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mathematics
calculus

Questions and Answers of Calculus

If a die is rolled 300 times, what is the probability that a 6 will occur 55 times? In the following problems use a normal approximation to the binomial.
If a fair coin is tossed 100 times, what is the probability that 50 heads will occur? In the following problems use a normal approximation to the binomial.
If a fair coin is tossed 100 times, what is the probability that no more than 50 heads will occur? In the following problems use a normal approximation to the binomial.
If a die is rolled 300 times, what is the probability that a 6 will occur at least 55 times? In the following problems use a normal approximation to the binomial.
Circuitown buys plasma televisions in lots of 1000. If one brand of set is defective with probability 0.01, what is the probability that at least 8 sets will be defective in a lot? In the following
The manager of a wholesale buyers' club has been told by the home office that 40% of all customers favor a new policy that gives nearly immediate checkout (by moving to the front of the line) for
An airport announced a plan to make a special security checkpoint so that first-class passengers would have priority passing through security. Airlines supported this plan, assuming that 25% of
Suppose that AudioMart has a secret sale released only on the Internet because management thinks that word-of-mouth publicity about the sale will result in 5% of U.S. households recognizing the
A report by U.S. News & World Report stated that 35% of all employee credentials examined were falsified. If this is true, what is the probability that at least 75 of 200 employees hired during a
A smoking cessation program advertises that 60% of all participants in its program quit smoking for at least 6 months. Suppose Consumer Checkup surveyed 400 "graduates" of the program and found that
If the ratio of boys born to girls born is 105 to 100, what is the probability that out of 2000 babies born in Santa Clara, 1030 will be girls?
It has been determined empirically that the probability that a certain cell will survive for a given period of time is 0.3. What is the probability that 600 of 2000 of these cells will survive for
Empirical records show that the probability is 0.06 that a driver insured with Riders Insurance Company will have an accident during 2016 in which he or she is at fault. If the company has 12,000
The Rent-to-Own Company estimates that 40% of its rentals result in a sale of the product. If the company rents 15,000 of its products in a year, what is the probability that it will sell at most
Suppose a final exam contains 120 multiple-choice questions with 5 choices for each answer, and a poorly prepared student knows the answers to 60 questions and guesses on the other 60. What is the
The probability that a certain drug will lower cholesterol significantly is 0.82. If the drug is administered to 1000 patients, what is the probability that more than 800 will have significantly
A standardized exam for admission to a special program has been passed by 53% of the students taking it over the last five years. (a) Suppose that 1120 of 2000 students taking the test this year
A television show rating of 6.2 means that 6.2% of the 110 million TV households watched the show. If a sample of 100 local TV households were queried, what is the probability that more than 10 would
The probability is 0.025 that a knee operation at Memorial Hospital has complications that warrant reopening the knee. If the hospital performs 1300 knee operations in a year, what is the probability
According to one of Mendel's hybridization experiments, when peas with green pods are crossed with peas with yellow pods, 25% of the offspring pods should be yellow. What is the probability that at
For the binomial experiments, find the normal approximation of the probability of 1. 91 successes in 150 trials if p = 0.6. 2. 740 successes in 900 trials if p = 0.8.
Find the mean, variance, and standard deviation of the following data: 4, 3, 4, 6, 8, 0, 2.
Find the mean, variance, and standard deviation of the following data: 3, 2, 1, 5, 1, 4, 0, 2, 1, 1.
Determine whether each function or table in Problems 15-18 represents a discrete probability distribution. 1. Pr(x) = x 15, x = 1, 2, 3, 4, 5 2. Pr(x) = x 55, x = 1, 2, 3, 4, 5, 6, 7, 8, 9 3. x
A bag contains 5 black balls and 7 white balls. If we draw 4 balls, with each one replaced before the next is drawn, what is the probability that (a) 2 balls are black? (b) At least 2 balls are black?
For the discrete probability distribution described by Pr(x) = x/16, x = 1, 2, 3, 4, 6, find the (a) Mean. (b) Variance. (c) Standard deviation.
