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mathematics
calculus with applications
Calculus With Applications 12th Edition Margaret L. Lial - Solutions
Identify which geometric series converge. Give the sum of each convergent series.27 + 9 + 3 + 1 +...
Use l’Hospital’s rule where applicable to find each limit. lim x-0 In(x + 1) X
For the functions defined as follows, find the Taylor polynomials of degree 4 at 0.ƒ(x) = √x + 9
The Taylor series for ex at 0 converges for all x.Determine whether each of the following statements is true or false, and explain why.
In Exercises, find the Taylor series for the functions defined as follows. Give the interval of convergence for each series. f(x) = -3 4 - x
List the first n terms of the geometric sequence satisfying the following conditions.a3 = 6, a4 = 12, n = 5
Find the amount of each ordinary annuity. (Interest is compounded annually.)R = $11,500, i = 0.055, n = 30
Use Newton’s method to find a solution for each equation in the given intervals. Find all solutions to the nearest hundredth.-3x3 + 5x2 + 3x + 2 = 0; [2, 3]
Identify which geometric series converge. Give the sum of each convergent series.64 + 16 + 4 + 1 +...
In Exercises, find the Taylor series for the functions defined as follows. Give the interval of convergence for each series. f(x) = 8x 1 + 3x
For the functions defined as follows, find the Taylor polynomials of degree 4 at 0.ƒ(x) = √x + 16
Use l’Hospital’s rule where applicable to find each limit. lim x-0 et - 1 -4 X
The Taylor series for ln(1 + x) at 0 converges for all x.Determine whether each of the following statements is true or false, and explain why.
List the first n terms of the geometric sequence satisfying the following conditions.a2 = 9, a3 = 3, n = 4
Find the amount of each ordinary annuity. (Interest is compounded annually.)R = $13,400, i = 0.045, n = 25
Use Newton’s method to find a solution for each equation in the given intervals. Find all solutions to the nearest hundredth.4x3 - 5x2 - 6x + 6 = 0; [1, 2]
Identify which geometric series converge. Give the sum of each convergent series.100 + 10 + 1 +...
In Exercises, find the Taylor series for the functions defined as follows. Give the interval of convergence for each series. f(x) = 7x 1 + 2x
For the functions defined as follows, find the Taylor polynomials of degree 4 at 0. f(x) = √√x + 8 X
Use l’Hospital’s rule where applicable to find each limit. e²x lim x→05r2 1 X
Newton’s method converges as long as there is a real root and the function is differentiable.Determine whether each of the following statements is true or false, and explain why.
Find a5 and an for the following geometric sequences.a1 = 4, r = 3
Find the amount of each ordinary annuity based on the information given.R = $10,500, 10% interest compounded semiannually for 7 years
Use Newton’s method to find a solution for each equation in the given intervals. Find all solutions to the nearest hundredth.2x4 - 2x3 - 3x2 - 5x - 8 = 0; [-2, -1], [2, 3]
Identify which geometric series converge. Give the sum of each convergent series.44 + 22 + 11 +...
Find a5 and an for the following geometric sequences.a1 = 8, r = 4
L’Hospital’s rule says that to take the derivative of a quotient, divide the derivative of the numerator by the derivative of the denominator.Determine whether each of the following statements is true or false, and explain why.
Identify which geometric series converge. Give the sum of each convergent series. + + 5100 + 5 14
Find the amount of each ordinary annuity based on the information given.R = $4200, 6% interest compounded semiannually for 11 years
Use Newton’s method to find a solution for each equation in the given intervals. Find all solutions to the nearest hundredth.3x4 + 4x3 - 6x2 - 2x - 12 = 0; [-3, -2], [1, 2]
In Exercises, find the Taylor series for the functions defined as follows. Give the interval of convergence for each series. f(x) = z.t 4 - x
For the functions defined as follows, find the Taylor polynomials of degree 4 at 0. I + XA = (x)ƒ
1. If X is a continuous random variable, P(a ≤ X ≤ b) is the same as P(a < X < b). Since these are different events, how can they have the same probability?2. Someone who rides the subway back and forth to work each weekday makes about 40 trips a month. On the exponential subway, how many
A continuous random variable can take on values greater than 1.Determine whether each of the following statements is true or false, and explain why.
A probability density function can take on values greater than 1.Determine whether each of the following statements is true or false, and explain why.
A continuous random variable can take on values less than 0.Determine whether each of the following statements is true or false, and explain why.
A probability density function can take on values less than 0.Determine whether each of the following statements is true or false, and explain why.
The expected value of a random variable must always be at least 0.Determine whether each of the following statements is true or false, and explain why.
The variance of a random variable must always be at least 0.Determine whether each of the following statements is true or false, and explain why.
Decide whether each function defined as follows is a probability density function for the given interval. f(x) = (2x + (2x + 4); [1,4] 27
The expected value of a uniform random variable is the average of the endpoints of the interval over which the density function is positive.Determine whether each of the following statements is true or false, and explain why.
For an exponential random variable, the expected value and standard deviation are always equal.Determine whether each of the following statements is true or false, and explain why.
