Exercises
 A machine for filling \(2\)liter bottles of soft drink delivers an amount to each bottle that varies from bottle to bottle according to a normal distribution with standard deviation \(0.002\) liter and mean whatever amount the machine is set to deliver.
 If the machine is set to deliver \(2\) liters (so the mean amount delivered is \(2\) liters) what proportion of the bottles will contain at least \(2\) liters of soft drink?
 Find the minimum setting of the mean amount delivered by the machine so that at least \(99\%\) of all bottles will contain at least \(2\) liters.
 A nursery has observed that the mean number of days it must darken the environment of a species poinsettia plant daily in order to have it ready for market is \(71\) days. Suppose the lengths of such periods of darkening are normally distributed with standard deviation \(2\) days. Find the number of days in advance of the projected delivery dates of the plants to market that the nursery must begin the daily darkening process in order that at least \(95\%\) of the plants will be ready on time. (Poinsettias are so longlived that once ready for market the plant remains salable indefinitely.)
Answers

 \(2.43\)
 \(2.17\)
 \(1.28\)
 \(2.29\)

 \(1.04\)
 \(0.67\)
 \(0.43\)
 \(0.84\)

 \(1.96\)
 \(0.84\)

 \(87.52\)
 \(89.58\)
 \(15.32\)

 \(382\)
 \(415\)
 \(7030.14,\; 7069,\; 7107.86\)
 \(85.90\)
 \(65,054\)
 \(1.58,\; 1.86\)
 \(66.5,\; 78.5\)

 \(0.5\)
 \(2.005\)
A
2.1.1. \(f (x, y) = x^ 2 + y^ 2 −1\)
2.1.2. \(f (x, y) = \frac{1}{ x^ 2 + y^ 2}\)
2.1.3. \(f (x, y) = \sqrt{ x^ 2 + y^ 2 −4}\)
2.1.4. \(f (x, y) = \frac{x^ 2 +1}{ y}\)
2.1.5. \(f (x, y, z) = \sin (x yz)\)
2.1.6. \(f (x, y, z) = \sqrt{ (x−1)(yz −1)}\)
For Exercises 718, evaluate the given limit.
2.1.7. \(\lim\limits_{(x,y)→(0,0)} \cos (x y)\)
2.1.8. \(\lim\limits_{(x,y)→(0,0)} e^{ x y}\)
2.1.9. \(\lim\limits_{(x,y)→(0,0)} \frac{x^ 2 − y^ 2}{ x^ 2 + y^ 2}\)
2.1.10. \(\lim\limits_{(x,y)→(0,0)} \frac{x y^2}{ x^ 2 + y^ 4}\)
2.1.11. \(\lim\limits_{(x,y)→(1,−1)} \frac{x^ 2 −2x y+ y^ 2}{ x− y}\)
2.1.12. \(\lim\limits_{(x,y)→(0,0)} \frac{x y^2}{ x^ 2 + y^ 2}\)
2.1.13. \(\lim\limits_{(x,y)→(1,1)} \frac{x^ 2 − y^ 2}{ x− y}\)
2.1.14. \(\lim\limits_{(x,y)→(0,0)} \frac{x^ 2 −2x y+ y^ 2}{ x− y}\)
2.1.15. \(\lim\limits_{(x,y)→(0,0)} \frac{y^ 4 \sin (x y)}{ x^ 2 + y^ 2}\)
2.1.16. \(\lim\limits_{(x,y)→(0,0)} (x^ 2 + y^ 2 )\cos \left ( \frac{1}{ x y}\right ) \)
2.1.17. \(\lim\limits_{(x,y)→(0,0)} \frac{x}{ y}\)
2.1.18. \(\lim\limits_{(x,y)→(0,0)} \cos \left ( \frac{1}{ x y}\right ) \)
B
2.1.19. Show that \(f (x, y) = \frac{1}{ 2πσ^2} e^{ −(x^ 2+y^ 2 )/2σ^ 2}\) , for \(σ > 0\), is constant on the circle of radius \(r > 0\) centered at the origin. This function is called a Gaussian blur, and is used as a filter in image processing software to produce a “blurred” effect.
2.1.20. Suppose that \(f (x, y) ≤ f (y, x) \text{ for all }(x, y)\) in \(\mathbb{R}^ 2\) . Show that \(f (x, y) = f (y, x) \text{ for all }(x, y)\) in \(\mathbb{R}^ 2\) .
