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are two available models, A and B. The dealer normally sells at least twice as many model A cars as model B cars. If the dealer makes a profit of $1300 on model A cars and $1700 on model B cars, how many of each car should the dealer have in the lot? 23. A 30-acre orchard is to contain two types of trees, peach and al... |
.......... 790 Section 12.2 Section 12.3 Section 12.4 Section 12.5 Matrix................................ 795 Augmented matrix....................... 795 Equivalent systems....................... 795 Elementary row operations................ 796 Reduced row-echelon form................ 797 Gauss-Jordan elimination........ |
......................... 818 Algebraic solutions of systems of nonlinear equations...................... 821 Graphical solution of systems of nonlinear equations...................... 822 Section 12.5.A Solutions of systems of linear inequalities.... 826 Linear programming..................... 829 834 Review Exercises... |
y z 2 6x 4y 14z 24 2x y 4z 7 z 0 z 1 x 2, x 1, x 1, y 3, y 1, y 3, is a solution. is a solution. a. b. c. d. The system has an infinite number of solutions. e. is not a solution. is a solution. z 1 x 2, y 5, z 3 15. Tickets to a lecture cost $1 for students, $1.50 for faculty, and $2 for others. Total attendance at the... |
z 6w 2 3x 4y 2z w 0 5x 4y 2z 5w 4 4x 4y 2z 3w 1 39. Find the equation of the parabola passing through the points, (2, 17), (8, 305). 3, 52 1 2 Chapter Review 837 38. 2x y 2z u 2 x 3y 4z 2u 2v 2 2x 3y 5z 4u v 1 x 3y 2z 4u 4v 4 2x 3y 6z 4u 5v 0 40. The table shows the number of hours spent per person per year on home vid... |
248 ft of space and can hold a maximum of 440 lb. If the store makes a profit of $20 on the standard model and $30 on the deluxe model, how many boxes of each model can the van carry to maximize the profit for each load? 3 C H A P T E R 12 Partial Fractions In calculus it is sometimes necessary to write a complicated ... |
in Example 1. x 2 x 1 2, 2 x 2 is a repeated factor, both as shown in Fig2 x 2 x 2 and 2 2 1 1 1 1 2 1 2 Multiply both sides of the equation by the common denominator, x 1 2, and collect like terms on the right side. C x 2 1 2 2 15x 10 A x 1 x 2 2 B x 2 2x 4x 4 A x x 1 1 2 Ax 2 Bx 2B Cx C 2 4Ax 4A Bx 4A 2B C 1 The pol... |
Bx 4C 2A 4C 2 2x 2 2 2 2x 2 x 2 A x 1 Ax 2 2Ax 2A Bx 2A 4B C Bx C A B 1 x Because there is no sponding coefficient must be 0. Therefore, 2 term in the original rational expression, the corre 4B C 1 2 A 4C 2 Coefficient of x 2 Coefficient of x Constant term Figure 12.C-5 Figure 12.C-5. The solution of the system is A 3 ... |
. For each linear factor of the form (px q)m, the partial fraction must include the following sum: A1 px q A2 (px q)2 p Am (px q)m (ax2 bx c)n, the 4. For each quadratic factor of the form partial fraction must include the following sum: B2x C2 (ax2 bx c)2 B1x C1 ax2 bx c Bnx Cn (ax2 bx c)n p Exercises In Exercises 1–7... |
from the data. Most statistical data is gathered by taking a random sample of the population. In a random sample, all members of the population and all groups of members of a given size have an equal chance of being in the sample. Data can be divided into two types: qualitative and quantitative. Quantitative data is n... |
value occurs is known as the frequency of that value. If the frequency is divided by the total number of responses, the result is the relative frequency of that value, which can be expressed as a fraction, a decimal, or a percent. A frequency table displays the categories with frequencies, relative frequencies, or bot... |
the data values. The most common distribution shapes are shown below. Uniform: All the data values have approximately the same frequency. Symmetric: The right and left sides of the distribution have frequencies that are mirror images of each other. Skewed right: The right side of the distribution has much lower freque... |
