CALCULUS (CLOTH)
CALCULUS (CLOTH)
4th Edition
ISBN: 9781319050733
Author: Rogawski
Publisher: MAC HIGHER
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Chapter 16.2, Problem 62E
To determine

The mathematical proof of the inequality 43D(x2+y2)dA203 where D is the domain bounded by y=x2+1 and y=2

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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 16 Solutions

CALCULUS (CLOTH)

Ch. 16.1 - Prob. 5ECh. 16.1 - Prob. 6ECh. 16.1 - Prob. 7ECh. 16.1 - Prob. 8ECh. 16.1 - Prob. 9ECh. 16.1 - Prob. 10ECh. 16.1 - Prob. 11ECh. 16.1 - Prob. 12ECh. 16.1 - Prob. 13ECh. 16.1 - Prob. 14ECh. 16.1 - Prob. 15ECh. 16.1 - Prob. 16ECh. 16.1 - Prob. 17ECh. 16.1 - Prob. 18ECh. 16.1 - Prob. 19ECh. 16.1 - Prob. 20ECh. 16.1 - Prob. 21ECh. 16.1 - Prob. 22ECh. 16.1 - Prob. 23ECh. 16.1 - Prob. 24ECh. 16.1 - Prob. 25ECh. 16.1 - Prob. 26ECh. 16.1 - Prob. 27ECh. 16.1 - Prob. 28ECh. 16.1 - Prob. 29ECh. 16.1 - Prob. 30ECh. 16.1 - Prob. 31ECh. 16.1 - Prob. 32ECh. 16.1 - Prob. 33ECh. 16.1 - Prob. 34ECh. 16.1 - Prob. 35ECh. 16.1 - Prob. 36ECh. 16.1 - Prob. 37ECh. 16.1 - Prob. 38ECh. 16.1 - Prob. 39ECh. 16.1 - Prob. 40ECh. 16.1 - Prob. 41ECh. 16.1 - Prob. 42ECh. 16.1 - Prob. 43ECh. 16.1 - Prob. 44ECh. 16.1 - Prob. 45ECh. 16.1 - Prob. 46ECh. 16.1 - Prob. 47ECh. 16.1 - Prob. 48ECh. 16.1 - Prob. 49ECh. 16.1 - Prob. 50ECh. 16.1 - Prob. 51ECh. 16.1 - Prob. 52ECh. 16.1 - Prob. 53ECh. 16.1 - Prob. 54ECh. 16.1 - Prob. 55ECh. 16.2 - Prob. 1PQCh. 16.2 - Prob. 2PQCh. 16.2 - Prob. 3PQCh. 16.2 - Prob. 4PQCh. 16.2 - Prob. 1ECh. 16.2 - Prob. 2ECh. 16.2 - Prob. 3ECh. 16.2 - Prob. 4ECh. 16.2 - Prob. 5ECh. 16.2 - Prob. 6ECh. 16.2 - Prob. 7ECh. 16.2 - Prob. 8ECh. 16.2 - Prob. 9ECh. 16.2 - Prob. 10ECh. 16.2 - Prob. 11ECh. 16.2 - Prob. 12ECh. 16.2 - Prob. 13ECh. 16.2 - Prob. 14ECh. 16.2 - Prob. 15ECh. 16.2 - Prob. 16ECh. 16.2 - Prob. 17ECh. 16.2 - Prob. 18ECh. 16.2 - Prob. 19ECh. 16.2 - Prob. 20ECh. 16.2 - Prob. 21ECh. 16.2 - Prob. 22ECh. 16.2 - Prob. 23ECh. 16.2 - Prob. 24ECh. 16.2 - Prob. 25ECh. 16.2 - Prob. 26ECh. 16.2 - Prob. 27ECh. 16.2 - Prob. 28ECh. 16.2 - Prob. 29ECh. 16.2 - Prob. 30ECh. 16.2 - Prob. 31ECh. 16.2 - Prob. 32ECh. 16.2 - Prob. 33ECh. 16.2 - Prob. 34ECh. 16.2 - Prob. 35ECh. 