PRECALCULUS: MATHEMATICS FOR CALCULUS
7th Edition
ISBN: 9781305718906
Author: Stewart
Publisher: CENGAGE L
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Chapter 12.3, Problem 9E
To determine
To find: The nth term of a sequence is if a = 7 and r = 4, find the fourth term.
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2. Answer the following questions.
(A) [50%] Given the vector field F(x, y, z) = (x²y, e", yz²), verify the differential identity
Vx (VF) V(V •F) - V²F
(B) [50%] Remark. You are confined to use the differential identities.
Let u and v be scalar fields, and F be a vector field given by
F = (Vu) x (Vv)
(i) Show that F is solenoidal (or incompressible).
(ii) Show that
G =
(uvv – vVu)
is a vector potential for F.
A driver is traveling along a straight road when a buffalo runs into the street. This driver has a reaction time of 0.75 seconds. When the driver sees the buffalo he is traveling at 44 ft/s, his car can decelerate at 2 ft/s^2 when the brakes are applied. What is the stopping distance between when the driver first saw the buffalo, to when the car stops.
Topic 2
Evaluate S
x
dx, using u-substitution. Then find the integral using
1-x2
trigonometric substitution. Discuss the results!
Topic 3
Explain what an elementary anti-derivative is. Then consider the following
ex
integrals: fed dx
x
1
Sdx
In x
Joseph Liouville proved that the first integral does not have an elementary anti-
derivative Use this fact to prove that the second integral does not have an
elementary anti-derivative. (hint: use an appropriate u-substitution!)
Chapter 12 Solutions
PRECALCULUS: MATHEMATICS FOR CALCULUS
Ch. 12.1 - A sequence is a function whose domain is...Ch. 12.1 - Prob. 2ECh. 12.1 - Terms of a Sequence Find the first four terms and...Ch. 12.1 - Prob. 4ECh. 12.1 - Prob. 5ECh. 12.1 - Terms of a Sequence Find the first four terms and...Ch. 12.1 - Prob. 7ECh. 12.1 - Prob. 8ECh. 12.1 - Terms of a Sequence Find the first four terms and...Ch. 12.1 - Prob. 10E
Ch. 12.1 - Prob. 11ECh. 12.1 - Prob. 12ECh. 12.1 - Prob. 13ECh. 12.1 - Prob. 14ECh. 12.1 - Recursive Sequences A sequence is defined...Ch. 12.1 - Recursive Sequences A sequence is defined...Ch. 12.1 - Prob. 17ECh. 12.1 - Prob. 18ECh. 12.1 - Prob. 19ECh. 12.1 - Prob. 20ECh. 12.1 - Prob. 21ECh. 12.1 - Prob. 22ECh. 12.1 - Terms of a Sequence Use a graphing calculator to...Ch. 12.1 - Prob. 24ECh. 12.1 - Prob. 25ECh. 12.1 - Prob. 26ECh. 12.1 - Prob. 27ECh. 12.1 - Prob. 28ECh. 12.1 - nth term of a Sequence Find the nth term of a...Ch. 12.1 - Prob. 30ECh. 12.1 - Prob. 31ECh. 12.1 - Prob. 32ECh. 12.1 - Prob. 33ECh. 12.1 - Prob. 34ECh. 12.1 - Prob. 35ECh. 12.1 - Prob. 36ECh. 12.1 - Prob. 37ECh. 12.1 - Prob. 38ECh. 12.1 - Prob. 39ECh. 12.1 - Prob. 40ECh. 12.1 - Partial Sums Find the first six partial sums S1,...Ch. 12.1 - Prob. 42ECh. 12.1 - nth Partial Sum Find the first four partial sums...Ch. 12.1 - Prob. 44ECh. 12.1 - nth Partial Sum Find the first four partial sums...Ch. 12.1 - nth Partial Sum Find the first four partial sums...Ch. 12.1 - Prob. 47ECh. 12.1 - Prob. 48ECh. 12.1 - Prob. 49ECh. 12.1 - Prob. 50ECh. 12.1 - Prob. 51ECh. 12.1 - Prob. 52ECh. 12.1 - Prob. 53ECh. 12.1 - Prob. 54ECh. 12.1 - Prob. 55ECh. 12.1 - Evaluating a Sum Use a graphing calculator to...Ch. 12.1 - Prob. 57ECh. 12.1 - Prob. 58ECh. 12.1 - Prob. 59ECh. 12.1 - Prob. 60ECh. 12.1 - Prob. 61ECh. 12.1 - Prob. 62ECh. 12.1 - Prob. 63ECh. 12.1 - Prob. 64ECh. 12.1 - Prob. 65ECh. 12.1 - Sigma Notation Write the sum without using sigma...Ch. 12.1 - Sigma Notation Write the sum using sigma notation....Ch. 12.1 - Prob. 68ECh. 12.1 - Prob. 69ECh. 12.1 - Sigma Notation Write the sum using sigma notation....Ch. 12.1 - Sigma Notation Write the sum using sigma notation....Ch. 12.1 - Prob. 72ECh. 12.1 - Prob. 73ECh. 12.1 - Prob. 74ECh. 12.1 - nth Term of a Sequence Find a formula for the nth...Ch. 12.1 - Compound Interest Julio deposits 2000 in a savings...Ch. 12.1 - Prob. 78ECh. 12.1 - Population of a City A city was incorporated in...Ch. 12.1 - Paying off a Debt Margarita borrows 10,000 from...Ch. 12.1 - Fish Farming A fish farmer has 5000 catfish in his...Ch. 12.1 - Prob. 82ECh. 12.1 - Prob. 83ECh. 12.1 - Concentration of a Solution A biologist is trying...Ch. 12.1 - Prob. 85ECh. 12.1 - DISCOVER PROVE: Different Sequences That Start the...Ch. 12.1 - Prob. 87ECh. 12.1 - Prob. 88ECh. 12.2 - An arithmetic sequence is a sequence in which the...Ch. 12.2 - Prob. 2ECh. 12.2 - Prob. 3ECh. 12.2 - Prob. 4ECh. 12.2 - Prob. 5ECh. 12.2 - Prob. 6ECh. 12.2 - Prob. 7ECh. 12.2 - Prob. 8ECh. 12.2 - Prob. 9ECh. 12.2 - Prob. 10ECh. 12.2 - Prob. 11ECh. 12.2 - Prob. 12ECh. 12.2 - Prob. 13ECh. 12.2 - Prob. 14ECh. 12.2 - Prob. 15ECh. 12.2 - Prob. 16ECh. 12.2 - Prob. 17ECh. 12.2 - Prob. 18ECh. 12.2 - Prob. 19ECh. 12.2 - Prob. 20ECh. 12.2 - Prob. 21ECh. 12.2 - Prob. 22ECh. 12.2 - Prob. 23ECh. 12.2 - Prob. 24ECh. 12.2 - Prob. 25ECh. 12.2 - Prob. 26ECh. 12.2 - Prob. 27ECh. 12.2 - Arithmetic Sequence? Find the first five terms of...Ch. 12.2 - Arithmetic Sequence? Find the first five terms of...Ch. 12.2 - Prob. 30ECh. 12.2 - Prob. 31ECh. 12.2 - Prob. 32ECh. 12.2 - Prob. 33ECh. 12.2 - Terms of an Arithmetic Sequence Determine the...Ch. 12.2 - Prob. 35ECh. 12.2 - Prob. 36ECh. 12.2 - Prob. 37ECh. 12.2 - Prob. 38ECh. 12.2 - Terms of an Arithmetic Sequence Determine the...Ch. 12.2 - Terms of an Arithmetic Sequence