The normal distribution and the exponential distribution have approximately the same shape.Determine whether each of the following statements is true or false, and explain why.
In the standard normal distribution, the expected value is 1 and the standard deviation is 0.Determine whether each of the following statements is true or false, and explain why.
In a probability function, the y-values (or function values) represent______________.
Define a continuous random variable.
The probability density function of a random variable X is defined byFind the following probabilities.(a) P(X ≥ 3) (b) P(X ≤ 4)(c) P(3 ≤ X ≤ 4) f(x) = 1 - - 1 Vx - 1 for x in [2, 5].
Give the two conditions that a probability density function for [a, b] must satisfy.
In a probability density function, the probability that X equals a specific value, P(X = c), is______________.
Decide whether each function defined as follows is a probability density function for the given interval.ƒ(x) = √x; [4, 9]
The probability density function of a random variable X is defined byFind the following probabilities.(a) P(X ≤ 12) (b) P(X ≥ 31/2)(c) P(10.8 ≤ X ≤ 16.2) f(x): = 1 10 for x in [10, 20].
Decide whether each function defined as follows is a probability density function for the given interval.ƒ(x) = 0.7e-0.7x; [0, ∞)
Decide whether each function defined as follows is a probability density function for the given interval.ƒ(x) = 0.4; [4, 6.5]
The probability density functions shown in the graphs have the same mean. Which has the smallest standard deviation?(a)(b)(c) 1 μ X
In Exercises, find a value of k that will make f(x) define a probability density function for the indicated interval.ƒ(x) = kx2; [1, 4]
For the probability density functions defined in Exercises, find (a) The expected value, (b) The variance, (c) The standard deviation, (d) The median, and (e) The cumulative distribution function. f(x) - 2/0 (1 + 3): 11.9 [1,9] Vx
For the probability density functions defined in Exercises, find (a) The expected value, (b) The variance, (c) The standard deviation, (d) The median, and (e) The cumulative distribution function. 2 f(x) = ²/² (x - (x - 2); [2,5]
In Exercises, find a value of k that will make f(x) define a probability density function for the indicated interval.ƒ(x) = k√x; [4, 9]
For the probability density functions defined in Exercises, find (a) The expected value, (b) The variance, (c) The standard deviation, (d) The median, and (e) The cumulative distribution function. f(x) = ; [4, 9] 5'
Describe what the expected value or mean of a probability distribution represents geometrically.
For Exercises, find (a) The mean of the distribution, (b) The standard deviation of the distribution, and (c) The probability that the value of the random variable is within 1 standard deviation of the mean. f(x) = 5 112 (1-x-3/2) for x in [1, 25]
For the probability density functions defined in Exercises, find (a) The expected value, (b) The variance, (c) The standard deviation, (d) The median, and (e) The cumulative distribution function.ƒ(x) = 5x-6; [1, ∞)
The probability density function of a random variable is defined by ƒ(x) = 4x - 3x2 for x in [0, 1]. Find the following for the distribution.(a) The mean(b) The standard deviation(c) The probability that the value of the random variable will be less than the mean(d) The probability that the value
Find the median of the random variable of Exercise 29. Then find the probability that the value of the random variable will lie between the median and the mean of the distribution.Exercise 29The probability density function of a random variable is defined by ƒ(x) = 4x - 3x2 for x in [0, 1]. Find
For Exercises, find (a) The mean of the distribution, (b) The standard deviation of the distribution, and (c) The probability that the value of the random variable is within 1 standard deviation of the mean.ƒ(x) = 0.01e-0.01x for x in [0, ∞)
In Exercises, find the proportion of observations of a standard normal distribution for each region.The region to the left of z = -0.43
The chi-square distribution is important in statistics for testing whether data come from a specified distribution and for testing the independence of two characteristics of a set of data. When a quantity called the degrees of freedom is equal to 4, the probability density function is given by(a)
In Exercises, find the proportion of observations of a standard normal distribution for each region.The region to the right of z = 1.62
In Exercises, find the proportion of observations of a standard normal distribution for each region.The region between z = -1.17 and z = -0.09
The topics in this short chapter involved much of the material studied earlier in this book, including functions, domain and range, exponential functions, area and integration, improper integrals, integration by parts, and numerical integration. For the following special probability density
In Exercises, find the proportion of observations of a standard normal distribution for each region.The region between z = -1.39 and z = 1.28
In Exercises, find the proportion of observations of a standard normal distribution for each region.The region that is 1.2 standard deviations or more below the mean
In Exercises, find the proportion of observations of a standard normal distribution for each region.The region that is up to 2.5 standard deviations above the mean
The time (in years) until a certain machine requires repairs is a random variable t with probability density function defined by(a) Find the probability that no repairs are required in the first three years by finding the probability that a repair will beneeded in years 4 through 25.(b) Find the
In Exercises, find the proportion of observations of a standard normal distribution for each region.Find a z-score so that 52% of the area under the normal curve is to the right of z.
