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Precalculus: Concepts Through Functions, A Unit Circle Approach to Trigonometry (4th Edition)
4th Edition
ISBN: 9780134686974
Author: Michael Sullivan, Michael Sullivan III
Publisher: PEARSON
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Textbook Question
Chapter 11.2, Problem 60AYU
How many terms must be added in an arithmetic sequence whose first term is 78 and whose common difference is to obtain a sum of 702?
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Chapter 11 Solutions
Precalculus: Concepts Through Functions, A Unit Circle Approach to Trigonometry (4th Edition)
Ch. 11.1 - For the function f( x )= x1 x , find f( 2 ) and f(...Ch. 11.1 - True or False A function is a relation between two...Ch. 11.1 - Prob. 3AYUCh. 11.1 - True or False The notation a5 represents the fifth...Ch. 11.1 - True or False If n2 is am integer, then...Ch. 11.1 - The sequence , is an example of...Ch. 11.1 - The notation a 1 + a 2 + a 3 ++ a n = k=1 n a k...Ch. 11.1 - ______.
(a) (b)
(c) (d)
...Ch. 11.1 - In Problems 11-16, evaluate each factorial...Ch. 11.1 - In Problems 11-16, evaluate each factorial...
Ch. 11.1 - In Problems 11-16, evaluate each factorial...Ch. 11.1 - In Problems 11-16, evaluate each factorial...Ch. 11.1 - In Problems 9 – 14, evaluate each factorial...Ch. 11.1 - In Problems 11-16, evaluate each factorial...Ch. 11.1 - In Problems 17-28, write down the first five terms...Ch. 11.1 - In Problems 17-28, write down the first five terms...Ch. 11.1 - In Problems 17-28, write down the first five terms...Ch. 11.1 - In Problems 17-28, write down the first five terms...Ch. 11.1 - In Problems 17-28, write down the first five terms...Ch. 11.1 - In Problems 17-28, write down the first five terms...Ch. 11.1 - In Problems 15 – 26, write down the first five...Ch. 11.1 - In Problems 17-28, write down the first five terms...Ch. 11.1 - In Problems 17-28, write down the first five terms...Ch. 11.1 - In Problems 17-28, write down the first five terms...Ch. 11.1 - In Problems 17-28, write down the first five terms...Ch. 11.1 - In Problems 17-28, write down the first five terms...Ch. 11.1 - In Problems 29-36, the given pattern continues....Ch. 11.1 - In Problems 29-36, the given pattern continues....Ch. 11.1 - In Problems 29-36, the given pattern continues....Ch. 11.1 - In Problems 29-36, the given pattern continues....Ch. 11.1 - In Problems 29-36, the given pattern continues....Ch. 11.1 - In Problems 29-36, the given pattern continues....Ch. 11.1 - In Problems 29-36, the given pattern continues....Ch. 11.1 - In Problems 29-36, the given pattern continues....Ch. 11.1 - In Problems 37-50, a sequence is defined...Ch. 11.1 - In Problems 37-50, a sequence is defined...Ch. 11.1 - In Problems 37-50, a sequence is defined...Ch. 11.1 - In Problems 37-50, a sequence is defined...Ch. 11.1 - In Problems 35 – 48, a sequence is defined...Ch. 11.1 - In Problems 37-50, a sequence is defined...Ch. 11.1 - In Problems 37-50, a sequence is defined...Ch. 11.1 - In Problems 37-50, a sequence is defined...Ch. 11.1 - In Problems 37-50, a sequence is defined...Ch. 11.1 - In Problems 37-50, a sequence is defined...Ch. 11.1 - In Problems 37-50, a sequence is defined...Ch. 11.1 - In Problems 37-50, a sequence is defined...Ch. 11.1 - In Problems 37-50, a sequence is defined...Ch. 11.1 - In Problems 37-50, a sequence is defined...Ch. 11.1 - In Problems 51-60, write out each sum.
Ch. 11.1 - In Problems 51-60, write out each sum. k=1 n (...Ch. 11.1 - In Problems 51-60, write out each sum. k=1 n k 2...Ch. 11.1 - In Problems 51-60, write out each sum. k=1 n (...Ch. 11.1 - In Problems 51-60, write out each sum. k=0 n 1 3...Ch. 11.1 - In Problems 51-60, write out each sum. k=0 n ( 3...Ch. 11.1 - In Problems 51-60, write out each sum. k=0 n1 1 3...Ch. 11.1 - In Problems 51-60, write out each sum. k=0 n1 (...Ch. 11.1 - In Problems 51-60, write out each sum.
