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study help
mathematics
calculus
Questions and Answers of
Calculus
Determine whether the statement is true or false. Justify your answer. 1. A sequence is geometric when the ratios of consecutive differences of consecutive terms are the same. 2. To find the nth term
1. Consider the graph of y = 1 − rx / 1 - r (a) Use g graphing utility to graph y = for r = 1 / 2, 3 / 2, and 4 / 5. What happens as x → ∞? (b) Use the graphing utility to graph y for r = 1.5,
1. The first step in proving a formula by ________ ________ is to show that the formula is true when n = 1. 2. To find the ________ differences of a sequence, subtract consecutive terms. 3. A
Use mathematical induction to prove the formula for all integers n ≥ 1. 1. 2 + 4 + 6 + 8 + . . . + 2n = n (n + 1) 2. 6 + 12 + 18 + 24 + . . . + 6n = 3n (n + 1)
Use mathematical induction to prove the inequality for the specified integer values of n. 1. n! > 2n, n ≥ 4 2. (4/3)n > n, n ≥ 7
Use mathematical induction to prove the property for all integers n ≥ 1. 1. A factor of n3 + 3n2 + 2n is 3. 2. A factor of n4 − n + 4 is 2.
Find a formula for the sum of the first n terms of the sequence. Prove the validity of your formula. 1, 5, 9, 13, ...
Find the sum using the formulas for the sums of powers of integers.1.2. 3. 4. 5.
Find the statement Pk+1 for the given statement Pk. 1. Pk = 5 / k (k + 1) 2. Pk = 1 / 2 (k + 2) 3. Pk = k2 (k + 3)2
Decide whether the sequence can be represented perfectly by a linear or a quadratic model. Then find the model. 1. 5, 14, 23, 32, 41, 50, . . . 2. 3, 9, 15, 21, 27, 33, . . . 3. 4, 10, 20, 34, 52,
Write the first six terms of the sequence beginning with the term a1. Then calculate the first and second differences of the sequence. State whether the sequence has a perfect linear model, a perfect
Find the quadratic model for the sequence with the given terms. 1. a0 = 3, a1 = 3, a4 = 15 2. a0 = 7, a1 = 6, a3 = 10
The table shows the numbers an (in thousands) of residents of Alabama from 2010 through 2015.(a) Find the first differences of the data shown in the table. Then find a linear model that approximates
Find a formula for the sum of the angles (in degrees) of a regular polygon. Then use mathematical induction to prove this formula for a general n-sided polygon.
Determine whether the statement is true or false. Justify your answer. 1. If the statement Pk is true and Pk implies Pk+1, then P1 is also true. 2. A sequence with n terms has n − 1 second
1. When you find the terms that result from raising a binomial to a power, you are ________ the binomial. 2. The coefficients of a binomial expansion are called ________ ________. 3. To find binomial
Evaluate using Pascal's Triangle. 1. 6C3 2. 4C2 3. 5 1 4. 7 4
Use the Binomial Theorem to write the expansion of the expression. 1. (x + 1)6 2. (x + 1)4 3. (y − 3)3 4. (y − 2)5
Expand the expression by using Pascal's Triangle to determine the coefficients. 1. (a + 6)4 2. (a + 5)5 3. (y − 1)6
Find the specified nth term in the expansion of the binomial. 1. (x + y)10, n = 4 2. (x − y)6, n = 2 3. (x − 6y)5, n = 3 4. (x + 2z)7, n = 4
Find the coefficient a of the term in the expansion of the binomial. Binomial ...................................... Term 1. (x + 2)6 ...................................... ax3 2. (x − 2)6
Find the binomial coefficient. 1. 5C3 2. 7C6 3. 12C0 4. 20C20
Use the Binomial Theorem to write the expansion of the expression. 1. (√x + 5)3 2. (2√t - 1)3 3. (x2/3 - y1/3)3
Simplify the difference quotient, using the Binomial Theorem if necessary. [f(x + y) - f(x)] / h 1. f(x) = x3 2. f(x) = x4
Use the Binomial Theorem to expand the complex number. Simplify your result. 1. (1 + i)4 2. (2 − i)5 3. (2 − 3i)6
Use the Binomial Theorem to approximate the quantity accurate to three decimal places. For example, in Exercise 73, use the expansion (1.02)8 = (1+ 0.02)8 = 1 + 8(0.02) + 28(0.02)2 + . .