5 6 7 8 9 10 Figure 13.1-1 6 0 3 The entry sents the scores 63, 64, and 67. There are 31 scores represented. represents the score 32. Similarly, the row 3 0 2 4 7 repre- 848 Chapter 13 Statistics and Probability Note that the score of 32 is far below the remaining data and so it could be an outlier in this distributio... |
distribution is cut off at 3, because no phone calls last longer than 3 minutes, the length of the entire message. ■ A histogram, which can be thought of as a bar graph with no gap between adjacent bars, is often used with large sets of quantitative data. First, the Figure 13.1-2 Section 13.1 Basic Statistics 849 data... |
500, 740, 510, 540, 560, 510, 430, 440, 590, 560, 510, 600, 460, 450, 510, 420, 430, 560, 680, 610, 600, 600, 520, 480, 490, 320, 450, 500, 490 850 Chapter 13 Statistics and Probability Solution The smallest value is 160 and the largest value is 740, so the range of the data is 580 points. A convenient choice for the ... |
examines every cracked eggs. carton to see how many cartons contain 50th 3. A survey is taken to determine the number of pets in the typical American family. A computer is used to randomly select 5 states, then 10 counties in each state, then 50 families in each county. Each of these families is asked how many pets th... |
. 25. 23, 45, 38, 41, 24, 67, 42, 46, 51, 33, 43, 47, 54, 49, 47, 36, 27, 33, 41, 29 26. 1.8, 2.0, 1.4, 5.6, 1.1, 2.6, 0.8, 1.5, 1.4, 2.6, 0.7, 1.6, 0.4, 1.1, 0.5, 1.3 27. 98, 87, 100, 86, 92, 78, 56, 100, 90, 88, 93, 99, 76, 83, 86, 91, 72, 85, 79, 81, 82, 91, 86, 70, 84 During summer semester, a community college sur... |
, 14, 20, 13, 23, 19, 13, 20, 14, 24, 10, 18, 30, 22, 16, 26, 10, 23, 22, 19, 23, 21, 16, 18, 18, 20, 25, 14, 19, 7, 16, 18, 31, 14, 7, 10, 16, 13, 18, 10 34. Create a histogram with a class interval of 5 for the data in the stem plot below 35. Critical Thinking How does the shape of the histogram you created in Exerci... |
x4 3 8 5 6 6 10 6 6, x5 6, x6 38 6 10 6.3 The data shows an average of 6.3 accidents per month at the given intersection. ■ One problem with the mean as a measure of center is that it may be distorted by extreme values, as shown in the following example. 854 Chapter 13 Statistics and Probability Example 2 Mean Home Pr... |
2 Notice that three values are less than the median and three values are greater than the median. Median Section 13.2 Measures of Center and Spread 855 p xn x3 x2, x1, If are ordered from smallest to largest, then the median is the middle entry when n is odd and the average of the two middle entries when n is even. fo... |
median mode y y mode mode median median x x x mean symmetric mean skewed left Figure 13.2-1 mean skewed right The mean is the balance point of a distribution. Notice that on a skewed distribution, the mean moves toward the tail to balance out the “weight” of the outlying data. The median divides the area under the dis... |
their deviations on a number line in Figure 13.2-2. 858 Chapter 13 Statistics and Probability NOTE The distance from xi x data value to the mean is the absolute value of the deviation. deviation = 0.6 deviation = −1.4 deviation = 1.6 deviation = −4.4 deviation = 3.6 0 1 2 3 4 5 6 6.4 7 8 9 10 Figure 13.2-2 The average... |
. Then use a calculator to find the population standard deviations of the data in the other two plots. Solution For the first stem plot, squared deviations. x 105 and n 9. The following table shows the xi xi 85 91 x 20 14 xi 1 x 2 2 400 196 99 6 36 101 4 16 105 109 111 119 125 0 0 4 16 6 36 14 20 196 400 s 400 196 36 1... |
, 23, 26, 28, 30, 33 Q1 Q3 Figure 13.2-4 The interquartile range is the difference between the quartiles, IQR Q3 Q1 which represents the spread of the middle 50% of the data. A value that is less than considered an outlier, as shown in Figure 13.2-5. IQR Q1 1 2 or greater than 1.5 Q3 1.5 IQR is 2 1 1.5 IQR IQR median 1... |
plot for the data in each of the other two stem plots on page 857. Exercises 13.2 In Exercises 1–4, find the mean of each data set. 6. Find the median of the data set in Exercise 2. 1. 23, 25, 38, 42, 54, 57, 65 7. Find the median of the data set in Exercise 3. 2. 3, 5, 6, 2, 10, 9, 7, 5, 11, 6, 4, 2, 5, 4 8. Find the... |