For the probability distribution shown, find the x Pr(x) 1...........................1/12 2.............................1/6 3.............................1/3 4...........................5/12 (a)
A coin has been altered so that the probability that a head will occur is 2/3. If the coin is tossed 6 times, give the mean and standard deviation of the distribution of the number of heads.
1. Suppose a pair of dice is thrown 18 times. How many times would we expect a sum of 7 to occur? 2. Expand (x + y)5. 3. What is the area under the standard normal curve between z = - 1.6 and z = 1.9?
1. If z is a standard normal score, find Pr(- 1 ≤ z ≤ -0.5). 2. Find Pr(1.23 ≤ z ≤ 2.55). 3. If a variable x is normally distributed, with μ = 25 and σ = 5, find Pr(25 ≤ x ≤ 30).
1. If a variable x is normally distributed, with μ = 25 and σ = 5, find Pr(20 ≤ x ≤ 30).2. If a variable x is normally distributed, with μ = 25 and σ = 5, find Pr(30 ≤ x ≤ 35).
If a die is rolled 4 times, what is the probability that a number greater than 4 is rolled at least 2 times?
Find the number A so that Pr(- A < z < A) = 0.98 for the standard normal z-score.
For the binomial distributions will the normal distribution be a good approximation if1. n = 25, p = 0.15?2. n = 15, p = 0.4?
50 successes in 200 trials when p = 0.2. For the binomial experiments find the normal approximation for the probability of
At least 43 successes in 60 trials when p = 0.8. For the binomial experiments find the normal approximation for the probability of
Between 19 and 36 successes (inclusive) in 80 trials when p = 0.4. For the binomial experiments find the normal approximation for the probability of
1. No more than 132 successes in 250 trials when p = 0.6. 2. Suppose the probability that a certain couple will have a blond child is 1/4. If they have 6 children, what is the probability that 2 of
Suppose 70% of a population opposes a proposal and a sample of size 5 is drawn from the population. What is the probability that the majority of the sample will favor the proposal?
Construct a frequency histogram for the data. Consider the data in the frequency table, and use these data. Score
One person in 100,000 develops a certain disease. Calculate (a) Pr(exactly 1 person in 100,000 has the disease). (b) Pr(at least 1 person in 100,000 has the disease).
Make a frequency histogram for these data. The distribution of farm families (as a percent of the population) in a 50-county survey is given in the table. Percent Number of
Find the mean percent of farm families in these 50 counties. The distribution of farm families (as a percent of the population) in a 50-county survey is given in the table. Percent Number of
Find the standard deviation of the percent of farm families in these 50 counties. The distribution of farm families (as a percent of the population) in a 50-county survey is given in the
1. The probability that a man with prostate cancer will test positive with the prostate specific antigen (PSA) test is 0.91. If 500 men with prostate cancer are tested, how many would we expect to
1. Suppose you paid $2 for a bet on a horse and as a result you will win $100 if your horse wins a given race. Find the expected value of the bet if the odds that the horse will win are 3 to 12.2. A
Refer to Problem 40. Suppose 20% of the population are victims of crime. Out of 5 people selected at random, find the probability that 2 are crime victims. (a) What is the expected number of people
1. What is the mode of the data? 2. What is the mean of the data? 3. What is the median of the data? Consider the data in the frequency table, and use these data. Score
Suppose the mean SAT score for students admitted to a university is 1500, with a standard deviation of 300. Suppose that a student is selected at random. If the scores are normally distributed, find
Suppose the mean net worth of the residents of Sun City, a retirement community, is $611,000. If their net worth is normally distributed with a standard deviation of $96,000, what percent of the
The length of time it takes to complete a college placement test is normally distributed with a mean of 39 minutes and a standard deviation of 6 minutes. How much time (to the nearest minute) is
The ratio of females to males in the U.S. population is 51 to 49. If 100 people are selected at random from the population, find the probability that at least 53 will be female.