Determine the...Ch. 12.2 - Terms of an Arithmetic Sequence Determine the...Ch. 12.2 - Terms of an Arithmetic Sequence Determine the...Ch. 12.2 - Terms of an Arithmetic Sequence Determine the...Ch. 12.2 - Prob. 44ECh. 12.2 - Finding Terms of an Arithmetic Sequence Find the...Ch. 12.2 - Prob. 46ECh. 12.2 - Prob. 47ECh. 12.2 - Finding Terms of an Arithmetic Sequence Find the...Ch. 12.2 - Prob. 49ECh. 12.2 - Prob. 50ECh. 12.2 - Prob. 51ECh. 12.2 - Prob. 52ECh. 12.2 - Prob. 53ECh. 12.2 - Prob. 54ECh. 12.2 - Prob. 55ECh. 12.2 - Partial Sums of an Arithmetic Sequence Find the...Ch. 12.2 - Prob. 57ECh. 12.2 - Prob. 58ECh. 12.2 - Prob. 59ECh. 12.2 - Prob. 60ECh. 12.2 - Prob. 61ECh. 12.2 - Prob. 62ECh. 12.2 - Prob. 63ECh. 12.2 - Prob. 64ECh. 12.2 - Prob. 65ECh. 12.2 - Prob. 66ECh. 12.2 - Prob. 67ECh. 12.2 - Prob. 68ECh. 12.2 - Prob. 69ECh. 12.2 - Prob. 70ECh. 12.2 - Depreciation The purchase value of an office...Ch. 12.2 - Poles in a Pile Telephone poles are being stored...Ch. 12.2 - Salary Increases A man gets a job with a salary of...Ch. 12.2 - Drive-In Theater A drive-in theater has spaces for...Ch. 12.2 - Theater Seating An architect designs a theater...Ch. 12.2 - Prob. 76ECh. 12.2 - The Twelve Days of Christmas In the well-known...Ch. 12.2 - DISCUSS: Arithmetic Means The arithmetic mean (or...Ch. 12.3 - A geometric sequence is a sequence in which the...Ch. 12.3 - Prob. 2ECh. 12.3 - True or False? If we know the first and second...Ch. 12.3 - Prob. 4ECh. 12.3 - Prob. 5ECh. 12.3 - Prob. 6ECh. 12.3 - Prob. 7ECh. 12.3 - Prob. 8ECh. 12.3 - Prob. 9ECh. 12.3 - Prob. 10ECh. 12.3 - nth Term of a Geometric Sequence Find the nth term...Ch. 12.3 - Prob. 12ECh. 12.3 - Prob. 13ECh. 12.3 - Prob. 14ECh. 12.3 - Prob. 15ECh. 12.3 - Prob. 16ECh. 12.3 - Prob. 17ECh. 12.3 - Prob. 18ECh. 12.3 - Prob. 19ECh. 12.3 - Prob. 20ECh. 12.3 - Geometric Sequence? The first four terms of a...Ch. 12.3 - Prob. 22ECh. 12.3 - Prob. 23ECh. 12.3 - Prob. 24ECh. 12.3 - Prob. 25ECh. 12.3 - Prob. 26ECh. 12.3 - Prob. 27ECh. 12.3 - Prob. 28ECh. 12.3 - Prob. 29ECh. 12.3 - Prob. 30ECh. 12.3 - Prob. 31ECh. 12.3 - Prob. 32ECh. 12.3 - Prob. 33ECh. 12.3 - Prob. 34ECh. 12.3 - Terms of a Geometric Sequence Determine the common...Ch. 12.3 - Prob. 36ECh. 12.3 - Prob. 37ECh. 12.3 - Terms of a Geometric Sequence Determine the common...Ch. 12.3 - Finding Terms of a Geometric Sequence Find the...Ch. 12.3 - Prob. 40ECh. 12.3 - Prob. 41ECh. 12.3 - Finding Terms of a Geometric Sequence Find the...Ch. 12.3 - Prob. 43ECh. 12.3 - Prob. 44ECh. 12.3 - Finding Terms of a Geometric Sequence Find the...Ch. 12.3 - Prob. 46ECh. 12.3 - Which Term? The first term of a geometric sequence...Ch. 12.3 - Prob. 48ECh. 12.3 - Prob. 49ECh. 12.3 - Prob. 50ECh. 12.3 - Prob. 51ECh. 12.3 - Prob. 52ECh. 12.3 - Prob. 53ECh. 12.3 - Prob. 54ECh. 12.3 - Prob. 55ECh. 12.3 - Prob. 56ECh. 12.3 - Prob. 57ECh. 12.3 - Prob. 58ECh. 12.3 - Prob. 59ECh. 12.3 - Prob. 60ECh. 12.3 - Prob. 61ECh. 12.3 - Prob. 62ECh. 12.3 - Prob. 63ECh. 12.3 - Prob. 64ECh. 12.3 - Infinite Geometric Series Determine whether the...Ch. 12.3 - Prob. 66ECh. 12.3 - Prob. 67ECh. 12.3 - Infinite Geometric Series Determine whether the...Ch. 12.3 - Prob. 69ECh. 12.3 - Prob. 70ECh. 12.3 - Prob. 71ECh. 12.3 - Prob. 72ECh. 12.3 - Prob. 73ECh. 12.3 - Prob. 74ECh. 12.3 - Prob. 75ECh. 12.3 - Prob. 76ECh. 12.3 - Prob. 77ECh. 12.3 - Prob. 78ECh. 12.3 - Prob. 79ECh. 12.3 - Prob. 80ECh. 12.3 - Prob. 81ECh. 12.3 - Prob. 82ECh. 12.3 - Prob. 83ECh. 12.3 - Prob. 84ECh. 12.3 - Prob. 85ECh. 12.3 - Prob. 86ECh. 12.3 - Depreciation A construction company purchases a...Ch. 12.3 - Prob. 88ECh. 12.3 - Bouncing Ball A ball is dropped from a height of...Ch. 12.3 - Bacteria Culture A culture initially has 5000...Ch. 12.3 - Mixing Coolant A truck radiator holds 5 gal and is...Ch. 12.3 - Musical Frequencies The frequencies of musical...Ch. 12.3 - Bouncing Ball A ball is dropped from a height of 9...Ch. 12.3 - Prob. 94ECh. 12.3 - St. Ives The following is a well-known childrens...Ch. 12.3 - Prob. 96ECh. 12.3 - Bouncing Ball A certain ball rebounds to half the...Ch. 12.3 - Prob. 98ECh. 12.3 - Geometry A circular disk of radius R is cut out of...Ch. 12.3 - Prob. 102ECh. 12.3 - PROVE: Logarithms of a Geometric Sequence If a1,...Ch. 12.3 - Prob. 104ECh. 12.3 - Geometry A yellow square of side 1 is divided into...Ch. 12.4 - An annuity is a sum of money that is paid in...Ch. 12.4 - Prob. 2ECh. 12.4 - Prob. 3ECh. 12.4 - Annuity Find the amount of an annuity that...Ch. 12.4 - Annuity Find the amount of an annuity that...Ch. 12.4 - Prob. 6ECh. 12.4 - Prob. 7ECh. 12.4 - Prob. 8ECh. 12.4 - Prob. 9ECh. 12.4 - Saving How much money should be invested monthly...Ch. 12.4 - Prob. 11ECh. 12.4 - Prob. 12ECh. 12.4 - Prob. 13ECh. 12.4 - Funding an Annuity A 55-year-old man deposits...Ch. 12.4 - Prob. 15ECh. 12.4 - Prob. 16ECh. 12.4 - Prob. 17ECh. 12.4 - Prob. 18ECh. 12.4 - Mortgage Dr. Gupta is considering a 30-year...Ch. 12.4 - Prob. 20ECh. 12.4 - Prob. 21ECh. 12.4 - Prob. 22ECh. 12.4 - Prob. 23ECh. 12.4 - Prob. 24ECh. 12.4 - Interest Rate John buys a stereo system for 640....Ch. 12.4 - Prob. 26ECh. 12.4 - Prob. 27ECh. 12.4 - Interest Rate A man purchases a 2000 diamond ring...Ch. 12.4 - Prob. 29ECh. 12.4 - Prob. 30ECh. 12.4 - DISCOVER: Amortizing a Mortgage When they bought...Ch. 12.5 - Mathematical induction is a method of proving that...Ch. 12.5 - Prob. 