2.1.21. Use the substitution \(r = \sqrt{ x^ 2 + y^ 2}\) to show that
\[\lim\limits_{(x,y)→(0,0)} \frac{\sin \sqrt{ x^ 2 + y^ 2}}{ \sqrt{ x^ 2 + y^ 2}} = 1 .\]
(Hint: You will need to use L’Hôpital’s Rule for singlevariable limits.)
C
2.1.22. Prove Theorem 2.1(a) in the case of addition. (Hint: Use Definition 2.1.)
2.1.23. Prove Theorem 2.1(b).
A
For Exercises 116, find \(\frac{∂f}{ ∂x} \text{ and }\frac{∂f}{ ∂y}\) .
2.2.1. \(f (x, y) = x^ 2 + y^ 2\)
2.2.2. \(f (x, y) = \cos (x+ y)\)
2.2.3. \(f (x, y) = \sqrt{ x^ 2 + y+4}\)
2.2.4. \(f (x, y) = \frac{x+1}{ y+1}\)
2.2.5. \(f (x, y) = e^{ x y} + x y\)
2.2.6. \(f (x, y) = x^ 2 − y^ 2 +6x y+4x−8y+2\)
2.2.7. \(f (x, y) = x^ 4\)
2.2.8. \(f (x, y) = x+2y\)
2.2.9. \(f (x, y) = \sqrt{ x^ 2 + y^ 2}\)
2.2.10. \(f (x, y) = \sin (x+ y)\)
2.2.11. \(f (x, y) = \sqrt[3]{ x^ 2 + y+4}\)
2.2.12. \(f (x, y) = \frac{x y+1}{ x+ y}\)
2.2.13. \(f (x, y) = e^{ −(x^ 2+y^ 2 )}\)
2.2.14. \(f (x, y) = \ln (x y)\)
2.2.15. \(f (x, y) = \sin (x y)\)
2.2.16. \(f (x, y) = \tan (x+ y)\)
For Exercises 1726, find \(\frac{∂^ 2 f}{ ∂x^ 2} ,\, \frac{∂^ 2 f}{ ∂y^ 2} \text{ and }\frac{∂^ 2 f}{ ∂y∂x}\) (use Exercises 18, 14, 15).
2.2.17. \(f (x, y) = x^ 2 + y^ 2\)
2.2.18. \(f (x, y) = \cos (x+ y)\)
2.2.19. \(f (x, y) = \sqrt{ x^ 2 + y+4}\)
2.2.20. \(f (x, y) = \frac{x+1}{ y+1}\)
2.2.21. \(f (x, y) = e^{ x y} + x y\)
2.2.22. \(f (x, y) = x^ 2 − y^ 2 +6x y+4x−8y+2\)
2.2.23. \(f (x, y) = x^ 4\)
2.2.24. \(f (x, y) = x+2y\)
2.2.25. \(f (x, y) = \ln (x y)\)
2.2.26. \(f (x, y) = \sin (x y)\)
B
2.2.27. Show that the function \(f (x, y) = \sin (x+ y)+\cos (x− y)\) satisfies the wave equation
\[\frac{∂^ 2 f}{ ∂x^ 2} − \frac{∂^ 2 f}{ ∂y^ 2} = 0 .\]
The wave equation is an example of a partial differential equation.
2.2.28 Let \(u \text{ and }v\) be twicedifferentiable functions of a single variable, and let \(c \neq 0\) be a constant. Show that \(f (x, y) = u(x+ c y)+v(x− c y)\) is a solution of the general onedimensional wave equation
\[\frac{∂^ 2 f}{ ∂x^ 2} − \frac{1}{ c^ 2} \frac{∂^ 2 f}{ ∂y^ 2} = 0\]
A
For Exercises 16, find the equation of the tangent plane to the surface \(z = f (x, y)\) at the point \(P\).
2.3.1. \(f (x, y) = x^ 2 + y^ 3 ,\, P = (1,1,2)\)
2.3.2. \(f (x, y) = x y,\, P = (1,−1,−1) \)
2.3.3. \(f (x, y) = x^ 2 y,\, P = (−1,1,1)\)
2.3.4. \(f (x, y) = xe^ y ,\, P = (1,0,1)\)
2.3.5. \(f (x, y) = x+2y,\, P = (2,1,4)\)
2.3.6. \(f (x, y) = \sqrt{ x^ 2 + y^ 2},\, P = (3,4,5)\)
For Exercises 710, find the equation of the tangent plane to the given surface at the point \(P\).