Consider n independent trials of an experiment in which each trial has two possible outcomes: "success" or "failure." The probability of a success on each trial is p, and the probability of a failure
Use a graphing utility to graph f and g in the same viewing window. What is the relationship between the two graphs? Use the Binomial Theorem to write the polynomial function g in standard form. 1. f
Describe the pattern formed by the sums of the numbers along the diagonal line segments shown in Pascal's Triangle (see figure).
Describe the error.(x - 3)3 = 3C0x3 + 3C1x2(3) + 3C2x(3)2 + 3C3(3)3= 1x3 + 3x2(3) + 3x(3)2 + 1(3)3= x2 + 9x2 + 27x + 27
The amounts f (t) (in billions of dollars) of child support collected in the United States from 2005 through 2014 can be approximated by the model f (t) = −0.056t2 + 1.62t + 16.4, 5 ≤ t ≤
The table shows the average prices f (t) (in cents per kilowatt-hour) of residential electricity in the United States from 2007 through 2014.(a) Use the regression feature of a graphing utility to
Determine whether the statement is true or false. Justify your answer. 1. The Binomial Theorem could be used to produce each row of Pascal's Triangle. 2. A binomial that represents a difference
1. Explain how to form the rows of Pascal's Triangle. 2. Forming Rows of Pascal's Triangle Form rows 8-10 of Pascal's Triangle.
Use a graphing utility to graph the functions in the same viewing window. Which two functions have identical graphs, and why? f (x) = (1 − x)3 g(x) = 1 − x3 h(x) = 1 + 3x + 3x2 + x3 k(x) = 1 −
The expansions of (x + y)4, (x + y)5, and (x + y)6 are shown below. (x + y)4 = 1x4 + 4x3y + 6x2y2 + 4xy3 + 1y4 (x + y)5 = 1x5 + 5x4y + 10x3y2 + 10x2y3 + 5xy4 + 1y5 (x + y)6 = 1x6 + 6x5y + 15x4y2 +
Prove the property for all integers r and n, where 0 ≤ r ≤ n. 1. nCr = nCn-r 2. nC0 - nC1 3. n+1Cr = nCr + nCr-1 4. The sum of the numbers in the nth row of Pascal's Triangle is 2n.
Complete the table. What characteristic of Pascal's Triangle does this table illustrate?
1. The ________ ________ ________ states that when there are m1 different ways for one event to occur and m2 different ways for a second event to occur, there are m1 · m2 ways for both events to
1. A customer can choose one of three amplifiers, one of two compact disc players, and one of five speaker models for an entertainment system. Determine the number of possible system
1. In how many ways can you answer a six-question true-false exam? (Assume that you do not omit any questions.) 2. In how many ways can you answer a 12-question true-false exam? (Assume that you do
1. A combination lock will open when you select the right choice of three numbers (from 1 to 40, inclusive). How many different lock combinations are possible? 2. A combination lock will open when
Evaluate nPr . 1. 5P2 2. 6P6 3. 12P2 4. 6P5
Use a graphing utility to evaluate nPr. 1. 15P3 2. 100P4 3. 50P4 4. 10P5
A patient with end-stage kidney disease has nine family members who are potential kidney donors. How many possible orders are there for a best match, a second-best match, and a third-best match?
1. From a pool of 12 candidates, the offices of president, vice-president, secretary, and treasurer need to be filled. In how many different ways can the offices be filled? 2. A baseball coach is
Find the number of distinguishable permutations of the group of letters. 1. A, A, G, E, E, E, M 2. B, B, B, T, T, T, T, T 3. A, L, G, E, B, R, A 4. M, I, S, S, I, S, S, I, P, P, I
1. Write all permutations of the letters A, B, C, and D. 2. Write all permutations of the letters A, B, C, and D when letters B and C must remain between A and D.