16.2 - Prob. 36ECh. 16.2 - Prob. 37ECh. 16.2 - Prob. 38ECh. 16.2 - Prob. 39ECh. 16.2 - Prob. 40ECh. 16.2 - Prob. 41ECh. 16.2 - Prob. 42ECh. 16.2 - Prob. 43ECh. 16.2 - Prob. 44ECh. 16.2 - Prob. 45ECh. 16.2 - Prob. 46ECh. 16.2 - Prob. 47ECh. 16.2 - Prob. 48ECh. 16.2 - Prob. 49ECh. 16.2 - Prob. 50ECh. 16.2 - Prob. 51ECh. 16.2 - Prob. 52ECh. 16.2 - Prob. 53ECh. 16.2 - Prob. 54ECh. 16.2 - Prob. 55ECh. 16.2 - Prob. 56ECh. 16.2 - Prob. 57ECh. 16.2 - Prob. 58ECh. 16.2 - Prob. 59ECh. 16.2 - Prob. 60ECh. 16.2 - Prob. 61ECh. 16.2 - Prob. 62ECh. 16.2 - Prob. 63ECh. 16.2 - Prob. 64ECh. 16.2 - Prob. 65ECh. 16.2 - Prob. 66ECh. 16.2 - Prob. 67ECh. 16.2 - Prob. 68ECh. 16.2 - Prob. 69ECh. 16.2 - Prob. 70ECh. 16.3 - Prob. 1PQCh. 16.3 - Prob. 2PQCh. 16.3 - Prob. 3PQCh. 16.3 - Prob. 1ECh. 16.3 - Prob. 2ECh. 16.3 - Prob. 3ECh. 16.3 - Prob. 4ECh. 16.3 - Prob. 5ECh. 16.3 - Prob. 6ECh. 16.3 - Prob. 7ECh. 16.3 - Prob. 8ECh. 16.3 - Prob. 9ECh. 16.3 - Prob. 10ECh. 16.3 - Prob. 11ECh. 16.3 - Prob. 12ECh. 16.3 - Prob. 13ECh. 16.3 - Prob. 14ECh. 16.3 - Prob. 15ECh. 16.3 - Prob. 16ECh. 16.3 - Prob. 17ECh. 16.3 - Prob. 18ECh. 16.3 - Prob. 19ECh. 16.3 - Prob. 20ECh. 16.3 - Prob. 21ECh. 16.3 - Prob. 22ECh. 16.3 - Prob. 23ECh. 16.3 - Prob. 24ECh. 16.3 - Prob. 25ECh. 16.3 - Prob. 26ECh. 16.3 - Prob. 27ECh. 16.3 - Prob. 28ECh. 16.3 - Prob. 29ECh. 16.3 - Prob. 30ECh. 16.3 - Prob. 31ECh. 16.3 - Prob. 32ECh. 16.3 - Prob. 33ECh. 16.3 - Prob. 34ECh. 16.3 - Prob. 35ECh. 16.3 - Prob. 36ECh. 16.3 - Prob. 37ECh. 16.3 - Prob. 38ECh. 16.3 - Prob. 39ECh. 16.3 - Prob. 40ECh. 16.3 - Prob. 41ECh. 16.3 - Prob. 42ECh. 16.3 - Prob. 43ECh. 16.3 - Prob. 44ECh. 16.3 - Prob. 45ECh. 16.3 - Prob. 46ECh. 16.4 - Prob. 1PQCh. 16.4 - Prob. 2PQCh. 16.4 - Prob. 3PQCh. 16.4 - Prob. 4PQCh. 16.4 - Prob. 5PQCh. 16.4 - Prob. 1ECh. 16.4 - Prob. 2ECh. 16.4 - Prob. 3ECh. 16.4 - Prob. 4ECh. 16.4 - Prob. 5ECh. 16.4 - Prob. 6ECh. 16.4 - Prob. 7ECh. 16.4 - Prob. 8ECh. 16.4 - Prob. 9ECh. 16.4 - Prob. 10ECh. 16.4 - Prob. 11ECh. 16.4 - Prob. 12ECh. 16.4 - Prob. 13ECh. 16.4 - Prob. 14ECh. 16.4 - Prob. 15ECh. 16.4 - Prob. 16ECh. 16.4 - Prob. 17ECh. 16.4 - Prob. 18ECh. 16.4 - Prob. 19ECh. 16.4 - Prob. 20ECh. 16.4 - Prob. 21ECh. 16.4 - Prob. 22ECh. 16.4 - Prob. 23ECh. 16.4 - Prob. 