2.3.7. \(\frac{x^ 2}{ 4} + \frac{y^ 2}{ 9} + \frac{z^ 2}{ 16} = 1,\, P = \left ( 1,2, \frac{2 \sqrt{ 11}}{ 3} \right ) \)
2.3.8. \(x^ 2 + y^ 2 + z^ 2 = 9,\, P = (0,0,3)\)
2.3.9. \(x^ 2 + y^ 2 − z^ 2 = 0,\, P = (3,4,5)\)
2.3.10. \(x^ 2 + y^ 2 = 4,\, P = ( \sqrt{ 3},1,0)\)
A
For Exercises 110, compute the gradient \(∇f\) .
2.4.1. \(f (x, y) = x^ 2 + y^ 2 −1\)
2.4.2. \(f (x, y) = \frac{1}{ x^ 2 + y^ 2}\)
2.4.3. \(f (x, y) = \sqrt{ x^ 2 + y^ 2 +4}\)
2.4.4. \(f (x, y) = x^ 2 e^ y\)
2.4.5. \(f (x, y) = \ln (x y)\)
2.4.6. \(f (x, y) = 2x+5y\)
2.4.7. \(f (x, y, z) = \sin (x yz)\)
2.4.8. \(f (x, y, z) = x^ 2 e^{ yz}\)
2.4.9. \(f (x, y, z) = x^ 2 + y^ 2 + z^ 2\)
2.4.10. \(f (x, y, z) = \sqrt{ x^ 2 + y^ 2 + z^ 2}\)
For Exercises 1114, find the directional derivative of \(f\) at the point \(P\) in the direction of \(v = \left ( \frac{1}{ \sqrt{ 2}} , \frac{1}{ \sqrt{ 2}} \right ) \) .
2.4.11. \(f (x, y) = x^ 2 + y^ 2 −1,\, P = (1,1)\)
2.4.12. \(f (x, y) = \frac{1}{ x^ 2 + y^ 2} ,\, P = (1,1)\)
2.4.13. \(f (x, y) = \sqrt{ x^ 2 + y^ 2 +4},\, P = (1,1)\)
2.4.14. \(f (x, y) = x^ 2 e^ y ,\, P = (1,1)\)
For Exercises 1516, find the directional derivative of \(f\) at the point \(P\) in the direction of \(v = \left ( \frac{1}{ \sqrt{ 3}} , \frac{1}{ \sqrt{ 3}} , \frac{1}{ \sqrt{ 3}} \right ) \) .
2.4.15. \(f (x, y, z) = \sin (x yz),\, P = (1,1,1)\)
2.4.16. \(f (x, y, z) = x^ 2 e^{ yz} ,\, P = (1,1,1)\)
2.4.17. Repeat Example 2.16 at the point \((2,3)\).
2.4.18. Repeat Example 2.17 at the point \((3,1,2)\).
B
For Exercises 1926, let \(f (x, y) \text{ and }g(x, y)\) be continuously differentiable realvalued functions, let \(c\) be a constant, and let \(v\) be a unit vector in \(\mathbb{R}^ 2\) . Show that:
2.4.19. \(∇(c f ) = c∇f \)
2.4.20. \(∇(f + g) = ∇f + ∇g\)
2.4.21. \(∇(f g) = f ∇g + g∇f\)
2.4.22. \(∇(f /g) = \frac{g∇f − f ∇g}{ g^ 2}\text{ if }g(x, y) \neq 0\)
2.4.23. \(D_{−v} f = −D_v f\)
2.4.24. \(D_v(c f ) = c D_v f\)
2.4.25. \(D_v(f + g) = D_v f + D_v g\)
2.4.26. \(D_v(f g) = f D_v g + g D_v f\)
2.4.27. The function \(r(x, y) = \sqrt{ x^ 2 + y^ 2}\) is the length of the position vector \(\textbf{r} = x\textbf{i} + y\textbf{j}\) for each point \((x, y)\) in \(\mathbb{R}^ 2\) . Show that \(∇r = \frac{1}{ r} \textbf{r}\) when \((x, y) \neq (0,0)\), and that \(∇(r^ 2 ) = 2\textbf{r}\).
A
For Exercises 110, find all local maxima and minima of the function \(f (x, y)\).