ains Dairy Meat Fat For each distribution shape, indicate whether the mean is larger, the median is larger, or the mean and median are equal. 14. symmetric 15. skewed left 16. skewed right 17. uniform Find the population standard deviation of the following data sets without using a calculator. 18. 8, 9, 10, 11, 12 19. ... |
3 players had 2 hits each, and 6 players had no hits. Find the mean number of hits per player. Find the sample standard deviation and the population standard deviation of the data, and interpret your results. 45. A teacher has two sections of the same course. The average on an exam was 94 for one class with 20 student... |
one or more observable outcomes. The set of all possible outcomes is called the sample space of the experiment. Some examples of experiments and their sample spaces are shown in the following table. Section 13.3 Basic Probability 865 3 Figure 13.3-1 Experiment tossing a coin Sample space heads and tails, written as {H... |
red marble, a 30% chance of drawing a blue marble, a 10% chance of drawing a yellow marble, and a 10% chance of drawing a green marble. b. Write out a reasonable probability distribution for this experiment, and verify that the sum of the probabilities of the outcomes is 1. c. What is the probability that a blue or gr... |
F 5 6 6 C, S 5 in the first five letters 1 of the alphabet C 2 b. What is the complement of the event {A, S}? c. What is the probability of the event “the spinner does not land on A?” Solution A E C S Figure 13.3-2 a. The events E {A, C, E} and F {C, S} are not mutually exclusive because they have a common outcome, C.... |
once? Solution a. The results of the two different trials are independent, so the probability of winning both times can be found by multiplying the probability of winning each time. winning both games P 1 2 0.1 0.1 0.01 b. Since losing is the complement of winning, the probability of losing 1 0.1 0.9. is by multiplyin... |
, 6) (5, 6) (6, 6) b. The smallest possible value is 2, which is assigned to the outcome (1, 1). The largest possible value is 12, which is assigned to the outcome (6, 6). The range is the set of integers from 2 to 12. c. The value 7 is assigned to the outcomes (1, 6), (2, 5), (3, 4), (4, 3), (5, 2), and (6, 1). ■ Expe... |
given below. Find the expected value and interpret the result. Solution Win $0 $3 $5 Probability 0.882746 0.06 0.04 $10 0.01 $20 $40 $100 $400 $2500 0.005 0.002 0.0002 0.00005 0.000004 0.882746 0 1 100 0.0002 2 3 0.06 1 400 1 2 0.04 5 2 0.00005 1 1 2 10 1 2 2500 1 20 0.01 2 0.000004 0.005 2 1 0.71 2 40 0.002 2 1 The a... |
-6 can be used to estimate the probability that a customer calling a company’s customer service line will have to wait for a given amount of time. The area of each square on the grid is and the total area under the curve is 1. Estimate the probability that a customer will have to wait between 2 and 3 minutes. 0.5 0.05 ... |
change condition worsens 55% 24% 17% 4% Suppose the procedure is performed on 5 patients. Assume that the procedure is independent for each patient. 9. What is the probability that all five patients will recover completely? 10. What is the probability that none of the patients will get worse? A bag contains red and bl... |
sprout. 26. Use the probability distribution to determine the probability that each seed will sprout, assuming that they are independent. Hint: if the probability that one seed will sprout is p, what is the probability that all four seeds will sprout? A random variable with a uniform distribution has a probability den... |
unlikely that it would occur fewer than 20 times or more than 40 times. Figure 13.4-1 Section 13.4 Determining Probabilities 875 The basis for experimental estimates of probability may be summarized as follows: As the number of trials of an experiment increases, the relative frequency of an outcome approaches the prob... |