24ECh. 16.4 - Prob. 25ECh. 16.4 - Prob. 26ECh. 16.4 - Prob. 27ECh. 16.4 - Prob. 28ECh. 16.4 - Prob. 29ECh. 16.4 - Prob. 30ECh. 16.4 - Prob. 31ECh. 16.4 - Prob. 32ECh. 16.4 - Prob. 33ECh. 16.4 - Prob. 34ECh. 16.4 - Prob. 35ECh. 16.4 - Prob. 36ECh. 16.4 - Prob. 37ECh. 16.4 - Prob. 38ECh. 16.4 - Prob. 39ECh. 16.4 - Prob. 40ECh. 16.4 - Prob. 41ECh. 16.4 - Prob. 42ECh. 16.4 - Prob. 43ECh. 16.4 - Prob. 44ECh. 16.4 - Prob. 45ECh. 16.4 - Prob. 46ECh. 16.4 - Prob. 47ECh. 16.4 - Prob. 48ECh. 16.4 - Prob. 49ECh. 16.4 - Prob. 50ECh. 16.4 - Prob. 51ECh. 16.4 - Prob. 52ECh. 16.4 - Prob. 53ECh. 16.4 - Prob. 54ECh. 16.4 - Prob. 55ECh. 16.4 - Prob. 56ECh. 16.4 - Prob. 57ECh. 16.4 - Prob. 58ECh. 16.4 - Prob. 59ECh. 16.4 - Prob. 60ECh. 16.5 - Prob. 1PQCh. 16.5 - Prob. 2PQCh. 16.5 - Prob. 3PQCh. 16.5 - Prob. 1ECh. 16.5 - Prob. 2ECh. 16.5 - Prob. 3ECh. 16.5 - Prob. 4ECh. 16.5 - Prob. 5ECh. 16.5 - Prob. 6ECh. 16.5 - Prob. 7ECh. 16.5 - Prob. 8ECh. 16.5 - Prob. 9ECh. 16.5 - Prob. 10ECh. 16.5 - Prob. 11ECh. 16.5 - Prob. 12ECh. 16.5 - Prob. 13ECh. 16.5 - Prob. 14ECh. 16.5 - Prob. 15ECh. 16.5 - Prob. 16ECh. 16.5 - Prob. 17ECh. 16.5 - Prob. 18ECh. 16.5 - Prob. 19ECh. 16.5 - Prob. 20ECh. 16.5 - Prob. 21ECh. 16.5 - Prob. 22ECh. 16.5 - Prob. 23ECh. 16.5 - Prob. 24ECh. 16.5 - Prob. 25ECh. 16.5 - Prob. 26ECh. 16.5 - Prob. 27ECh. 16.5 - Prob. 28ECh. 16.5 - Prob. 29ECh. 16.5 - Prob. 30ECh. 16.5 - Prob. 31ECh. 16.5 - Prob. 32ECh. 16.5 - Prob. 33ECh. 16.5 - Prob. 34ECh. 16.5 - Prob. 35ECh. 16.5 - Prob. 36ECh. 16.5 - Prob. 37ECh. 16.5 - Prob. 38ECh. 16.5 - Prob. 39ECh. 16.5 - Prob. 40ECh. 16.5 - Prob. 41ECh. 16.5 - Prob. 42ECh. 16.5 - Prob. 43ECh. 16.5 - Prob. 44ECh. 16.5 - Prob. 45ECh. 16.5 - Prob. 46ECh. 16.5 - Prob. 47ECh. 16.5 - Prob. 48ECh. 16.5 - Prob. 49ECh. 16.5 - Prob. 50ECh. 16.5 - Prob. 51ECh. 16.5 - Prob. 52ECh. 16.5 - Prob. 53ECh. 16.5 - Prob. 54ECh. 16.5 - Prob. 55ECh. 16.5 - Prob. 56ECh. 16.5 - Prob. 57ECh. 16.5 - Prob. 58ECh. 16.5 - Prob. 59ECh. 16.5 - Prob. 60ECh. 16.5 - Prob. 61ECh. 16.5 - Prob. 62ECh. 16.5 - Prob. 63ECh. 16.5 - Prob. 64ECh. 16.5 - Prob. 65ECh. 16.5 - Prob. 66ECh. 16.6 - Prob. 1PQCh. 16.6 - Prob. 2PQCh. 16.6 - Prob. 3PQCh. 16.6 - Prob. 4PQCh. 16.6 - Prob. 1ECh. 16.6 - Prob. 2ECh. 16.6 - Prob. 3ECh. 