2ECh. 12.5 - Prob. 3ECh. 12.5 - Prob. 4ECh. 12.5 - Prob. 5ECh. 12.5 - Prob. 6ECh. 12.5 - Prob. 7ECh. 12.5 - Prob. 8ECh. 12.5 - Prob. 9ECh. 12.5 - Proving a Formula Use mathematical induction to...Ch. 12.5 - Prob. 11ECh. 12.5 - Proving a Formula Use mathematical induction to...Ch. 12.5 - Prob. 13ECh. 12.5 - Prob. 14ECh. 12.5 - Prob. 15ECh. 12.5 - Prob. 16ECh. 12.5 - Proving a Statement Use mathematical induction to...Ch. 12.5 - Proving a Statement Use mathematical induction to...Ch. 12.5 - Prob. 19ECh. 12.5 - Prob. 20ECh. 12.5 - Prob. 21ECh. 12.5 - Prob. 22ECh. 12.5 - Prob. 23ECh. 12.5 - Prob. 24ECh. 12.5 - Prob. 25ECh. 12.5 - Formula for a Recursive Sequence A sequence is...Ch. 12.5 - Prob. 27ECh. 12.5 - Prob. 28ECh. 12.5 - Prob. 29ECh. 12.5 - Prob. 30ECh. 12.5 - Fibonacci Sequence Fn denotes the nth term of the...Ch. 12.5 - Prob. 32ECh. 12.5 - Prob. 33ECh. 12.5 - Prob. 34ECh. 12.5 - Prob. 35ECh. 12.5 - Prob. 36ECh. 12.5 - Prob. 37ECh. 12.5 - DISCUSS: All Cats Are Black? What is wrong with...Ch. 12.6 - An algebraic expression of the form a + b, which...Ch. 12.6 - Prob. 2ECh. 12.6 - The binomial coefficients can be calculated...Ch. 12.6 - Prob. 4ECh. 12.6 - Prob. 5ECh. 12.6 - Prob. 6ECh. 12.6 - Prob. 7ECh. 12.6 - Prob. 8ECh. 12.6 - Prob. 9ECh. 12.6 - Prob. 10ECh. 12.6 - Prob. 11ECh. 12.6 - Prob. 12ECh. 12.6 - Prob. 13ECh. 12.6 - Prob. 14ECh. 12.6 - Prob. 15ECh. 12.6 - Prob. 16ECh. 12.6 - Prob. 17ECh. 12.6 - Prob. 18ECh. 12.6 - Calculating Binomial Coefficients Evaluate the...Ch. 12.6 - Prob. 20ECh. 12.6 - Prob. 21ECh. 12.6 - Prob. 22ECh. 12.6 - Prob. 23ECh. 12.6 - Prob. 24ECh. 12.6 - Prob. 25ECh. 12.6 - Prob. 26ECh. 12.6 - Prob. 27ECh. 12.6 - Binomial Theorem Use the Binomial Theorem to...Ch. 12.6 - Prob. 29ECh. 12.6 - Prob. 30ECh. 12.6 - Prob. 31ECh. 12.6 - Prob. 32ECh. 12.6 - Prob. 33ECh. 12.6 - Prob. 34ECh. 12.6 - Terms of a Binomial Expansion Find the indicated...Ch. 12.6 - Terms of a Binomial Expansion Find the indicated...Ch. 12.6 - Prob. 37ECh. 12.6 - Prob. 38ECh. 12.6 - Prob. 39ECh. 12.6 - Prob. 40ECh. 12.6 - Prob. 41ECh. 12.6 - Prob. 42ECh. 12.6 - Prob. 43ECh. 12.6 - Prob. 44ECh. 12.6 - Prob. 45ECh. 12.6 - Prob. 46ECh. 12.6 - Prob. 47ECh. 12.6 - Simplifying a Difference Quotient Simplify using...Ch. 12.6 - Prob. 49ECh. 12.6 - Prob. 50ECh. 12.6 - Prob. 51ECh. 12.6 - Prob. 52ECh. 12.6 - Prob. 53ECh. 12.6 - Prob. 54ECh. 12.6 - Difference in Volumes of Cubes The volume of a...Ch. 12.6 - Probability of Hitting a Target The probability...Ch. 12.6 - Prob. 57ECh. 12.6 - Prob. 58ECh. 12.6 - Prob. 59ECh. 12 - (a) What is a sequence? What notation do we use to...Ch. 12 - Prob. 2RCCCh. 12 - Prob. 3RCCCh. 12 - Prob. 4RCCCh. 12 - Prob. 5RCCCh. 