(K, 1, N, 1) sum 1 If T 0 A 1 S A If T 1 B 1 S B If T 2 C 1 S C If T 3 D 1 S D End N is the number of trials. T is the outcome of a single trial. A is the number of trials with 0 heads. B is the number of trials with 1 head. C is the number of trials with 2 heads. D is the number of trials with 3 heads. {A/N, B/N, C/N... |
Write the probability distribution for the experiment. b. Find the probability of the event that an even number is rolled. Solution a. The sample space consists of the 6 outcomes {1, 2, 3, 4, 5, 6}. If the outcomes are equally likely, then the probability of each is probability distribution for the experiment is 1 6. ... |
determined, but some experiments require more sophisticated counting techniques. The basis of most counting techniques is the Fundamental Counting Principle, which is also known as the Multiplication Principle. n1 Consider a set of k experiments. Suppose the first experiment outcomes, and so on. Then outcomes, the sec... |
outcomes in each case. With replacement Order important Without replacement Order important Without replacement Any order 26 26 26 17,576 26 25 24 15,600 26 25 24 3 2 1 2600 Example 6 3 Coin Toss Use the Fundamental Counting Principle to verify the probability distribution in Example 4. Solution There are two possible... |
utations and combinations are all equally likely for a given value of r. Note: may also be written as nPr Pn,r or P(n, r), and may be nCr written as Cn,r, C(n, r), or n r b a. Example 7 Matching Problem Suppose you have four personalized letters and four addressed envelopes. If the letters are randomly placed in the en... |
2 2 2 Exercises 13.4 For Exercises 1–4, an experiment consists of drawing a marble out of a bag, observing the color, and then placing it back in the bag. Suppose the experiment is repeated 75 times, with the following results: 4. Suppose it is known that there is a total of 300 marbles in the bag. Estimate the number... |
variable. 12. A bag contains 3 red marbles and 4 blue marbles. Suppose each marble is equally likely to be chosen. What is the probability of the event of drawing a red marble? 13. Suppose that a person’s birthday is equally likely to be any day of the year. What is the probability that a randomly chosen person has th... |
), and asks each participant to guess the numbers in order. What is the probability of guessing all 6 correctly? A botanist is testing two kinds of seeds. She divides a plot of land into 16 equal areas numbered from 1 to 16. She then randomly chooses 8 of these areas to plant seed A, and she plants seed B in the remain... |
winning or losing, heads or tails, boy or girl. These experiments determine a group of problems called binomial or Bernoulli experiments, named after Jacob Bernoulli, a Swiss mathematician who studied these distributions extensively in the late 1600’s. Binomial Experiments Here is a typical binomial experiment: in a b... |
note that the probability of SSF is the same as the probability of SFS or FSS. In general, the probability of any outcome with r successes and n–r failures in n trials is qq p q 2 n r times 21 1 ⎧⎨⎩ ⎧⎨⎩ r times prqnr pp p p 1 2 1 2 To develop a general formula for the probability of r successes in n trials, it is nece... |
999 2 1 2 0.999 0.999 0.999 0.001 2 1 0.001 2 0.001 0.001 0.001 1 0.3677 0.3681 0.1840 0.0613 0.0153 0.0036 Outcome 0 wins 1 win 2 wins 3 wins 4 wins 5 or more wins Probability 0.3677 0.3681 0.1840 0.0613 0.0153 0.0036 Section 13.4.A Excursion: Binomial Experiments 887 b. In order to break even or better, you must win ... |
and the probability of heads on each toss is A probability distribution for. 1 2 the number of heads is shown below. Number of heads Probability 0 1 16 16 A probability density function that represents this distribution is shown in Figure 13.4.A-2. Notice that the shape of the graph is symmetric. The expected value of... |
ises 13.4.A For Exercises 1–4, a binomial experiment consists of planting 4 seeds. The probability of success (that a given seed will sprout) is The sample outcome SFSS means that the first seed sprouted, the second seed did not sprout, and the third and fourth seeds sprouted. p 0.65. 1. What is the probability of fail... |