16.6 - Prob. 4ECh. 16.6 - Prob. 5ECh. 16.6 - Prob. 6ECh. 16.6 - Prob. 7ECh. 16.6 - Prob. 8ECh. 16.6 - Prob. 9ECh. 16.6 - Prob. 10ECh. 16.6 - Prob. 11ECh. 16.6 - Prob. 12ECh. 16.6 - Prob. 13ECh. 16.6 - Prob. 14ECh. 16.6 - Prob. 15ECh. 16.6 - Prob. 16ECh. 16.6 - Prob. 17ECh. 16.6 - Prob. 18ECh. 16.6 - Prob. 19ECh. 16.6 - Prob. 20ECh. 16.6 - Prob. 21ECh. 16.6 - Prob. 22ECh. 16.6 - Prob. 23ECh. 16.6 - Prob. 24ECh. 16.6 - Prob. 25ECh. 16.6 - Prob. 26ECh. 16.6 - Prob. 27ECh. 16.6 - Prob. 28ECh. 16.6 - Prob. 29ECh. 16.6 - Prob. 30ECh. 16.6 - Prob. 31ECh. 16.6 - Prob. 32ECh. 16.6 - Prob. 33ECh. 16.6 - Prob. 34ECh. 16.6 - Prob. 35ECh. 16.6 - Prob. 36ECh. 16.6 - Prob. 37ECh. 16.6 - Prob. 38ECh. 16.6 - Prob. 39ECh. 16.6 - Prob. 40ECh. 16.6 - Prob. 41ECh. 16.6 - Prob. 42ECh. 16.6 - Prob. 43ECh. 16.6 - Prob. 44ECh. 16.6 - Prob. 45ECh. 16.6 - Prob. 46ECh. 16.6 - Prob. 47ECh. 16.6 - Prob. 48ECh. 16.6 - Prob. 49ECh. 16.6 - Prob. 50ECh. 16.6 - Prob. 51ECh. 16.6 - Prob. 52ECh. 16 - Prob. 1CRECh. 16 - Prob. 2CRECh. 16 - Prob. 3CRECh. 16 - Prob. 4CRECh. 16 - Prob. 5CRECh. 16 - Prob. 6CRECh. 16 - Prob. 7CRECh. 16 - Prob. 8CRECh. 16 - Prob. 9CRECh. 16 - Prob. 10CRECh. 16 - Prob. 11CRECh. 16 - Prob. 12CRECh. 16 - Prob. 13CRECh. 16 - Prob. 14CRECh. 16 - Prob. 15CRECh. 16 - Prob. 16CRECh. 16 - Prob. 17CRECh. 16 - Prob. 18CRECh. 16 - Prob. 19CRECh. 16 - Prob. 20CRECh. 16 - Prob. 21CRECh. 16 - Prob. 22CRECh. 16 - Prob. 23CRECh. 16 - Prob. 24CRECh. 16 - Prob. 25CRECh. 16 - Prob. 26CRECh. 16 - Prob. 27CRECh. 16 - Prob. 28CRECh. 16 - Prob. 29CRECh. 16 - Prob. 30CRECh. 16 - Prob. 31CRECh. 16 - Prob. 32CRECh. 16 - Prob. 33CRECh. 16 - Prob. 34CRECh. 16 - Prob. 35CRECh. 16 - Prob. 36CRECh. 16 - Prob. 37CRECh. 16 - Prob. 38CRECh. 16 - Prob. 39CRECh. 16 - Prob. 40CRECh. 16 - Prob. 41CRECh. 16 - Prob. 42CRECh. 16 - Prob. 43CRECh. 16 - Prob. 44CRECh. 16 - Prob. 45CRECh. 16 - Prob. 46CRECh. 16 - Prob. 47CRECh. 16 - Prob. 48CRECh. 16 - Prob. 49CRECh. 16 - Prob. 50CRECh. 16 - Prob. 51CRECh. 16 - Prob. 52CRECh. 16 - Prob. 53CRECh. 16 - Prob. 54CRECh. 16 - Prob. 55CRECh. 16 - Prob. 56CRECh. 16 - Prob. 57CRECh. 16 - Prob. 58CRECh. 16 - Prob. 59CRECh. 16 - Prob. 60CRECh. 16 - Prob. 61CRE
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