2.5.1. \(f (x, y) = x^ 3 −3x+ y^ 2\)
2.5.2. \(f (x, y) = x^ 3 −12x+ y^ 2 +8y\)
2.5.3. \(f (x, y) = x^ 3 −3x+ y^ 3 −3y\)
2.5.4. \(f (x, y) = x^ 3 +3x^ 2 + y^ 3 −3y^ 2\)
2.5.5. \(f (x, y) = 2x^ 3 +6x y+3y^ 2\)
2.5.6. \(f (x, y) = 2x^ 3 −6x y+ y^ 2\)
2.5.7. \(f (x, y) = \sqrt{ x^ 2 + y^ 2}\)
2.5.8. \(f (x, y) = x+2y\)
2.5.9. \(f (x, y) = 4x^ 2 −4x y+2y^ 2 +10x−6y\)
2.5.10. \(f (x, y) = −4x^ 2 +4x y−2y^ 2 +16x−12y\)
B
2.5.11. For a rectangular solid of volume 1000 cubic meters, find the dimensions that will minimize the surface area. (Hint: Use the volume condition to write the surface area as a function of just two variables.)
2.5.12. Prove that if \((a,b)\) is a local maximum or local minimum point for a smooth function \(f (x, y)\), then the tangent plane to the surface \(z = f (x, y)\) at the point \((a,b, f (a,b))\) is parallel to the \(x y\)plane. (Hint: Use Theorem 2.5.)
C
2.5.13. Find three positive numbers \(x, y, z\) whose sum is 10 such that \(x^ 2 y^ 2 z\) is a maximum.
C
2.6.1. Recall Example 2.21 from the previous section, where we showed that the point \((2,1)\) was a global minimum for the function \(f (x, y) = (x −2)^4 +(x −2y)^ 2\) . Notice that our computer program can be modified fairly easily to use this function (just change the return values in the fx, fy, fxx, fyy and fxy function definitions to use the appropriate partial derivative). Either modify that program or write one of your own in a programming language of your choice to show that Newton’s algorithm does lead to the point \((2,1)\). First use the initial point \((0,3)\), then use the initial point \((3,2)\), and compare the results. Make sure that your program attempts to do 100 iterations of the algorithm. Did anything strange happen when your program ran? If so, how do you explain it? (Hint: Something strange should happen.)
2.6.2. There is a version of Newton’s algorithm for solving a system of two equations
\[f_1(x, y) = 0 \quad \text{ and }\quad f_2(x, y) = 0 ,\]
where \(f_1(x, y) \text{ and }f_2(x, y)\) are smooth realvalued functions:
Pick an initial point \((x_0 , y_0)\). For \(n = 0,1,2,3,...,\) define:
\[x_{n+1} = x_n  \frac{\begin{vmatrix} f_1(x_n, y_n) & f_2(x_n, y_n) \\ \frac{∂f_1}{ ∂y} (x_n, y_n) & \frac{∂f_2}{ ∂y} (x_n, y_n) \\ \end{vmatrix}}{D(x_n, y_n)},\quad y_{n+1} = y_n + \frac{\begin{vmatrix} f_1(x_n, y_n) & f_2(x_n, y_n) \\ \frac{∂f_1}{ ∂x} (x_n, y_n) & \frac{∂f_2}{ ∂x} (x_n, y_n) \\ \end{vmatrix}}{D(x_n, y_n)},\text{ where} \]
\[D(x_n, y_n) = \frac{∂f_1}{ ∂x} (x_n, y_n) \frac{∂f_2}{ ∂y} (x_n, y_n)− \frac{∂f_1}{ ∂y} (x_n, y_n) \frac{∂f_2}{ ∂x} (x_n, y_n) .\]
Then the sequence of points \((x_n, y_n)_{n=1}^{\infty}\) converges to a solution. Write a computer program that uses this algorithm to find approximate solutions to the system of equations
\[f_1(x, y) = \sin (x y)− x− y = 0 \quad \text{ and }\quad f_2(x, y) = e^{ 2x} −2x+3y = 0 .\]
Show that you get two different solutions when using \((0,0) \text{ and }(1,1)\) for the initial point \((x_0 , y_0)\).
A
2.7.1. Find the constrained maxima and minima of \(f (x, y) = 2x+ y\) given that \(x^ 2 + y^ 2 = 4\).
2.7.2. Find the constrained maxima and minima of \(f (x, y) = x y\) given that \(x^ 2 +3y^ 2 = 6\).
2.7.3. Find the points on the circle \(x^ 2+ y^ 2 = 100\) which are closest to and farthest from the point \((2,3)\).
B
2.7.4. Find the constrained maxima and minima of \(f (x, y, z) = x + y^ 2 +2z\) given that \(4x^ 2 +9y^ 2 − 36z^ 2 = 36\).
2.7.5. Find the volume of the largest rectangular parallelepiped that can be inscribed in the ellipsoid
\[\frac{x^ 2}{ a^ 2} + \frac{y^ 2}{ b^ 2} + \frac{z^ 2}{ c^ 2} = 1 .\]