it is also a valuable tool for predicting if an outcome is statistically significant or just caused by chance. 890 Chapter 13 Statistics and Probability Properties of the Normal Curve A normal distribution is bell-shaped and symmetric about its mean. The x-axis is a horizontal asymptote, and the area under the curve a... |
e Most of the time, the mean and standard deviation of an entire population cannot be measured. The population mean and standard deviation are often estimated by using a sample. Example 1 Using Sample Information A paper in Animal Behavior gives 11 sample distances, in cm, from which a bat can first detect a nearby in... |
function corresponds to the probability, the probability that a pair of shoes will last between 400 and 500 miles is about 0.68. b. The area under the curve between 350 and 550 miles is 95% of the total area, which leaves 5% for the area less than 350 and greater than 550. Since the normal curve is symmetric, the area... |
z-values correspond to values on the standard normal curve, as shown in Figure 13.5-7. z-values: 200 −3 300 −2 400 −1 500 0 600 1 700 2 Figure 13.5-7 z-Values The z-value of the value x in a data set with mean standard deviation is S x 800 3 M and z x M S The area under a normal curve between x a and x b is equal to t... |
cdf (lower bound, upper ).sm, bound, 0.16 y 0.23 x −0.4 0.6 Figure 13.5-9 ■ Example 5 Response Times The EMT response time for an emergency is the difference between the time the call is received and the time the ambulance arrives on the scene. Suppose the response times for a given city have a normal distribution 896 ... |
67, 65, 72, 71, 67, 73, 68, 72, 61, 75, 66, 78, 65, 71, 68, 76, 67, 68 Section 13.5 Normal Distributions 897 9. Compute the mean and standard deviation of this 23. the probability that the wait time for a table is sample. between 25 and 38 minutes 10. Draw a normal curve that represents the 24. the probability that th... |
and 30 minutes Daytime high temperatures in New York in February are normally distributed with an average of and a standard deviation of 30.2 8.5. 25. Estimate the probability that the temperature on a given day in February is 39° or higher. 26. Estimate the probability that the temperature on a given day in February ... |
............... 845 Uniform, symmetric, and skewed distributions............................ 846 Stem plot............................... 847 Histogram.............................. 849 Mean.................................. 853 Median................................ 854 Mode.................................. 855 Stand... |
............... 866 Complement............................ 866 Independent events....................... 867 Random variable......................... 869 Expected value.......................... 869 Probability density function................ 871 Experimental estimate of probability......... 875 Probability simulation... |
to largest, then the median If is the middle entry when n is odd and the average of the two middle entries when n is even. for n odd, the value in the n 1 2 position median • for n even, the average of the values in the n 2 and n 2 1 positions Population standard deviation s B © xi 1 2 x n 2 Sample standard deviation ... |
qualitative or quantitative? 2. Create a frequency table for the data. 3. Create a bar graph for the data. 4. Create a pie chart for the data. Exercises 5–17 refer to the following description: a study is done to determine the average commuting time of employees at a company. A total of 34 employees are surveyed, with... |
1 2 Outcome WWW WWR WRW RWW WRR RWR RRW RRR Probability 0.512 0.128 0.128 0.128 0.032 0.032 0.032 0.008 A random variable is assigned to the number of times the spinner lands on red. 23. What is the range of the random variable? 24. What is the probability that the value of the random variable is 2? 902 Chapter Review... |
matching all 5 numbers? What is the probability of matching any 4 numbers? 35. Suppose there are 4 people on a subcommittee, and you do not know their last names. If you have a list of 10 last names of all of the people in the committee, what is the probability of correctly guessing the last names of the people in the... |