In Exercises, find the proportion of observations of a standard normal distribution for each region.Find a z-score so that 21% of the area under the normal curve is to the left of z.
The number of new outlets for a clothing manufacturer is an exponential distribution with probability density function defined byFind the following for this distribution.(a) The mean(b) The standard deviation(c) The probability that the number of new outlets will be greater than the mean
When the degrees of freedom in the chi-square distribution (see the previous exercise) is 1, the probability density function is given byCalculating probabilities is now complicated by the fact that the density function cannot be antidifferentiated. Numerical integration is complicated because the
The topics in this short chapter involved much of the material studied earlier in this book, including functions, domain and range, exponential functions, area and integration, improper integrals, integration by parts, and numerical integration. For the following special probability density
The topics in this short chapter involved much of the material studied earlier in this book, including functions, domain and range, exponential functions, area and integration, improper integrals, integration by parts, and numerical integration. For the following special probability density
The price per share (in dollars) of a particular mutual fund is a random variable x with probability density function defined by(a) Find the probability that the price will be less than $8.50.(b) Find the expected value of the price.(c) Find the standard deviation. 3 f(x) = 2³/(x² - 16x + 65) for
The weight gain (in grams) of rats fed a certain vitamin supplement is a continuous random variable with probability density function defined by(a) Find the mean of the distribution.(b) Find the standard deviation of the distribution.(c) Find the probability that the value of the random variable is
The number of repairs required by a new product each month is exponentially distributed with an average of 8.(a) What is the probability density function for this distribution?(b) Find the expected number of repairs per month.(c) Find the standard deviation.(d) What is the probability that the
The useful life of a certain appliance part (in hundreds of hours) is 46.2, with a standard deviation of 15.8. Find the probability that one such part would last for at least 6000 (60 hundred) hours. Assume a normal distribution.
The body temperature (in degrees Celsius) of a particular species of bird is a continuous random variable with probability density function defined by(a) What is the expected body temperature of this species?(b) Find the probability of a body temperature below the mean. f(x) = 3 19,696 (x² + x)
A piece of equipment is being insured against early failure. The time from purchase until failure of the equipment is exponentially distributed with mean 10 years. The insurance will pay an amount x if the equipment fails during the first year, and it will pay 0.5x if failure occurs during the
The distance (in meters) that a certain animal moves away from a release point is exponentially distributed, with a mean of 100 m. Find the probability that the animal will move no farther than 100 m away.
The snowfall (in inches) in a certain area is uniformly distributed over the interval [2, 30].(a) What is the expected snowfall?(b) What is the probability of getting more than 20 inches of snow?
The average birth weight of infants in the United States is 7.8 lb, with a standard deviation of 1.1 lb. Assuming a normal distribution, what is the probability that a newborn will weigh more than 9 lb?
In a pilot study on tension of the heart muscle in dogs, the mean tension was 2.2 g, with a standard deviation of 0.4 g. Find the probability of a tension of less than 1.9 g. Assume a normal distribution.
According to the National Center for Health Statistics, the life expectancy for a 65-year-old American male is 17.0 years. Assuming that, from age 65, the survival of American males follows an exponential distribution, determine the following probabilities.(a) The probability that a randomly
According to the National Center for Health Statistics, the life expectancy for a 50-year-old American female is 32.5 years. Assuming that, from age 50, the survival of American females follows an exponential distribution, determine the following probabilities.(a) The probability that a randomly
The number of deaths in the United States caused by assault (murder) for each age group is given in the table on the next page.(a) Plot the data. What type of function appears to best match these data?(b) Use the regression feature on your graphing calculator to find a quartic equation that models
The time between major earthquakes in the Taiwan region is a random variable with probability density function defined bywhere t is measured in days. Find the expected value and standard deviation of this probability density function. f(t) = 1 3650.1 e-t/3650.1
The average state “take” on lotteries is 40%, with a standard deviation of 13%. Assuming a normal distribution, what is the probability that a state-run lottery will have a “take” of more than 50%?
Show that each function defined as follows is a probability density function on the given interval; then find the indicated probabilities. (Round probabilities to 4 decimal places.)(a) P(0 ≤ X ≤ 2) (b) P(X ≥ 2)(c) P(1 ≤ X ≤ 3) f(x) = 12 16 3x3 if 0 ≤ x ≤ 2 if x > 2
In Exercise 5, how long will it take Patricia to accumulate $30,000 in her retirement account?.Exercise 5Patricia deposits $5000 in an IRA at 6% interest compounded continuously for her retirement in 10 years. She intends to make continuous deposits at the rate of $3000 a year until she retires.
Find the general solution for each differential equation. dy dx + 2xy = 4x
1. Calculate the number of years to reduce pollution to 50%, 25%, and 1% of its current level in Lake Erie, which has a volume of 458 cm3 and an output flow rate of 5,550,720 liter/sec.2. Repeat Exercise 1 for Lake Michigan, which has a volume of 4871 cm3 and an output flow rate of 5,012,640
The expected value of a continuous random variable X can be negative.Determine whether each statement is true or false, and explain why.
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