...Ch. 11.1 - In Problems 51-60, write out each sum. k=3 n ( 1...Ch. 11.1 - In Problems 61-70, express each sum using...Ch. 11.1 - In Problems 61-70, express each sum using...Ch. 11.1 - In Problems 61-70, express each sum using...Ch. 11.1 - In Problems 61-70, express each sum using...Ch. 11.1 - In Problems 61-70, express each sum using...Ch. 11.1 - In Problems 61-70, express each sum using...Ch. 11.1 - In Problems 61-70, express each sum using...Ch. 11.1 - In Problems 61-70, express each sum using...Ch. 11.1 - In Problems 61-70, express each sum using...Ch. 11.1 - In Problems 61-70, express each sum using...Ch. 11.1 - In Problems 71-82, find the sum of each...Ch. 11.1 - In Problems 71-82, find the sum of each sequence. ...Ch. 11.1 - In Problems 71-82, find the sum of each...Ch. 11.1 - In Problems 71-82, find the sum of each sequence. ...Ch. 11.1 - In Problems 71-82, find the sum of each sequence. ...Ch. 11.1 - In Problems 71-82, find the sum of each...Ch. 11.1 - In Problems 71-82, find the sum of each...Ch. 11.1 - In Problems 71-82, find the sum of each...Ch. 11.1 - In Problems 71-82, find the sum of each sequence. ...Ch. 11.1 - In Problems 71-82, find the sum of each sequence. ...Ch. 11.1 - In Problems 71-82, find the sum of each...Ch. 11.1 - In Problems 71-82, find the sum of each...Ch. 11.1 - Credit Card Debt John has a balance of 3000 on his...Ch. 11.1 - Trout Population A pond currently contains 2000...Ch. 11.1 - Car Loans Phil bought a car by taking out a loan...Ch. 11.1 - Environmental Control The Environmental Protection...Ch. 11.1 - Growth of a Rabbit Colony A colony of rabbits...Ch. 11.1 - The Pascal Triangle The triangular array shown,...Ch. 11.1 - Fibonacci Sequence Use the result of Problem 86 to...Ch. 11.1 - Triangular Numbers A triangular number is a term...Ch. 11.1 - Challenge Problem For the sequence given in...Ch. 11.1 - Challenge Problem For the sequence given in...Ch. 11.1 - Write a paragraph that explains why the numbers...Ch. 11.1 - If $2500 is invested at 3% compounded monthly,...Ch. 11.1 - Write the complex number 1i in polar form. Express...Ch. 11.1 - For v=2ij and w=i+2j , find the dot product vw .Ch. 11.1 - Find an equation of the parabola with vertex and...Ch. 11.2 - In a(n) _________ sequence, the difference between...Ch. 11.2 - True or False For an arithmetic sequence whose...Ch. 11.2 - If the 5th term of an arithmetic sequence is 12...Ch. 11.2 - True or False The sum S n of the first n terms of...Ch. 11.2 - An arithmetic sequence can always be expressed as...Ch. 11.2 - If is the nth term of an arithmetic sequence, the...Ch. 11.2 - In Problems 7-16, show that each sequence is...Ch. 11.2 - In Problems 7-16, show that each sequence is...Ch. 11.2 - In Problems 7-16, show that each sequence is...Ch. 11.2 - In Problems 7-16, show that each sequence is...Ch. 11.2 - In Problems 7-16, show that each sequence is...Ch. 11.2 - In Problems 7-16, show that each sequence is...Ch. 11.2 - In Problems 7-16, show that each sequence is...Ch. 11.2 - In Problems 7-16, show that each sequence is...Ch. 11.2 - In Problems 7-16, show that each sequence is...Ch. 11.2 - In Problems 7-16, show that each sequence is...Ch. 11.2 - In Problems 17-24, find the nth term of the...Ch. 11.2 - In Problems 17-24, find the nth term of the...Ch. 11.2 - In Problems 17 – 24, find the nth term of the...Ch. 11.2 - In Problems 17-24, find the nth term of the...Ch. 11.2 - In Problems 17-24, find the nth term of the...Ch. 11.2 - In Problems 17-24, find the nth term of the...Ch. 11.2 - In Problems 17-24, find the nth term of the...Ch. 11.2 - In Problems 17-24, find the nth term of the...Ch. 11.2 - In Problems 25-30, find the indicated term in each...Ch. 11.2 - In Problems 25-30, find the indicated term in each...Ch. 11.2 - In Problems 25 30, find the indicated term in...Ch. 11.2 - In Problems 25-30, find the indicated term in each...Ch. 11.2 - In Problems 25-30, find the indicated term in each...Ch. 11.2 - In Problems 25-30, find the indicated term in each...Ch. 11.2 - In Problems 31-38, find the first term and the...Ch. 11.2 - In Problems 31-38, find the first term and the...Ch. 11.2 - Prob. 33AYUCh. 11.2 - Prob. 34AYUCh. 11.2 - Prob. 35AYUCh. 11.2 - Prob. 36AYUCh. 11.2 - In Problems 31-38, find the first term and the...Ch. 11.2 - In Problems 31-38, find the first term and the...Ch. 11.2 - In Problems 39-56, find each sum. 1+3+5++( 2n1 )Ch. 11.2 - In Problems 39-56, find each sum. 2+4+6++2nCh. 11.2 - In Problems 39-56, find each sum.
