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mathematics
applied calculus

Calculus And Its Applications 14th Edition Larry Goldstein, David Lay, David Schneider, Nakhle Asmar - Solutions

When a car’s brakes are slammed on at a speed of x miles per hour, the stopping distance is 1/20x2 feet. Show that when the speed is doubled the stopping distance increases fourfold.
A service contract on a computer costs $100 per year. The contract covers all necessary maintenance and repairs on the computer. Suppose that the actual cost to the manufacturer for providing this service is a random variable X (measured in hundreds of dollars) whose probability density function is
The probability of a negative test on the mixture of 5 samples is (.95)5 = .774. Thus, Table 2 gives the probabilities for the number X of tests required.(a) Find E(X ).(b) If the laboratory uses this procedure on 200 blood samples (that is, 40 batches of 5 samples), about how many tests can it
The amount of time required to serve a customer at a bank has an exponential density function with mean 3 minutes.Find the probability that serving a customer will require more than 5 minutes.
During a certain part of the day, an average of five automobiles arrives every minute at the tollgate on a turnpike. Let X be the number of automobiles that arrive in any 1-minute interval selected at random. Let Y be the interarrival time between any two successive arrivals. (The average
How is the expected value of a continuous random variable computed?
A newspaper publisher estimates that the proportion X of space devoted to advertising on a given day is a random variable with the beta probability density f (x) = 30x2(1 - x)2, 0 ≤ x ≤ 1.(a) Find the cumulative distribution function for X.(b) Find the probability that less than 25% of the
A citrus grower anticipates a profit of $100,000 this year if the nightly temperatures remain mild. Unfortunately, the weather forecast indicates a 25% chance that the temperatures will drop below freezing during the next week. Such freezing weather will destroy 40% of the crop and reduce the
A certain gas station sells X thousand gallons of gas each week. Suppose that the cumulative distribution function for X is F (x) = 1 - 1/4 (2 - x)2, 0 ≤ x ≤ 2.(a) If the tank contains 1.6 thousand gallons at the beginning of the week, find the probability that the gas station will have enough
During a certain part of the day, the time between arrivals of automobiles at the tollgate on a turnpike is an exponential random variable with expected value 20 seconds.Find the probability that the time between successive arrivals is more than 60 seconds.
Find the value of k that makes the given function a probability density function on the specified interval.f (x) = k, 5 ≤ x ≤ 20
A bakery makes gourmet cookies. For a batch of 4800 oatmeal and raisin cookies, how many raisins should be used so that the probability of a cookie having no raisins is .01?
Let X be the proportion of new restaurants in a given year that make a profit during their first year of operation, and suppose that the density function for X is f (x) = 20x3(1 - x), 0 ≤ x ≤ 1.(a) Find E(X ) and give an interpretation of this quantity.(b) Compute Var(X ).
Give two ways to compute the variance of a continuous random variable.
The density function f (x) for the lifetime of a certain battery is shown in Fig. 1. Each battery lasts between 3 and 10 hours.(a) Sketch the graph of the corresponding cumulative distribution function F (x).(b) What is the meaning of the number F (7) - F (5)?(c) Formulate the number in part (b) in
Find the expected values and the standard deviations (by inspection) of the normal random variables with the density functions given in Exercises. 1 3V2п e-(1/18)x²
Suppose that the weather forecast in Exercise 9 indicates a 10% chance that cold weather will reduce the citrus grower’s profit from $100,000 to $85,000 and a 10% chance that cold weather will reduce the profit to $75,000. Should the grower spend $5000 to protect the citrus fruit against the
During a certain part of the day, the time between arrivals of automobiles at the tollgate on a turnpike is an exponential random variable with expected value 20 seconds.Find the probability that the time between successive arrivals is greater than 10 seconds and less than 30 seconds.
Find the value of k that makes the given function a probability density function on the specified interval.f (x) = k>√x, 1 ≤ x ≤ 4
Find the expected values and the standard deviations (by inspection) of the normal random variables with the density functions given in Exercises. 1 5√2TT e-(1/2)[(x-3)/5]²
If X is a random variable with density function f (x) on A ≤ x ≤ B, the median of X is that number M such thatIn other words, Pr (X ≤ M ) = 1/2.Find the median of the random variable whose density function is f (x) = 1/18 x, 0 ≤ x ≤ 6. M ΤΑ A f(x) dx = -la 1 2
Suppose that a large number of persons become infected by a particular strain of staphylococcus that is present in food served by a fast-food restaurant and that the germ usually produces a certain symptom in 5% of the persons infected. What is the probability that, when customers are examined, the
Let X be a continuous random variable with density function f (x) = 3x-4, x ≥ 1. Compute E(X ) and Var(X ).