Evaluate nCr using the formula from this section. 1. 6C4 2. 5C4 3. 9C9 4. 12C0
Use a graphing utility to evaluate nCr . 1. 16C2 2. 17C5 3. 20C6 4. 50C8 5. Write all combinations of two letters that can be formed from the letters A, B, C, D, E, and F. (Order is not
1. In how many different ways can a jury of 12 people be randomly selected from a group of 40 people? 2. A U.S. Senate Committee has 14 members. Assuming party affiliation is not a factor in
A shipment of 25 television sets contains three defective units. In how many ways can a vending company purchase four of these units and receive (a) all good units, (b) two good units, and (c) at
The complexity of interpersonal relationships increases dramatically as the size of a group increases. Determine the numbers of different two-person relationships in groups of people of sizes (a) 3,
You are dealt five cards from a standard deck of 52 playing cards. In how many ways can you get (a) a full house and (b) a five-card combination containing two jacks and three aces? (A full house
1. An employer interviews 12 people for four openings at a company. Five of the 12 people are women. All 12 applicants are qualified. In how many ways can the employer fill the four positions when
Find the number of diagonals of the polygon. (A diagonal is a line segment connecting any two nonadjacent vertices of a polygon.) 1. Pentagon 2. Hexagon 3. Octagon 4. Decagon (10 sides)
Determine the number of ways a computer can randomly generate one or more such integers from 1 through 12. 1. An odd integer 2. An even integer 3. A prime integer 4. An integer that is greater than
1. Three points that are not collinear determine three lines. How many lines are determined by nine points, no three of which are collinear? 2. Powerball is a lottery game that is operated by the
Solve for n. 1. 4 · n+1P2 = n+2P3 2. 5 · n−1P1 = nP2
1. The number of letter pairs that can be formed in any order from any two of the first 13 letters in the alphabet (A-M) is an example of a permutation. 2. The number of permutations of n elements
Without calculating, determine whether the value of nPr is greater than the value of nCr for the values of n and r given in the table. Complete the table using yes (Y) or no (N). Is the value of nPr
Prove the identity. 1. nPn−1 = nPn 2. nCn = nC0 3. nCn-1 = nC1 4. nCr = nPr / r! 5. Can your graphing utility evaluate 100P80? If not, explain why.
1. An ________ is any happening for which the result is uncertain, and the possible results are called ________. 2. The set of all possible outcomes of an experiment is the ________ ________. 3. The
Find the probability for the experiment of tossing a coin three times. 1. The probability of getting exactly one tail 2. The probability of getting exactly two tails 3. The probability of getting a
Find the probability for the experiment of drawing a card at random from a standard deck of 52 playing cards. 1. The card is a face card. 2. The card is not a face card. 3. The card is a red face
Find the probability for the experiment of tossing a six-sided die twice. 1. The sum is 6. 2. The sum is at least 8. 3. The sum is less than 11.