Ch. 11.2 - In Problems 39-56, find each sum. 1+3+7++( 4n5 )Ch. 11.2 - In Problems 39-56, find each sum.
Ch. 11.2 - In Problems 39-56, find each sum. 1+3+5++59Ch. 11.2 - In Problems 39 – 56, find each sum.
45.
Ch. 11.2 - In Problems 39-56, find each sum.
Ch. 11.2 - In Problems 39 – 56, find each sum.
47.
Ch. 11.2 - In Problems 39-56, find each sum.
Ch. 11.2 - In Problems 39-56, find each sum.
Ch. 11.2 - In Problems 39-56, find each sum. 8+8 1 4 +8 1 2...Ch. 11.2 - In Problems 39 56, find each sum. n=180(2n5)Ch. 11.2 - In Problems 39-56, find each sum. n=1 90 ( 32n )Ch. 11.2 - In Problems 39-56, find each sum.
Ch. 11.2 - In Problems 39-56, find each sum.
Ch. 11.2 - In Problems 39-56, find each sum.
The sum of the...Ch. 11.2 - In Problems 39-56, find each sum. The sum of the...Ch. 11.2 - Find x so that , , and are consecutive terms of...Ch. 11.2 - Find x so that , and are consecutive terms of an...Ch. 11.2 - How many terms must be added in an arithmetic...Ch. 11.2 - How many terms must be added in an arithmetic...Ch. 11.2 - Drury Lane Theater The Drury Lane Theater has 25...Ch. 11.2 - Football Stadium The corner section of a football...Ch. 11.2 - Seats in an Amphitheater An outdoor amphitheater...Ch. 11.2 - Constructing a Brick Staircase A brick staircase...Ch. 11.2 - Salary If you take a job with a starting salary of...Ch. 11.2 - Stadium Construction How many rows are in the...Ch. 11.2 - Creating a Mosaic A mosaic is designed in the...Ch. 11.2 - Cooling Air As a parcel of air rises (for example,...Ch. 11.2 - Prob. 66AYUCh. 11.2 - Make up an arithmetic sequence. Give it to a...Ch. 11.2 - Describe the similarities and differences between...Ch. 11.2 - Problems 72-75 are based on material learned...Ch. 11.2 - Prob. 73AYUCh. 11.2 - Problems 72-75 are based on material learned...Ch. 11.2 - Problems 72-75 are based on material learned...Ch. 11.3 - Prob. 1AYUCh. 11.3 - How much do you need to invest now at 5 per annum...Ch. 11.3 - In a(n) _____________ sequence, the ratio of...Ch. 11.3 - If , the sum of the geometric series is...Ch. 11.3 - 5. If a series does not converge, it is called...Ch. 11.3 - True or False A geometric sequence may be defined...Ch. 11.3 - True or False In a geometric sequence, the common...Ch. 11.3 - True or False For a geometric sequence with first...Ch. 11.3 - Prob. 9AYUCh. 11.3 - Prob. 10AYUCh. 11.3 - Prob. 11AYUCh. 11.3 - Prob. 12AYUCh. 11.3 - Prob. 13AYUCh. 11.3 - Prob. 14AYUCh. 11.3 - Prob. 15AYUCh. 11.3 - Prob. 16AYUCh. 11.3 - Prob. 17AYUCh. 11.3 - Prob. 18AYUCh. 11.3 - Prob. 19AYUCh. 11.3 - Prob. 20AYUCh. 11.3 - Prob. 21AYUCh. 11.3 - Prob. 22AYUCh. 11.3 - Prob. 23AYUCh. 11.3 - Prob. 24AYUCh. 11.3 - Prob. 25AYUCh. 11.3 - Prob. 26AYUCh. 11.3 - Prob. 27AYUCh. 11.3 - Prob. 28AYUCh. 11.3 - Prob. 29AYUCh. 11.3 - Prob. 30AYUCh. 11.3 - Prob. 31AYUCh. 11.3 - Prob. 32AYUCh. 11.3 - Prob. 33AYUCh. 11.3 - Prob. 34AYUCh. 11.3 - Prob. 35AYUCh. 11.3 - Prob. 36AYUCh. 11.3 - Prob. 37AYUCh. 11.3 - Prob. 38AYUCh. 11.3 - Prob. 39AYUCh. 11.3 - Prob. 40AYUCh. 11.3 - Prob. 41AYUCh. 11.3 - Prob. 42AYUCh. 11.3 - Prob. 43AYUCh. 11.3 - In problems 41-46, find each sum.
Ch. 11.3 - In problems 41-46, find each sum. 1248( 2 n1 )Ch. 11.3 - In problems 41-46, find each sum.