, and h for the probability that the point is in the shaded region. Solution If the point is chosen randomly, then all points in the sample space are equally likely to be chosen. The sample space is all points in the rectangle, and the event above can be described as the set of all points in the shaded area, both of wh... |
height of the rectangle. N is the number of points in the rectangle. E is the number of points in the shaded region. This command generates an x-coordinate, X. This command generates a y-coordinate, Y. This command displays the point (X, Y). The point is tested to determine whether it is under the curve. This command ... |
a 2, b 10 a 3, b 5 a 7, b 9 the graph has a vertical asymptote. However, calculus can be used to show that the area under the curve from 0 to 1 is finite. Use a probability a 0 simulation to estimate the area by letting b 1. h 3, 4, for each value of h to be sure you have a good estimate. and Compare the value of the ... |
Many mathematical problems involve the behavior of a function at a particular value: What is the value of the function f when x c? x 1 2 rather than at The underlying idea of limit, however, is the behavior of the function near x c You have dealt with limits informally in previous chapters, but this section will discu... |
as you want to 2. x f 1 2 For instance, 3.99999 f 1 2 1.999984 and f 4.00001 1 2 2.000016 Notation The statement above is usually expressed by saying “ The limit of f x 2 1 as x approaches 4 is 2, ” which is written symbolically as Section 14.1 Limits of Functions 911 lim xS4 f x 1 2 2 or lim xS4 0.1x4 0.8x3 1.6x2 2x ... |
a fact that will be proved in calculus. ■ 2 Figure 14.1-2 Example 2 Limit of a Function Find lim xS2 f, where f is the function given by the following two-part rule. x 1 2 f x 1 2 0 if x is an integer 1 if x is not an integer e Solution A calculator is not much help here, but the function is easily graphed by hand. y ... |
5. 2 is the same as the limit as x ■ NOTE Whenever a calculator was used in preceding examples, it was said that the information provided by the calculator suggested that the limit of the function was a particular number. Although such calculator explorations provide strong evidence, they do not constitute a proof and ... |
If x approaches 0 from the right, that is, through positive values, then the Section 14.1 Limits of Functions 915 f x corresponding value of that is, from negative values, then the corresponding value of always ding values of the definition of limit. Therefore, the limit does not exist. is always 1. If x approaches 0 ... |
2 9. lim xS0 cos x 1 x 0.1 0.01 0.001 0.001 0.01 0.1 x f x 1 2 0.1 0.01 0.001 0.001 0.01 0.1 10. lim xSp 4 tan. lim xS0 2x 5 25 x 0.1 0.01 0.001 0.001 0.01 0.1 x f x 2 1 5. lim xS7 22 x 3 x 7 x 7.1 7.01 7.001 6.999 6.99 6.9 f x 1 2 6. lim xS0 21 x 1 x x f x 2 1 7. lim xS1 0.1 0.01 0.001 0.001 0.01 0..78 0.785 0.7853 0... |
2 −4 −3 −2 −4 −3 − Section 14.1 Limits of Functions 917 y y y 4 3 2 1 0 −1 −1 −2 −3 −4 4 3 2 1 0 −1 −1 −2 −3 −4 4 3 2 1 0 −1 −1 −2 −3 − 31. 32. −4 −3 −2 −4 −3 −2 33. a. Graph the function f whose rule is if x 6 2 if 2 x 6 2 if x 2 if x 7 2 Use the graph in part a to evaluate the following limits. x lim xS2 d. lim xS2 l... |
To find Thus, lim xS3 5 lim xS3 lim xS3 2 thing is true for any constant function. 1 1 2 lim xS3 1 2 Limit of a Constant If d is a constant, then lim xSc d d. The same phenomenon occurs with the identity function, which is given If c is any real number, then the statement “x by the rule approaches c” is exactly the sa... |
xSc lim xSc and 1 x g g 21 Limits of Polynomial Functions Properties 1–3, together with the facts about limits of constants and the identity function presented at the beginning of the section, now make it easy to find the limit of any polynomial function. Example 1 Limit of a Polynomial Function If f x 1 2 x2 2x 3, fi... |