Find the probability for the experiment of drawing two marbles at random (without replacement) from a bag containing one green, two yellow, and three red marbles. 1. Both marbles are red. 2. Both
In 2015, there were approximately 8.3 million unemployed workers in the United States. The circle graph shows the age profile of these unemployed workers.(a) Estimate the number of unemployed workers
An independent polling organization interviewed 100 college students to determine their political party affiliations and whether they favor a balanced-budget amendment to the Constitution. The table
In a high school graduating class of 128students, 52 are on the honor roll. Of these, 48 are going on to college. Of the 76 students not on the honor roll, 56 are going on to college. What is the
1. Three people are running for president of a class. The results of a poll show that the first candidate has an estimated 37% chance of winning and the second candidate has an estimated 44%chance
A payroll clerk addresses five paychecks and envelopes to five different people and randomly inserts the paychecks into the envelopes. Find the probability of each event. (a) Exactly one paycheck is
1. On a game show, you are given five digits to arrange in the proper order to form the price of a car. If you are correct, you win the car. What is the probability of winning, given the following
1. You draw one card at random from a standard deck of 52 playing cards. Find the probability that (a) The card is an even-numbered card, (b) The card is a heart or a diamond, (c) The card is a nine
A shipment of 12 microwave ovens contains three defective units. A vending company purchases four units at random. What is the probability that (a) All four units are good, (b) Exactly two units are
1. ATM personal identification number (PIN) codes typically consist of four-digit sequences of numbers. Find the probability that if you forget your PIN, you can guess the correct sequence (a) At
1. In a recent survey, people were asked whether they would prefer to work flexible hours-even when it meant slower career advancement-so they could spend more time with their families. The figure
You are given the probability that an event will not happen. Find the probability that the event will happen. 1. P (E') = 0.29 2. P (E') = 0.89 3. P (E') = 14 / 25 4. P (E') = 79 / 100
A space vehicle has an independent backup system for one of its communication networks. The probability that either system will function satisfactorily during a flight is 0.985. What is the
A fire department keeps two rescue vehicles. Due to the demand on the vehicles and the chance of mechanical failure, the probability that a specific vehicle is available when needed is 90%. The
American roulette is a game in which a wheel turns on a spindle and is divided into 38 pockets. Thirty-six of the pockets are numbered 1-36, of which half are red and half are black. Two of the
Assume that the probability of the birth of a child of a particular sex is 50%. In a family with four children, find the probability of each event. (a) All the children are boys. (b) All the children
1. You and a friend agree to meet at your favorite restaurant between 5:00 P.M. and 6:00 P.M. The one who arrives first will wait 15 minutes for the other, and then will leave (see figure). What is
1. If A and B are independent events with nonzero probabilities, then A can occur when B occurs. 2. Rolling a number less than 3 on a normal six-sided die has a probability of 1 / 3. The complement
Consider a group of n people.(a) Explain why the pattern below gives the probabilities that the n people have distinct birthdays.n = 2: 365 / 365 · 364 / 365 = 365 · 364 / 3652n = 3:
The circle graphs show the percents of undergraduate students by class level at two colleges. A student is chosen at random from the combined undergraduate population of the two colleges. The
1. Match the probability formula with the correct probability name. (a) Probability of the union of two events (i) P (A∪B) = P (A) + P (B) (b) Probability of the union of two events (ii) P (A') = 1
Write the first five terms of the sequence. (Assume that n begins with 1.) 1. an = 3 + 12/n 2. an = (-1)n5n/2n - 1
Determine whether the statement is true or false. Justify your answer.1. (n + 2)! / n! = n + 2 / n2.3. 4.
1. An infinite sequence beginning with a1 is a function. What is the domain of the function? 2. How do the two sequences differ? (a) an = (-1)n / n (b) an = (-1)n + 1 / n 3. Explain what is meant by
Find the sum.1.2.
Use sigma notation to write the sum. 1. 1/2(1) + 1/2(2) + 1/2(3) + . . . . + 1/2(20) 2. 1/2 + 2/3 + ¾ + . . . . + 9/10
Find the sum of the infinite series.1.2.
An investor deposits $10,000 in an account that earns 2.25% interest compounded monthly. The balance in the account after n months is given by An = 10,000 (1 + 0.0225/12)n n = 1, 2, 3, . . . . (a)
The population an (in thousands) of Miami, Florida, from 2010 through 2014 can be approximated by an = −0.34n2 + 14.8n + 288, n = 10, 11, . . . , 14 where n is the year with n = 10 corresponding to
Determine whether the sequence is arithmetic. If so, find the common difference. 1. 5, −1, −7, −13, −19, . . . 2. 0, 1, 3, 6, 10, . . . 3. 1, 8, 1/4, ½, 1, 2, . . . . 4. 1, 15/16, 7/8,
Find a formula for an for the arithmetic sequence. 1. a1 = 7, d = 12 2. a1 = 34, d = −4 3. a3 = 96, a7 = 24
Write the first five terms of the arithmetic sequence. 1. a1 = 4, d = 17 2. a1 = 25, an+1 = an + 3
1. Find the sum of the first 100 positive multiples of 9.
Find the sum1.2. 3.
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