Records over 10 years indicate that 5% of Acer televisions will have some defective components. If the latest models have the same number of defects, what is the probability that 6 or more of 100
Construct a frequency histogram for the following set of test scores: 14, 16, 15, 14, 17, 16, 12, 12, 13, 14.
1. What is the median of the scores in Problem 8? 2. What is the mode of the scores in Problem 8? 3. What is the mean of the scores in Problem 8? Construct a frequency histogram for the following set
A graph of y = f (x) is shown and a c-value is given. For each problem, use the graph to find the following, whenever they exist.(a) limx†’c f (x) and (b) f (c)1. c = 42. c = 6 3. c = 20
Complete each table and predict the limit, if it exists.1. f(x) = 2 - x - x2 / x - 1x f(x)0.90.990.9991.0011.011.12. f(x) = 2x + 1 /1/4 - x2x f (x)- 0.51- 0.501- 0.5001- 0.4999- 0.499- 0.49
Use properties of limits and algebraic methods to find the limits, if they exist. 1. limx→-35 (34 + x) 2. limx→80 (82 - x) 3. limx→-1 (4x3 - 2x2 + 2) 4. limx→3 (2x3 - 12x2 + 5x + 3)
Graph each function with a graphing calculator and use it to predict the limit. Check your work either by using the table feature of the calculator or by finding the limit algebraically.1.2.3.4.
Use the table feature of a graphing calculator to predict each limit. Check your work by using either a graphical or an algebraic approach.1.2.3.
Use values 0.1, 0.01, 0.001, 0.0001, and 0.00001 to Approximateto three decimal places. This limit equals the special number e that is discussed in Section 5.1, "Exponential Functions," and Section
1. The total revenue for a product is given byR(x) = 1600x -x22. If the profit function for a product is given by P(x) = 92x - x2 - 1760
The average monthly sales volume (in thousands of dollars) for a firm depends on the number of hours x of training of its sales staff, according to
During the first 4 months of employment, the monthly sales S (in thousands of dollars) for a new salesperson depend on the number of hours x of training, as follows:
Suppose that the daily sales S (in dollars) t days after the end of an advertising campaign are(a) Find S(0).
Sales y (in thousands of dollars) are related to advertising expenses x (in thousands of dollars) according to
During an 8-hour shift, the rate of change of productivity (in units per hour) of infant activity centers assembled after t hours on the job is(c) Is the rate of productivity higher near the lunch
If the revenue for a product is R(x) = 100x - 0.1x2, and the average revenue per unit is(a) Find (b) Find
Suppose that the cost C of obtaining water that contains p percent impurities is given byC(p) = 120,000 / p - 1200(c) Is complete purity possible? Explain.
Suppose that the cost C of removing p percent of the particulate pollution from the smokestacks of an industrial plant is given by(c) Can 100% of the particulate pollution be removed? Explain.
The following table shows part of the April 15, 2015, tax rate schedule for single filers. Use this schedule and create a table of values that could be used to find the following limits, if they
The Ace Parking Garage charges $5.00 for parking for 2 hours or less and $1.50 for each extra hour or part of an hour after the 2-hour minimum. The parking charges for the first 5 hours could be
The Corner Water Corp. of Shippenville, Pennsylvania, has the following rates per 1000 gallons of water used. Usage (x) Cost per 1000 Gallons (C(x)) First 10,000
Long-term parking at Savannah Airport is free for the first half-hour and costs $1.00 for each hour or part of an hour thereafter. If C = C(t) is the charge for t hours in Savannah's long-term
The graph in the figure shows the Dow Jones Industrial Average (DJIA) on a particularly tumultuous day in August 2011.if it exists. Explain what this limit corresponds to.if it exists. Explain what
(a) Find limx†’60 S(x), if it exists.(b) What does this limit predict?(c) Does this prediction seem plausible? Explain.The percent of obese American adults who are severely obese can be
Use the graph of y = f (x) and the given c-value to find the following, whenever they exist.(a) limx†’c f(x)(b) limx†’c f(x)(c) limx†’c f(x)(d) f(c)1. c = - 102. c = 2
(a) Find limx†’40 S(x), if it exists.(b) What does this limit estimate?(c) Is the function accurate as x †’ 40? Explain.The percent of obese American adults who are severely obese
Refer to the figure. For each given x-value, use the figure to determine whether the function is continuous or discontinuous at that x-value. If the function is discontinuous, state which of the
Use the trace and table features of a graphing calculator to investigate whether each of the following functions has any discontinuities.1.2.3.4.