the functions at Therefore, x 2. Section 14.2 Properties of Limits 921 lim xS2 f x 1 2 2 lim xS2 lim 1 xS2 lim xS2 x3 3x2 10 x2 6x 1 b Q x3 3x2 10 x2 6x 1 1 23 3 22 10 22 6 2 1 6 7 6 7 0.857 2 limit property 4 limits of polynomial functions Note that the limit of f x 1 2 as x approaches 2 is the number f 2. 2 1 ■ The ... |
x approaches c. x c, If f and g are functions that have limits as x approaches c and f(x) g(x) for all x c, then lim xSc f(x) lim xSc g(x). Recall that the difference quotient of a function f is given by The difference quotient can be evaluated for a specific value of x, say x c, to obtain a new form Limits of the dif... |
2 x2 x2 x 1 25. 27. f f x x 1 1 2 2 x3 2x In Exercises 9–23, find the limit, if it exists. If the limit does not exist, explain why. 9. lim xS2 1 6x3 2x2 5x 3 2 10. lim xS11 x7 2x5 x4 3x 4 2 11. lim xS2 3x 1 2x 3 12. lim xS3 x2 x 1 x2 2x 13. lim xS1 2x3 6x2 2x 5 14. lim xS2 2x2 x 3 15. ° lim xS0 ¢ 16. ° lim xS0 17. li... |
, but lim xSc 1 f g x 2 2 1 lim 2 xSc does exist. 1 14.2.A Excursion: One-Sided Limits Objectives • Find one-sided limits The function whose graph is shown in Figure 14.2.A-1 is defined for all values of x except x 4. y 4 2 −2 0 2 −2 −4 Figure 14.2.A-1 x 6 As x approaches 4 from the right, that is, takes values larger ... |
Limit Find xS3 lim 1x 3 1 A B. Solution f 2x 3 1 The function x 3 and values of x to the right of 3. The graph of ure 14.2.A-2a, and a table of values is shown in 14.2.A-2b. is defined only when x x f 2 1 2 1 x 3, that is, for is shown in Fig- 2.2 2.9 0 4 Figure 14.2.A-2a Figure 14.2.A-2b The values of approach 1 as x... |
hand and right-hand limit at If the can be made arbitrarily close to L by taking x close enough values of to c on both sides of c, then the same thing is true if you take only values of x on the left of c or on the right of c. Conversely, if a function has the same left-hand and right-hand limits at then this number mu... |
−1 −2 −3 −4 −4 −3 −2 −4 −3 −2 −4 −3 −. a. b. xS2 f lim 1 x xS2 f lim 11. In Exercise 33a of Section 14.1, find xS2 f lim 1 x xS2 f lim 12. In Exercise 34a of Section 14.1, find xS1 g lim 1 x xS1 g lim xS1 g lim 1 x xS1 g lim d. d. b. c. a In Exercises 13–22, find the limit. 13. xS1 lim A 2x 1 3 B 15. xS4 lim x 4 x2 16... |
and theorems. This section takes the first step in building this rigorous foundation by developing a formal definition of limit. In order to keep the discussion as concrete as possible, suppose f is a funcYou do not need to know the rule of f or tion such that 12. f x lim xS5 1 2 anything else about it to understand t... |
, between 11.99 and 12.01, you can find how close x must be to 5 to guarantee that x f 1 2 2 1 Any value of x in the interval around 5 shown in Figure 14.3-1 will produce 11.99 6 f 6 12.01. x 11.99 6 f x 6 12.01. 1 2 But “arbitrarily close” implies much more. You must be able to find how close x is to 5 regardless of h... |
e. Similarly, saying that x is within of 5 and not equal to 5 means that the distance from x to 5 is less than but greater than 0, that is 0 6 d d x 5 6 d. 0 0 e, NOTE d, Epsilon, and are Greek letters delta, that are often used to represent small amounts. Section 14.3 The Formal Definition of Limit 931 Using this not... |
0 e 4 d e 4 y 12 + ε 12 12 − ε x 5 5 + δ 5 − δ worry about how ment is valid. Figure 14.3-3 932 Chapter 14 Limits and Continuity If 0 x 5 0 6 d, then 4x 20 6 e 0 12 6 e 0 12 6 e 0 e 7 0, 6 d, then 12 lim xS5 1 2 0 1 4x 8 2 x f Multiply both sides by 4. 4 a 0 0 b ab 4 0 0 Rewrite 20 as 8 12. 4x 8 This verifies that for... |
2 Figure 14.3-4 Therefore, d e 2 has the required property, and the proof is complete. ■ Proving Limit Properties Once the algebraic scratch work was done in Examples 1 and 2, the limit proofs were relatively easy. In most cases, however, a more involved argument is required. In fact, it can be quite difficult to prov... |
6 x c 0 0 d Then Now let be the smaller of the two numbers d d2. x c x c and it must be true that x c and 0 6 if 0 6 0 6 6 d, 6 d1 d1 0 0 0 0 6 d2. 0 d2, so that d d1 and Therefore, Consequently, if L 6 e 2 6 d˛ 1 2 0 6 g˛1 x 0 22 2 0 x f ˛1 2 0 1 0 1 and x g ˛˛1 1 x g˛1 2 M 2 M 0 2 0 0 then x f ˛1 0 1 x f ˛1 0 2 6 e ... |