12 - Prob. 6RCCCh. 12 - (a) Write 13 + 23 + 33 + 43 + 53 using sigma...Ch. 12 - Prob. 8RCCCh. 12 - Prob. 9RCCCh. 12 - Prob. 10RCCCh. 12 - Prob. 11RCCCh. 12 - Prob. 12RCCCh. 12 - Prob. 1RECh. 12 - Prob. 2RECh. 12 - Prob. 3RECh. 12 - Prob. 4RECh. 12 - Prob. 5RECh. 12 - Prob. 6RECh. 12 - Prob. 7RECh. 12 - Prob. 8RECh. 12 - Prob. 9RECh. 12 - Prob. 10RECh. 12 - Prob. 11RECh. 12 - Prob. 12RECh. 12 - Prob. 13RECh. 12 - Prob. 14RECh. 12 - Prob. 15RECh. 12 - Arithmetic or Geometric? The first four terms of a...Ch. 12 - Prob. 17RECh. 12 - Prob. 18RECh. 12 - Prob. 19RECh. 12 - Prob. 20RECh. 12 - Prob. 21RECh. 12 - Prob. 22RECh. 12 - Prob. 23RECh. 12 - nth Term of a Geometric Sequence Find the nth term...Ch. 12 - Prob. 25RECh. 12 - Prob. 26RECh. 12 - Prob. 27RECh. 12 - Prob. 28RECh. 12 - Salary A teacher makes 32,000 in his first year at...Ch. 12 - Prob. 30RECh. 12 - Prob. 31RECh. 12 - Prob. 32RECh. 12 - Prob. 33RECh. 12 - Prob. 34RECh. 12 - Prob. 35RECh. 12 - Prob. 36RECh. 12 - Prob. 37RECh. 12 - Prob. 38RECh. 12 - Prob. 39RECh. 12 - Prob. 40RECh. 12 - Prob. 41RECh. 12 - Sigma Notation Write the sum without using sigma...Ch. 12 - Prob. 43RECh. 12 - Prob. 44RECh. 12 - Prob. 45RECh. 12 - Prob. 46RECh. 12 - Prob. 47RECh. 12 - Prob. 48RECh. 12 - Prob. 49RECh. 12 - Sums of Arithmetic and Geometric Sequences...Ch. 12 - Prob. 51RECh. 12 - Prob. 52RECh. 12 - Prob. 53RECh. 12 - Prob. 54RECh. 12 - Prob. 55RECh. 12 - Prob. 56RECh. 12 - Prob. 57RECh. 12 - Prob. 58RECh. 12 - Infinite Geometric Series Determine whether the...Ch. 12 - Prob. 60RECh. 12 - Terms of an Arithmetic Sequence The first term of...Ch. 12 - Prob. 62RECh. 12 - Ancestors A person has two parents, four...Ch. 12 - Prob. 64RECh. 12 - Prob. 65RECh. 12 - Prob. 66RECh. 12 - Prob. 67RECh. 12 - Prob. 68RECh. 12 - Prob. 69RECh. 12 - Prob. 70RECh. 12 - Prob. 71RECh. 12 - Prob. 72RECh. 12 - Prob. 73RECh. 12 - Prob. 74RECh. 12 - Prob. 75RECh. 12 - Prob. 76RECh. 12 - Prob. 77RECh. 12 - Prob. 78RECh. 12 - Prob. 79RECh. 12 - Prob. 80RECh. 12 - Prob. 81RECh. 12 - Prob. 82RECh. 12 - Prob. 83RECh. 12 - Prob. 1TCh. 12 - Prob. 2TCh. 12 - Prob. 3TCh. 12 - A geometric sequence begins 12,3,34,316,364,. (a)...Ch. 12 - The first term of a geometric sequence is 25, and...Ch. 12 - The first term of an arithmetic sequence is 10,...Ch. 12 - Prob. 7TCh. 12 - Write the expression without using sigma notation,...Ch. 12 - Prob. 9TCh. 12 - Prob. 11TCh. 12 - Prob. 12TCh. 12 - A puppy weighs 0.85 lb at birth, and each week he...Ch. 12 - Prob. 1PCh. 12 - Fitness Program Sheila decides to embark on a...Ch. 12 - Monthly Savings Program Alice opens a savings...Ch. 12 - Prob. 4PCh. 12 - Prob. 5P
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