The condenser motor in an air conditioner costs $300 to replace, but a home air-conditioning service will guarantee to replace it free when it burns out if you will pay an annual insurance premium of $25. The life span of the motor is an exponential random variable with an expected life of 10
If X is a random variable with density function f (x) on A ≤ x ≤ B, the median of X is that number M such thatIn other words, Pr (X ≤ M ) = 1/2.Find the median of the random variable whose density function is f (x) = 2(x - 1), 1 ≤ x ≤ 2. M ΤΑ A f(x) dx = -la 1 2
Suppose that you toss a fair coin until a head appears and count the number X of consecutive tails that precede it.(a) Determine the probability that exactly n consecutive tails occur.(b) Determine the average number of consecutive tails that occur.(c) Write down the infinite series that gives the
An exponential random variable X has been used to model the relief times (in minutes) of arthritic patients who have taken an analgesic for their pain. Suppose that the density function for X is f (x) = ke-kx and that a certain analgesic provides relief within 4 minutes for 75% of a large group of
At a certain supermarket, the amount of wait time at the express lane is a random variable with density function f (x) = 11/[10(x + 1)2], 0 ≤ x ≤ 10. (See Fig. 8.) Find the probability of having to wait less than 4 minutes at the express lane. 1.1 y 11 10(x + 1)² 0 10 Figure 8 A density
Show that the function f (x) = e-x2/2 has a relative maximum at x = 0.
Suppose that the lifetime X (in hours) of a certain type of flashlight battery is a random variable on the interval 30 ≤ x ≤ 50 with density function f (x) = 1/20, 30 ≤ x ≤ 50. Find the probability that a battery selected at random will last at least 35 hours.
Extensive records are kept of the life spans (in months) of a certain product, and a relative frequency histogram is constructed from the data, using areas to represent relative frequencies. It turns out that the upper boundary of the relative frequency histogram is approximated closely by the
Exercises illustrate a technique from statistics (called the method of maximum likelihood) that estimates a parameter for a probability distribution.In a production process, a box of fuses is examined and found to contain two defective fuses. Suppose that the probability of having two defective
A piece of new equipment has a useful life of X thousand hours, where X is a random variable with the density function f (x) = .01xe-x/10, x ≥ 0. A manufacturer expects the machine to generate $5000 of additional income for every thousand hours of use, but the machine costs $60,000. Should the
Show that the function f (x) = e-(1/2)[(x-μ)/σ]2 has a relative maximum at x = μ.
The machine component described in Exercise 11 has a 50% chance of lasting at least how long?Exercise 11The useful life (in hundreds of hours) of a certain machine component is a random variable X with the cumulative distribution function F (x) = 19x2, 0 ≤ x ≤ 3.(a) Find E(X ), and give an
Show that the function f (x) = e-x2/2 has inflection points at x = ±1.
Suppose that the police force in Exercise 23 maintains the same height requirements for women as men and that the heights of women in the city are normally distributed, with μ = 65 inches and σ = 1.6 inches. What percentage of the women are eligible for recruitment?Exercise 23The men hired by a
Calculate the area under the standard normal curve for values of z(a) Between .5 and 1.5,(b) Between -.75 and .75,(c) To the left of -.3,(d) To the right of -1.
The cumulative distribution function for a random variable X on the interval 1 ≤ x ≤ 5 is F (x) = 1/2 √x - 1. (See Fig. 9.) Find the corresponding density function. 1 y F(x)=√x-1 1 2 3 4 5 Figure 9 A cumulative distribution function. X
Let X be a geometric random variable with parameter p. Derive the formula for E(X) by using the power series formula 1 + 2x + 3x² + 1 (1-x)² for x < 1.
In Exercise 12, find the length of time T such that half of the assemblies are completed in T minutes or less.Exercise 12The time (in minutes) required to complete an assembly on a production line is a random variable X with the cumulative distribution function F (x) = 1/125x3, 0 ≤ x ≤ 5.(a)
Find the number M such that, half of the time, the dairy in Exercise 16 sells M thousand gallons of milk or less.Exercise 16The amount of milk (in thousands of gallons) that a dairy sells each week is a random variable X with the density function f (x) = 4(x - 1)3, 1 ≤ x ≤ 2. (See Fig. 4.)(a)
A certain machine part has a nominal length of 80 millimeters, with a tolerance of ±.05 millimeter. Suppose that the actual length of the parts supplied is a normal random variable with mean 79.99 millimeters and standard deviation .02 millimeter. How many parts in a lot of 1000 should you expect
Show that the function f (x) = e-(1/2)[(x-μ)/σ]2 has inflection points at x = μ ± s.