Ch. 11.3 - For Problems 47-52, use a graphing utility to find...Ch. 11.3 - For Problems 47-52, use a graphing utility to find...Ch. 11.3 - For Problems 47-52, use a graphing utility to find...Ch. 11.3 - For Problems 47-52, use a graphing utility to find...Ch. 11.3 - For Problems 47-52, use a graphing utility to find...Ch. 11.3 - For Problems 47-52, use a graphing utility to find...Ch. 11.3 - In Problems 53-68, determine whether each infinite...Ch. 11.3 - In Problems 53-68, determine whether each infinite...Ch. 11.3 - In Problems 53-68, determine whether each infinite...Ch. 11.3 - In Problems 53-68, determine whether each infinite...Ch. 11.3 - In Problems 53-68, determine whether each infinite...Ch. 11.3 - In Problems 53-68, determine whether each infinite...Ch. 11.3 - In Problems 53-68, determine whether each infinite...Ch. 11.3 - In Problems 53-68, determine whether each infinite...Ch. 11.3 - In Problems 53-68, determine whether each infinite...Ch. 11.3 - In Problems 53-68, determine whether each infinite...Ch. 11.3 - In Problems 53-68, determine whether each infinite...Ch. 11.3 - In Problems 53-68, determine whether each infinite...Ch. 11.3 - In Problems 53-68, determine whether each infinite...Ch. 11.3 - In Problems 53-68, determine whether each infinite...Ch. 11.3 - In Problems 53-68, determine whether each infinite...Ch. 11.3 - In Problems 53-68, determine whether each infinite...Ch. 11.3 - In Problems 69-82, determine whether the given...Ch. 11.3 - In Problems 69-82, determine whether the given...Ch. 11.3 - In Problems 69-82, determine whether the given...Ch. 11.3 - In Problems 69-82, determine whether the given...Ch. 11.3 - In Problems 69-82, determine whether the given...Ch. 11.3 - In Problems 69-82, determine whether the given...Ch. 11.3 - In Problems 69-82, determine whether the given...Ch. 11.3 - In Problems 69-82, determine whether the given...Ch. 11.3 - In Problems 69-82, determine whether the given...Ch. 11.3 - In Problems 69-82, determine whether the given...Ch. 11.3 - In Problems 69-82, determine whether the given...Ch. 11.3 - In Problems 69-82, determine whether the given...Ch. 11.3 - In Problems 69-82, determine whether the given...Ch. 11.3 - In Problems 69-82, determine whether the given...Ch. 11.3 - Find x so that x,x+2 , and x+3 are consecutive...Ch. 11.3 - Find x so that are consecutive terms of a...Ch. 11.3 - Salary Increases If you have been hired at an...Ch. 11.3 - Equipment Depreciation A new piece of equipment...Ch. 11.3 - Pendulum Swings Initially, a pendulum swings...Ch. 11.3 - Bouncing Balls A ball is dropped from a height of...Ch. 11.3 - 89. Retirement Christine contributes $100 each...Ch. 11.3 - Saving for a Home Jolene wants to purchase a new...Ch. 11.3 - Tax-Sheltered Annuity Don contributes $500 at the...Ch. 11.3 - 92. Retirement Ray contributes $ 1000 to an...Ch. 11.3 - Sinking Fund Scott and Alice want to purchase a...Ch. 11.3 - 94. Sinking Fund For a child born in 2017, the...Ch. 11.3 - Grains of Wheat on a Chess Board In an old fable,...Ch. 11.3 - Look at the figure. What fraction of the square is...Ch. 11.3 - Multiplier