S0 lim xS2 lim xS7 lim xS1 lim xS2 lim xS4 10. lim xS1 3x 2 lim xS3 1 4x 6x 3 1 1 2 15 2 5 2 2x 19 1 4 4 p p x 6 2 2x 7 1 1 2 5 2 11. 12. lim xS2 lim xS3 1 1 2x 5 2x 4 2 2 1 10 In Exercises 13 and 14, use the formal definition of limit to prove the statement. 13. 14. lim xS0 lim xS0 x2 0 x3 0 In Exercises 15 and 16, le... |
-2 is continuous at Try to draw one of these graphs at c and near without lifting your pencil from the paper. x c. c, f ˛1 1 22 (c, f(c)) y c a. y c. Section 14.4 Continuity 937 (c, f(c)) c y b. y x x x x c c Figure 14.4-2 d. Figure 14.4-2 shows that a function is discontinuous, that is, not continx c. uous, at if the ... |
graph. See graphs c and d of Figure 14.4-2 for functions that are not The situation can be described more precisely by saying: defined at when t is any number near c 1 1 x c. t, f 22 t t, 2 f(x) is defined for all x in some open interval containing c. In other words, there are numbers a and b with such that f(x) is de... |
f˛ 1 3 f ˛1 2 2 show that 2x2 x 1 x 5. lim xS3 213 8 940 Chapter 14 Limits and Continuity By the properties of limits given in Section 14.2, lim xS3 f x 2 1 lim xS3 2x2 x 1 x 5 2x2 x 1 x 5 2 1 x2 lim xS3 lim xS3 2 lim xS3 lim xS3 3 2 1 1 213 8 3 f˛ 1 2 x 2 limit of a quotient limit of a root 1 2 limit of a polynomial ... |
xSa f(x) f(a). lim A function f is continuous from the left at x b provided that xSb f˛(x) f(b). lim Example 2 Continuity at an Endpoint Show that 2x x f ˛1 2 is continuous from the right at x 0. Solution x The function 2 from the right at f ˛1 2x, x 0 which is not defined when and because 20 0, f 0 2 1 xS0 1x 0 f lim... |
x c: x c, then each of 1. the sum function f g 2. the difference function f g 3. the product function fg f g, 4. the quotient function g (c) 0 Proof By the definition of the sum function, Because f and g are continuous at Therefore, by the first property of limits, lim xSc x c 2 1 2 f lim xSc 1 f g x 2 21 x c, and lim... |
2x 2lim xS5 c 2 f and x 25 g c 5, 5. 2 1 1 2 the composite function x3 3x2 x 7 2 2x3 3x2 x 7 g f, ■ The Intermediate Value Theorem This section’s introduction to continuity will close by mentioning, without proof, a very important property of continuous functions. The Intermediate Value Theorem If the function f is co... |
by the Intermediate Value Theorem, with 2 2 x c 0. 0. f So when a calculator shows that the graph of a continuous function f has points above and below the x-axis, there really is an x-intercept between x these points, that is, a solution of Zoom-in uses this fact by look2 ing at smaller and smaller viewing windows th... |
24, determine whether or not the function is continuous at the given number. 19. f x 1 2 2x 4 if x 2 2x 4 if x 7 2 e, x 2 20. x g˛1 2 2x 5 if x 6 1 2x 1 if x 1 e, x 1 21. f x 1 2 x2 x if x 0 2x2 if x 7 0 e, x 0 22. g x 2 1 x3 x 1 if x 6 2 3x2 2x 1 if x 2 e, x 2 Section 14.4 Continuity 947 26. x g˛1 2 µ if x 2 x2 x 6 x2... |
infinity • Use properties of limits at infinity • Use the Limit Theorem NOTE Excursion 14.2.A is a prerequisite for some of the material that follows. q, There is no real number called “infinity,” and the symbol usually read “infinity,” does not represent any real number. which is q Nevertheless, the word “infinity” a... |
x approaches 5 from the right is negative infinity.” There are many cases like the ones illustrated above in which the language of limits and the word “infinity” can be useful for describing the behavior of a function that actually does not have a limit in the sense of Section 14.1. Example 1 Infinite Limits Describe ... |
.5-4. Graphing Exploration Produce the graph shown in Figure 14.5-4 and use the trace feature as x gets larger and larger. Are the values to find values of approaching a single value? x f 1 2 As you move to the right, the graph gets very close to the horizontal In other words, as x gets larger and larger, the correspon... |
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