The cumulative distribution function for a random variable X on the interval 1 ≤ x ≤ 2 is F (x) = 4/3 - 4/(3x2). Find the corresponding density function.
Find the length of time T such that about half of the time you wait only T minutes or less in the express lane at the supermarket.
Let Z be a standard normal random variable. Calculate(a) Pr (-1.3 ≤ Z ≤ 0) (b) Pr (.25 ≤ Z)(c) Pr (-1 ≤ Z ≤ 2.5) (d) Pr (Z ≤ 2)
The number of times a printing press breaks down each month is Poisson distributed with λ = 4. What is the probability that the printing press breaks down between 2 and 8 times during a particular month?
Compute the cumulative distribution function corresponding to the density function f (x) = 1/2 (3 - x), 1 ≤ x ≤ 3.
Use the laws of exponents to simplify the algebraic expressions. Xp3 -5,,6 4-5 20
When a drug is injected into a person’s muscle tissue, the concentration y of the drug in the blood is a function of the time elapsed since the injection. The graph of a typical time–concentration function f is given in Fig. 13, where t = 0 corresponds to the time of the injection.What is the
Letand graph the function f (f (x)) in the window [ -15, 15] by [ -10, 10]. Trace to examine the coordinates of several points on the graph and then determine the formula for f (f (x)). f(x) = X x-1
Evaluate each of the functions at the given value of x.f (x) = |x|, x = -2/3
The profit function in Fig. 19.Translate the task “solve P(x) = 30,000” into a task involving the graph of the function. 52,500 y y = P(x) 2500 Figure 19 A profit function. x
Solve the equations x² x2 + 14x + 49 1²+1 0
Solve the equations x²8x + 16 1 + √x = 0
Use the laws of exponents to simplify the algebraic expressions. Your answer should not involve parentheses or negative exponents(x1/3)6
Use the laws of exponents to simplify the algebraic expressions. Your answer should not involve parentheses or negative exponents 4 X sª . xp2
When a drug is injected into a person’s muscle tissue, the concentration y of the drug in the blood is a function of the time elapsed since the injection. The graph of a typical time–concentration function f is given in Fig. 13, where t = 0 corresponds to the time of the injection.What is the
Use the laws of exponents to simplify the algebraic expressions. x3/2 √x
The profit function in Fig. 19.Translate the task “find P(2000)” into a task involving the graph. 52,500 y y = P(x) 2500 Figure 19 A profit function. x
Use the laws of exponents to simplify the algebraic expressions. Your answer should not involve parentheses or negative exponents 1 -3 X
Use the laws of exponents to simplify the algebraic expressions. F (8x2/3)
Use your graphing calculator to find the value of the given function at the indicated values of x.f (x) = 3x3 + 8; x = -11, x = 10
Use your graphing calculator to find the value of the given function at the indicated values of x. f(x) = x² + √√3x − π; x = -2, x = 20
Use your graphing calculator to find the value of the given function at the indicated values of x.f (x) = x4 + 2x3 + x - 5; x = -1/2, x = 3
A ball is thrown straight up into the air. The function h(t) gives the height of the ball (in feet) after t seconds. Translate the task into both a statement involving the function and a statement involving the graph of the function.Find the height of the ball after 3 seconds.
Suppose that the cable television company’s cost function in Example 4 changes to C(x) = 275 + 12x. Determine the new breakeven points.
Use the laws of exponents to simplify the algebraic expressions. Your answer should not involve parentheses or negative exponents.(x3 · y6)1/3
When a drug is injected into a person’s muscle tissue, the concentration y of the drug in the blood is a function of the time elapsed since the injection. The graph of a typical time–concentration function f is given in Fig. 13, where t = 0 corresponds to the time of the injection.What is the
Use your graphing calculator to find the value of the given function at the indicated values of x. f(x) = 2x - 1 43 x³ + 3x² + 4x + 1' x = 2, x = 6
When a drug is injected into a person’s muscle tissue, the concentration y of the drug in the blood is a function of the time elapsed since the injection. The graph of a typical time–concentration function f is given in Fig. 13, where t = 0 corresponds to the time of the injection.At what time
Use the laws of exponents to simplify the algebraic expressions. Your answer should not involve parentheses or negative exponents. 4\ 3 X 2
Use the laws of exponents to simplify the algebraic expressions. Your answer should not involve parentheses or negative exponents.x-1/2
Suppose that $15,000 is deposited in a savings account that pays 4% per annum, compounded monthly, for t years.(a) Express the account balance A(t) as a function of t, the number of years that the principal has been in the account.(b) Calculate the account balance at the end of 2 years and at the
A ball is thrown straight up into the air. The function h(t) gives the height of the ball (in feet) after t seconds. Translate the task into both a statement involving the function and a statement involving the graph of the function.Find the time at which the ball attains its greatest height.