Suppose that, throughout the U.S....Ch. 11.3 - Multiplier Refer to Problem 97. Suppose that the...Ch. 11.3 - Stock Price One method of pricing a stock is to...Ch. 11.3 - Stock Price Refer to Problem 99. Suppose that a...Ch. 11.3 - A Rich Mans Promise A rich man promises to give...Ch. 11.3 - Seating Revenue A special section in the end zone...Ch. 11.3 - Equal Pay You are offered two jobs. Job A has a...Ch. 11.3 - Fractal Area: A fractal known as the Koch Curve is...Ch. 11.3 - Critical Thinking You are interviewing for a job...Ch. 11.3 - Critical Thinking Which of the following choices,...Ch. 11.3 - Critical Thinking You have just signed a 7year...Ch. 11.3 - Critical Thinking Suppose you were offered a job...Ch. 11.3 - Can a sequence be both arithmetic and geometric?...Ch. 11.3 - Make up a geometric sequence. Give it to a friend...Ch. 11.3 - Make up two infinite geometric series, one that...Ch. 11.3 - Describe the similarities and differences between...Ch. 11.3 - Use the ChangeofBase Formula and a calculator to...Ch. 11.3 - Prob. 114AYUCh. 11.3 - Problems 112-115 are based on material learned...Ch. 11.3 - Problems 112-115 are based on material learned...Ch. 11.4 - In Problems 1-22, use the Principle of...Ch. 11.4 - In Problems 1-22, use the Principle of...Ch. 11.4 - In Problems 1-22, use the Principle of...Ch. 11.4 - In Problems 1-22, use the Principle of...Ch. 11.4 - In Problems 1-22, use the Principle of...Ch. 11.4 - In Problems 1-22, use the Principle of...Ch. 11.4 - In Problems 1-22, use the Principle of...Ch. 11.4 - In Problems 1-22, use the Principle of...Ch. 11.4 - In Problems 1-22, use the Principle of...Ch. 11.4 - In Problems 1-22, use the Principle of...Ch. 11.4 - In Problems 1-22, use the Principle of...Ch. 11.4 - In Problems 1-22, use the Principle of...Ch. 11.4 - In Problems 1-22, use the Principle of...Ch. 11.4 - In Problems 1-22, use the Principle of...Ch. 11.4 - In Problems 1-22, use the Principle of...Ch. 11.4 - In Problems 1-22, use the Principle of...Ch. 11.4 - In Problems 1-22, use the Principle of...Ch. 11.4 - In Problems 1-22, use the Principle of...Ch. 11.4 - In Problems 1-22, use the Principle of...Ch. 11.4 - In Problems 1-22, use the Principle of...Ch. 11.4 - In Problems 1-22, use the Principle of...Ch. 11.4 - In Problems 1-22, use the Principle of...Ch. 11.4 - In Problems 23-27, prove each statement.
If , then...Ch. 11.4 - In Problems 23-27, prove each statement. If 0x1 ,...Ch. 11.4 - In Problems 23-27, prove each statement. ab is a...Ch. 11.4 - In Problems 23-27, prove each statement. a+b is a...Ch. 11.4 - In Problems 23-27, prove each statement. ( 1+a ) n...Ch. 11.4 - Show that the statement n 2 n+41 is a prime...Ch. 11.4 - Show that the formula
obeys Condition II of the...Ch. 11.4 - Use mathematical induction to prove that if r1 ,...Ch. 11.4 - Use mathematical induction to prove that
Ch. 11.4 - Extended Principle of Mathematical Induction The...Ch. 11.4 - Geometry Use the Extended Principle of...Ch. 11.4 - How would you explain the Principle of...Ch. 11.4 - Solve: log 2 x+5 =4Ch. 11.4 - Solve the system:
Ch. 11.4 - A mass of 500 kg is suspended from two cables, as...Ch. 11.4 - For , find .
