Pearson eText for Calculus & Its Applications -- Instant Access (Pearson+)
14th Edition
ISBN: 9780137400096
Author: Larry Goldstein, David Lay
Publisher: PEARSON+
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Chapter 11.4, Problem 5E
To determine
Theinfinite series
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Students have asked these similar questions
1. 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.
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?
Question 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.
Chapter 11 Solutions
Pearson eText for Calculus & Its Applications -- Instant Access (Pearson+)
Ch. 11.1 - Determine the third Taylor polynomial of f(x)=cosx...Ch. 11.1 - Prob. 2CYUCh. 11.1 - Prob. 1ECh. 11.1 - Prob. 2ECh. 11.1 - Prob. 3ECh. 11.1 - Prob. 4ECh. 11.1 - Prob. 5ECh. 11.1 - Prob. 6ECh. 11.1 - Prob. 7ECh. 11.1 - Prob. 8E
Ch. 11.1 - Prob. 9ECh. 11.1 - Prob. 10ECh. 11.1 - Prob. 11ECh. 11.1 - Prob. 12ECh. 11.1 - Prob. 13ECh. 11.1 - Prob. 14ECh. 11.1 - Prob. 15ECh. 11.1 - Prob. 16ECh. 11.1 - Prob. 17ECh. 11.1 - Prob. 18ECh. 11.1 - Determine the third and fourthTaylor polynomial...Ch. 11.1 - Prob. 20ECh. 11.1 - Prob. 21ECh. 11.1 - Prob. 22ECh. 11.1 - Prob. 23ECh. 11.1 - Prob. 24ECh. 11.1 - Prob. 25ECh. 11.1 - Prob. 26ECh. 11.1 - Prob. 27ECh. 11.1 - Prob. 28ECh. 11.1 - Prob. 29ECh. 11.1 - Prob. 30ECh. 11.1 - Graph the function Y1=11x and its fourth Taylor...Ch. 11.1 - Prob. 32ECh. 11.1 - Prob. 33ECh. 11.1 - Prob. 34ECh. 11.2 - Prob. 1CYUCh. 11.2 - Prob. 2CYUCh. 11.2 - In Exercises 18, use three repetitions of the...Ch. 11.2 - In Exercises 18, use three repetitions of the...Ch. 11.2 - Prob. 3ECh. 11.2 - Prob. 4ECh. 11.2 - In Exercises 18, use three repetitions of the...Ch. 11.2 - Prob. 6ECh. 11.2 - Prob. 7ECh. 11.2 - Prob. 8ECh. 11.2 - Sketch the graph of y=x3+2x+2, and use the...Ch. 11.2 - Prob. 10ECh. 11.2 - Prob. 11ECh. 11.2 - Prob. 12ECh. 11.2 - Prob. 13ECh. 11.2 - Internet Rate of Return An investor buys a bond...Ch. 11.2 - Prob. 15ECh. 11.2 - Prob. 16ECh. 11.2 - Prob. 17ECh. 11.2 - Prob. 18ECh. 11.2 - Prob. 19ECh. 11.2 - Prob. 20ECh. 11.2 - Prob. 21ECh. 11.2 - Figure 9contains the graph of the function...Ch. 11.2 - Prob. 23ECh. 11.2 - Prob. 24ECh. 11.2 - Exercises 25 and 26 present two examples in which...Ch. 11.2 - Prob. 26ECh. 11.2 - Prob. 27ECh. 11.2 - Prob. 28ECh. 11.2 - Prob. 29ECh. 11.2 - Prob. 30ECh. 11.3 - Determine the sum of the geometric series...Ch. 11.3 - Prob. 2CYUCh. 11.3 - Determine the sums of the following geometric...Ch. 11.3 - Prob. 2ECh. 11.3 - Prob. 3ECh. 11.3 - Determine the sums of the following geometric...Ch. 11.3 - Prob. 5ECh. 11.3 - Determine the sums of the following geometric...Ch. 11.3 - Prob. 7ECh. 11.3 - Prob. 8ECh. 11.3 - Prob. 9ECh. 11.3 - Prob. 10ECh. 11.3 - Prob. 11ECh. 11.3 - Prob. 12ECh. 11.3 - Prob. 13ECh. 11.3 - Prob. 14ECh. 11.3 - Prob. 15ECh. 11.3 - Sum an appropriate infinite series to find the...Ch. 11.3 - Prob. 17ECh. 11.3 - Sum an appropriate infinite series to find the...Ch. 11.3 - Prob. 19ECh. 11.3 - Prob. 20ECh. 11.3 - Prob. 21ECh. 11.3 - Prob. 22ECh. 11.3 - Prob. 23ECh. 11.3 - The Multiplier Effect Compute the effect of a 20...Ch. 11.3 - Perpetuity Consider a perpetuity that promises to...Ch. 11.3 - Prob. 26ECh. 11.3 - Bonus plus Taxes on Taxes A generous corporation...Ch. 11.3 - Total Distance Travelled by a Bouncing Ball The...Ch. 11.3 - Elimination of a Drug A patient receives 6 mg of a...Ch. 11.3 - Elimination of a Drug A patient receives 2 mg of a...Ch. 11.3 - Drug Dosage A patient receives M mg of a certain...Ch. 11.3 - Drug Dosage A patient receives M mg of a certain...Ch. 11.3 - Prob. 33ECh. 11.3 - The infinite series a1+a2+a3+ has partial sums...Ch. 11.3 - Prob. 35ECh. 11.3 - Prob. 36ECh. 11.3 - Prob. 37ECh. 11.3 - Determine the sums of the following infinite...Ch. 11.3 - Prob. 39ECh. 11.3 - Prob. 40ECh. 11.3 - Prob. 41ECh. 11.3 - Prob. 42ECh. 11.3 - Prob. 43ECh. 11.3 - Prob. 44ECh. 11.3 - Prob. 45ECh. 11.3 - Prob. 46ECh. 11.3 - Prob. 47ECh. 11.3 - Prob. 48ECh. 11.3 - Prob. 49ECh. 11.3 - In Exercises 49 and 50, convince yourself that the...Ch. 11.4 - What is the improper integral associated with the...Ch. 11.4 - Prob. 2CYUCh. 