When a car is moving at x miles per hour and the driver decides to slam on the brakes, the car will travel x + 1/20 x2 feet. (The general formula is f (x) = ax + bx2, where the constant a depends on the driver’s reaction time and the constant b depends on the weight of the car and the type of
Use the laws of exponents to simplify the algebraic expressions. Your answer should not involve parentheses or negative exponents -2
Suppose that $7000 is deposited in a savings account that pays 9% per annum, compounded biannually, for t years.(a) Express the account balance A(t) as a function of t, the number of years that the principal has been in the account.(b) Calculate the account balance at the end of 10 years and at the
A ball is thrown straight up into the air. The function h(t) gives the height of the ball (in feet) after t seconds. Translate the task into both a statement involving the function and a statement involving the graph of the function.Find the greatest height attained by the ball.
Is the point (3, 12) on the graph of the function f (x) = (x - 1/2)(x + 2)?
Suppose that $15,000 is deposited in a savings account that pays a rate of interest r per annum, compounded annually, for 10 years.(a) Express the account balance A(r) as a function of r.(b) Calculate the account balance for r = 0.04 and r = 0.06
A ball is thrown straight up into the air. The function h(t) gives the height of the ball (in feet) after t seconds. Translate the task into both a statement involving the function and a statement involving the graph of the function.Determine when the ball will hit the ground.
Is the point (-2, 12) on the graph of the function f (x) = x(5 + x)(4 - x)?
Suppose that $7000 is deposited in a savings account that pays a rate of interest r per annum, compounded annually, for 20 years.(a) Express the account balance A(r) as a function of r.(b) Calculate the account balance for r = .07 and r = .12.
Use the laws of exponents to simplify the algebraic expressions. Your answer should not involve parentheses or negative exponents 7/ε(x+1)x+ IA
A ball is thrown straight up into the air. The function h(t) gives the height of the ball (in feet) after t seconds. Translate the task into both a statement involving the function and a statement involving the graph of the function.Determine when the height of the ball is 100 feet.
Is the point (1, 1) on the graph of the function g(x) = (3x - 1)/(x2 + 1)?
Use the laws of exponents to simplify the algebraic expressions. Your answer should not involve parentheses or negative exponents.(x3y5)4
Use the laws of exponents to simplify the algebraic expressions. Your answer should not involve parentheses or negative exponents 45 2 3 X
A ball is thrown straight up into the air. The function h(t) gives the height of the ball (in feet) after t seconds. Translate the task into both a statement involving the function and a statement involving the graph of the function.Find the height of the ball when it is first released.
Is the point (4, 1/4) on the graph of the function g(x) = (x2 + 4)/(x + 2)?
A ball thrown straight up into the air has height -16x2 + 80x feet after x seconds.(a) Graph the function in the window [0, 6] by [ -30, 120].(b) What is the height of the ball after 3 seconds?(c) At what times will the height be 64 feet?(d) At what time will the ball hit the ground?(e) When will
Use the laws of exponents to simplify the algebraic expressions. Your answer should not involve parentheses or negative exponents. -3x 15x4
Find the y-coordinate of the point (a + 1, ?) if this point lies on the graph of the function f (x) = x3.
Compute f (1), f (2), and f (3). f(x) = Vx 1+x for 0 ≤ x < 2 for 2 ≤x≤5
Use the laws of exponents to simplify the algebraic expressions. Your answer should not involve parentheses or negative exponents. -x3y -xy
Compute f (1), f (2), and f (3). f(x) = [1/x L.x² for 1 ≤x≤2 for 2 < x
The daily cost (in dollars) of producing x units of a certain product is given by the function C(x) = 225 + 36.5x - .9x2 + .01x3.(a) Graph C(x) in the window [0, 70] by [ -400, 2000].(b) What is the cost of producing 50 units of goods?(c) Consider the situation as in part (b). What is the
Find the y-coordinate of the point (2 + h, ?) if this point lies on the graph of the function f (x) = (5/x) - x.

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