Ch. 11.5 - The ______ ______ is a triangular display of the...Ch. 11.5 - .
Ch. 11.5 - True or False ( n j )= j! ( nj )!n!Ch. 11.5 - The ______ ________ can be used to expand...Ch. 11.5 - In Problems 5-16, evaluate each expression. ( 5 3...Ch. 11.5 - In Problems 5-16, evaluate each expression.
Ch. 11.5 - In Problems 5-16, evaluate each expression.
Ch. 11.5 - In Problems 5-16, evaluate each expression.
Ch. 11.5 - In Problems 5-16, evaluate each expression. ( 50...Ch. 11.5 - In Problems 5-16, evaluate each expression. ( 100...Ch. 11.5 - In Problems 5-16, evaluate each expression.
Ch. 11.5 - In Problems 5-16, evaluate each expression.
Ch. 11.5 - In Problems 5-16, evaluate each expression.
Ch. 11.5 - In Problems 5-16, evaluate each expression. ( 60...Ch. 11.5 - In Problems 5-16, evaluate each expression. ( 47...Ch. 11.5 - In Problems 5-16, evaluate each expression. ( 37...Ch. 11.5 - In Problems 17-28, expand each expression using...Ch. 11.5 - In Problems 17-28, expand each expression using...Ch. 11.5 - In Problems 17-28, expand each expression using...Ch. 11.5 - In Problems 17-28, expand each expression using...Ch. 11.5 - In Problems 17-28, expand each expression using...Ch. 11.5 - In Problems 17-28, expand each expression using...Ch. 11.5 - In Problems 17-28, expand each expression using...Ch. 11.5 - In Problems 17-28, expand each expression using...Ch. 11.5 - In Problems 17-28, expand each expression using...Ch. 11.5 - In Problems 17-28, expand each expression using...Ch. 11.5 - In Problems 17-28, expand each expression using...Ch. 11.5 - In Problems 17-28, expand each expression using...Ch. 11.5 - In Problems 29-42, use the Binomial Theorem to...Ch. 11.5 - In Problems 29-42, use the Binomial Theorem to...Ch. 11.5 - In Problems 29-42, use the Binomial Theorem to...Ch. 11.5 - In Problems 29-42, use the Binomial Theorem to...Ch. 11.5 - In Problems 29-42, use the Binomial Theorem to...Ch. 11.5 - In Problems 29-42, use the Binomial Theorem to...Ch. 11.5 - In Problems 29-42, use the Binomial Theorem to...Ch. 11.5 - In Problems 29-42, use the Binomial Theorem to...Ch. 11.5 - In Problems 29-42, use the Binomial Theorem to...Ch. 11.5 - In Problems 29-42, use the Binomial Theorem to...Ch. 11.5 - In Problems 29-42, use the Binomial Theorem to...Ch. 11.5 - In Problems 29-42, use the Binomial Theorem to...Ch. 11.5 - Prob. 41AYUCh. 11.5 - Prob. 42AYUCh. 11.5 - Prob. 43AYUCh. 11.5 - Prob. 44AYUCh. 11.5 - Prob. 45AYUCh. 11.5 - Show that if n and j are integers with 0jn, then...Ch. 11.5 - Prob. 47AYUCh. 11.5 - Prob. 48AYUCh. 11.5 - Prob. 49AYUCh. 11.5 - Prob. 50AYUCh. 11.5 - Prob. 51AYUCh. 11.5 - Prob. 52AYUCh. 11.5 - Prob. 53AYUCh. 11.5 - Prob. 54AYUCh. 11 - In Problems 14, list the five terms of each...Ch. 11 - Prob. 2RECh. 11 - Prob. 3RECh. 11 - Prob. 4RECh. 11 - Prob. 5RECh. 11 - Prob. 6RECh. 11 - Prob. 7RECh. 11 - Prob. 8RECh. 11 - Prob. 9RECh. 11 - Prob. 10RECh. 11 - Prob. 11RECh. 11 - Prob. 12RECh. 11 - Prob. 13RECh. 11 - Prob. 14RECh. 11 - Prob. 15RECh. 11 - Prob. 16RECh. 11 - Prob. 17RECh. 11 - Prob. 18RECh. 11 - Prob. 19RECh. 11 - Prob. 20RECh. 11 - Prob. 21RECh. 11 - Prob. 22RECh. 11 - Prob. 23RECh. 11 - Prob. 24RECh. 11 - Prob. 25RECh. 11 - In Problems 2628, use the Principle of...Ch. 11 - Prob. 27RECh. 11 - Prob. 28RECh. 