11.4 - Prob. 1ECh. 11.4 - Prob. 2ECh. 11.4 - Prob. 3ECh. 11.4 - Prob. 4ECh. 11.4 - Prob. 5ECh. 11.4 - Prob. 6ECh. 11.4 - Prob. 7ECh. 11.4 - Prob. 8ECh. 11.4 - Prob. 9ECh. 11.4 - Prob. 10ECh. 11.4 - In Exercises 116, use the integral test to...Ch. 11.4 - Prob. 12ECh. 11.4 - Prob. 13ECh. 11.4 - Prob. 14ECh. 11.4 - Prob. 15ECh. 11.4 - In Exercises 116, use the integral test to...Ch. 11.4 - Prob. 17ECh. 11.4 - Prob. 18ECh. 11.4 - Prob. 19ECh. 11.4 - Prob. 20ECh. 11.4 - In Excercises 2126, use the comparison test to...Ch. 11.4 - Prob. 22ECh. 11.4 - Prob. 23ECh. 11.4 - Prob. 24ECh. 11.4 - Prob. 25ECh. 11.4 - Prob. 26ECh. 11.4 - Prob. 27ECh. 11.4 - Prob. 28ECh. 11.4 - Prob. 29ECh. 11.4 - Prob. 30ECh. 11.4 - Use Exercise 29 to show that the series...Ch. 11.4 - Use Exercise 30 to show that the series k=13k2 is...Ch. 11.5 - Find the Taylor series expansion of sinx at x=0.Ch. 11.5 - Find the Taylor series expansion of cosx at x=0.Ch. 11.5 - Prob. 3CYUCh. 11.5 - Prob. 4CYUCh. 11.5 - Prob. 1ECh. 11.5 - Prob. 2ECh. 11.5 - Prob. 3ECh. 11.5 - In Exercises 14, find the Taylor series at x=0 of...Ch. 11.5 - Prob. 5ECh. 11.5 - Prob. 6ECh. 11.5 - Prob. 7ECh. 11.5 - Prob. 8ECh. 11.5 - Prob. 9ECh. 11.5 - In Exercises 520, find the Taylor series at x=0 of...Ch. 11.5 - Prob. 11ECh. 11.5 - Prob. 12ECh. 11.5 - Prob. 13ECh. 11.5 - Prob. 14ECh. 11.5 - In Exercises 520, find the Taylor series at x=0 of...Ch. 11.5 - Prob. 16ECh. 11.5 - Prob. 17ECh. 11.5 - Prob. 18ECh. 11.5 - Prob. 19ECh. 11.5 - In Exercises 520, find the Taylor series at x=0 of...Ch. 11.5 - Find the Taylor series of xex2 at x=0.Ch. 11.5 - Prob. 22ECh. 11.5 - Prob. 23ECh. 11.5 - Prob. 24ECh. 11.5 - Prob. 25ECh. 11.5 - Prob. 26ECh. 11.5 - Prob. 27ECh. 11.5 - Prob. 28ECh. 11.5 - Prob. 29ECh. 11.5 - Prob. 30ECh. 11.5 - Prob. 31ECh. 11.5 - Prob. 32ECh. 11.5 - Prob. 33ECh. 11.5 - The Taylor series at x=0 for 1+x21x is...Ch. 11.5 - Prob. 35ECh. 11.5 - Prob. 36ECh. 11.5 - Prob. 37ECh. 11.5 - Prob. 38ECh. 11.5 - In Exercises 3840, find the infinite series that...Ch. 11.5 - Prob. 40ECh. 11.5 - Prob. 41ECh. 11.5 - Prob. 42ECh. 11.5 - Prob. 43ECh. 11.5 - Prob. 44ECh. 11.5 - Prob. 45ECh. 11.5 - Prob. 46ECh. 11 - Prob. 1CCECh. 11 - Prob. 2CCECh. 11 - Prob. 3CCECh. 11 - Prob. 4CCECh. 11 - Prob. 5CCECh. 11 - Prob. 6CCECh. 11 - What is meant by the sum of a convergent infinite...Ch. 11 - Prob. 8CCECh. 11 - Prob. 9CCECh. 11 - Prob. 10CCECh. 11 - Prob. 11CCECh. 11 - Prob. 1RECh. 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 - Use the third Taylor polynomial of ln(1x) at x=0...Ch. 11 - Prob. 11RECh. 11 - Prob. 12RECh. 11 - In Exercise 1320, find the sum of the given...Ch. 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 - Prob. 26RECh. 11 - Prob. 27RECh. 11 - Prob. 28RECh. 11 - Prob. 29RECh. 11 - In Exercise 2932, find the Taylor series at x=0 of...Ch. 11 - Prob. 31RECh. 11 - Prob. 32RECh. 11 - Fine the Taylor series of cos2x at x=0, either by...Ch. 11 - Prob. 34RECh. 11 - Prob. 35RECh. 11 - Prob. 36RECh. 11 - Prob. 37RECh. 11 - Prob. 38RECh. 11 - Prob. 39RECh. 11 - Prob. 40RECh. 11 - Prob. 41RECh. 11 - Prob. 42RECh. 11 - Prob. 43RECh. 11 - Prob. 44RECh. 11 - Prob. 45RE
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