Contains a function and its graph. For each problem, answer parts (a) and (b).(a) Use the graph to determine, as well as you can,(i) Vertical asymptotes.(iv) Horizontal asymptotes.(b) Check your
Complete (a) and (b).(a) Use analytic methods to evaluate each limit.(b) What does the result from part (a) tell you about horizontal asymptotes?You can verify your conclusions by graphing the
Determine whether each function is continuous or discontinuous at the given x-value. Examine the three conditions in the definition of continuity.1.2.3.
Use a graphing calculator to complete (a) and (b).1.2.
Complete (a)-(c). Use analytic methods to find(a) Any points of discontinuity and(b) Limits as x → ∞ and x → - ∞.(c) Then explain why, for these functions, a graphing calculator is better as
Suppose that the weekly sales volume (in thousands of units) for a product is given by y = 32 / (p + 8)2/5 where p is the price in dollars per unit. Is this function continuous? (a) For all values of
Suppose that the average number of minutes M that it takes a new employee to assemble one unit of a product is given by M = 40 + 30t / 2t + 1 where t is the number of days on the job. Is this
The sales volume y (in thousands of dollars) is related to advertising expenditures x (in thousands of dollars) according to Y = 200x / x + 10 (a) Is this function discontinuous at any points? (b)
If an annuity makes an infinite series of equal payments at the end of the interest periods, it is called a perpetuity. If a lump sum investment of An is needed to result in n periodic payments of R
1. Experimental evidence suggests that the response y of the body to the concentration x of injected adrenalin is given byy = x/a + bxwhere a and b are experimental constants.(a) Is this function
The percent p of particulate pollution that can be removed from the smokestacks of an industrial plant by spending C dollars is given by P = 100C / 7300 + C Find the percent of the pollution that
The percent p of impurities that can be removed from the waste water of a manufacturing process at a cost of C dollars is given by P = 100C / 8100 + C Find the percent of the impurities that could be
The tax owed by a married couple filing jointly and their tax rates can be found in the following tax rate schedule.From this schedule, the tax rate R(x) is a function of taxable income x, as
The monthly charge in dollars for x kilowatt-hours (kWh) of electricity used by a residential consumer of Excelsior Electric Membership Corporation from November through June is given by the
First-class postage for a standard letter is 49 cents for the first ounce or part of an ounce that a letter weighs plus an additional 21 cents for each additional ounce or part of an ounce. Use the
Since 1950 the U.S. workforce has seen the arrival of the Baby Boomers and the changing role of women, among other influences. The data in the table show the millions of men and women in the U.S.
Using data from 2010 and projected to 2016, the fraction of U.S. cell phone users who are smart phone users can be modeled by the functionwhere t is equal to the number of years after 2010 (a) Is the
Determine whether the given function is continuous. If it is not, identify where it is discontinuous and which condition fails to hold. You can verify your conclusions by graphing each function with
For each given function find the average rate of change over each specified interval. 1. f (x) = x2 + x + 12 over (a) [0, 5] and (b) [-3, 10] 2. f (x) = 6 - x - x2 over (a) [- 1, 2] and (b) [1, 10]
Let f(x) = 9 - 1/2 x2. (a) Use the definition of derivative and the Procedure/Example box in this section to verify that f(x) = - x. (b) Find the instantaneous rate of change of f (x) at x = 2. (c)

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