11 - Prob. 29RECh. 11 - Prob. 30RECh. 11 - Prob. 31RECh. 11 - Prob. 32RECh. 11 - Prob. 33RECh. 11 - Prob. 34RECh. 11 - Prob. 35RECh. 11 - Prob. 36RECh. 11 - Prob. 37RECh. 11 - Prob. 38RECh. 11 - Prob. 1CTCh. 11 - Prob. 2CTCh. 11 - Prob. 3CTCh. 11 - Prob. 4CTCh. 11 - Prob. 5CTCh. 11 - Prob. 6CTCh. 11 - Prob. 7CTCh. 11 - Prob. 8CTCh. 11 - Prob. 9CTCh. 11 - Prob. 10CTCh. 11 - Prob. 11CTCh. 11 - Prob. 12CTCh. 11 - Prob. 13CTCh. 11 - Prob. 14CTCh. 11 - Prob. 15CTCh. 11 - Prob. 16CTCh. 11 - Prob. 1CRCh. 11 - Prob. 2CRCh. 11 - Prob. 3CRCh. 11 - Prob. 4CRCh. 11 - Prob. 5CRCh. 11 - Prob. 6CRCh. 11 - Prob. 7CRCh. 11 - Prob. 8CRCh. 11 - Prob. 9CRCh. 11 - Prob. 10CRCh. 11 - Prob. 11CRCh. 11 - Prob. 12CR
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- 12. Evaluate ſ √9-x2 -dx. x2 a) C 9-x2 √9-x2 - x2 b) C - x x arcsin ½-½ c) C + √9 - x² + arcsin x d) C + √9-x2 x2 13. Find the indefinite integral S cos³30 √sin 30 dᎾ . 2√√sin 30 (5+sin²30) √sin 30 (3+sin²30) a) C+ √sin 30(5-sin²30) b) C + c) C + 5 5 5 10 d) C + 2√√sin 30 (3-sin²30) 2√√sin 30 (5-sin²30) e) C + 5 15 14. Find the indefinite integral ( sin³ 4xcos 44xdx. a) C+ (7-5cos24x)cos54x b) C (7-5cos24x)cos54x (7-5cos24x)cos54x - 140 c) C - 120 140 d) C+ (7-5cos24x)cos54x e) C (7-5cos24x)cos54x 4 4 15. Find the indefinite integral S 2x2 dx. ex - a) C+ (x²+2x+2)ex b) C (x² + 2x + 2)e-* d) C2(x²+2x+2)e¯* e) C + 2(x² + 2x + 2)e¯* - c) C2x(x²+2x+2)e¯*arrow_forward4. Which substitution would you use to simplify the following integrand? S a) x = sin b) x = 2 tan 0 c) x = 2 sec 3√√3 3 x3 5. After making the substitution x = = tan 0, the definite integral 2 2 3 a) ៖ ស្លឺ sin s π - dᎾ 16 0 cos20 b) 2/4 10 cos 20 π sin30 6 - dᎾ c) Π 1 cos³0 3 · de 16 0 sin20 1 x²√x²+4 3 (4x²+9)2 π d) cos²8 16 0 sin³0 dx d) x = tan 0 dx simplifies to: de 6. In order to evaluate (tan 5xsec7xdx, which would be the most appropriate strategy? a) Separate a sec²x factor b) Separate a tan²x factor c) Separate a tan xsecx factor 7. Evaluate 3x x+4 - dx 1 a) 3x+41nx + 4 + C b) 31n|x + 4 + C c) 3 ln x + 4+ C d) 3x - 12 In|x + 4| + C x+4arrow_forward1. Abel's Theorem. The goal in this problem is to prove Abel's theorem by following a series of steps (each step must be justified). Theorem 0.1 (Abel's Theorem). If y1 and y2 are solutions of the differential equation y" + p(t) y′ + q(t) y = 0, where p and q are continuous on an open interval, then the Wronskian is given by W (¥1, v2)(t) = c exp(− [p(t) dt), where C is a constant that does not depend on t. Moreover, either W (y1, y2)(t) = 0 for every t in I or W (y1, y2)(t) = 0 for every t in I. 1. (a) From the two equations (which follow from the hypotheses), show that y" + p(t) y₁ + q(t) y₁ = 0 and y½ + p(t) y2 + q(t) y2 = 0, 2. (b) Observe that Hence, conclude that (YY2 - Y1 y2) + P(t) (y₁ Y2 - Y1 Y2) = 0. W'(y1, y2)(t) = yY2 - Y1 y2- W' + p(t) W = 0. 3. (c) Use the result from the previous step to complete the proof of the theorem.arrow_forward
- 2. Observations on the Wronskian. Suppose the functions y₁ and y2 are solutions to the differential equation p(x)y" + q(x)y' + r(x) y = 0 on an open interval I. 1. (a) Prove that if y₁ and y2 both vanish at the same point in I, then y₁ and y2 cannot form a fundamental set of solutions. 2. (b) Prove that if y₁ and y2 both attain a maximum or minimum at the same point in I, then y₁ and Y2 cannot form a fundamental set of solutions. 3. (c) show that the functions & and t² are linearly independent on the interval (−1, 1). Verify that both are solutions to the differential equation t² y″ – 2ty' + 2y = 0. Then justify why this does not contradict Abel's theorem. 4. (d) What can you conclude about the possibility that t and t² are solutions to the differential equation y" + q(x) y′ + r(x)y = 0?arrow_forwardQuestion 4 Find an equation of (a) The plane through the point (2, 0, 1) and perpendicular to the line x = y=2-t, z=3+4t. 3t, (b) The plane through the point (3, −2, 8) and parallel to the plane z = x+y. (c) The plane that contains the line x = 1+t, y = 2 − t, z = 4 - 3t and is parallel to the plane 5x + 2y + z = 1. (d) The plane that passes through the point (1,2,3) and contains the line x = 3t, y = 1+t, and z = 2-t. (e) The plane that contains the lines L₁: x = 1 + t, y = 1 − t, z = 2t and L2 : x = 2 − s, y = s, z = 2.arrow_forwardPlease find all values of x.arrow_forward
- 3. Consider the initial value problem 9y" +12y' + 4y = 0, y(0) = a>0: y′(0) = −1. Solve the problem and find the value of a such that the solution of the initial value problem is always positive.arrow_forward5. Euler's equation. Determine the values of a for which all solutions of the equation 5 x²y" + axy' + y = 0 that have the form (A + B log x) x* or Ax¹¹ + Bä” tend to zero as a approaches 0.arrow_forward4. Problem on variable change. The purpose of this problem is to perform an appropriate change of variables in order to reduce the problem to a second-order equation with constant coefficients. ty" + (t² − 1)y'′ + t³y = 0, 0arrow_forward
- 4. Some psychologists contend that the number of facts of a certain type that are remembered after t hours is given by f(t)== 90t 951-90 Find the rate at which the number of facts remembered is changing after 1 hour and after 10 hours. Interpret.arrow_forward12:05 MA S 58 58. If f(x) = ci.metaproxy.org 25 2xon [0, 10] and n is a positive integer, then there is some Riemann sum Sthat equals the exact area under the graph of ƒ from x = Oto x = 10. 59. If the area under the graph of fon [a, b] is equal to both the left sum L, and the right sum Rfor some positive integer n, then fis constant on [a, b]. 60. If ƒ is a decreasing function on [a, b], then the area under the graph of fis greater than the left sum Land less than the right sum R₂, for any positive integer n. Problems 61 and 62 refer to the following figure showing two parcels of land along a river: River Parcel 2 Parcel 1 h(x) 500 ft 1,000 ft. Figure for 61 and 62 61. You want to purchase both parcels of land shown in the figure and make a quick check on their combined area. There is no equation for the river frontage, so you use the average of the left and right sums of rectangles covering the area. The 1,000-foot baseline is divided into 10 equal parts. At the end of each…arrow_forwardIf a snowball melts so that its surface area decreases at a rate of 10 cm²/min, find the rate (in cm/min) at which the diameter decreases when the diameter is 12 cm. (Round your answer to three decimal places.) cm/minarrow_forward
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