Applied Calculus, 6e WileyPLUS + Loose-leaf
Applied Calculus, 6e WileyPLUS + Loose-leaf
6th Edition
ISBN: 9781119408901
Author: Deborah Hughes-Hallett, Andrew M. Gleason, Patti Frazer Lock, Daniel E. Flath
Publisher: Wiley (WileyPLUS Products)
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Chapter 4.7, Problem 2P
To determine

To find the largest number of cases in country.

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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 4 Solutions

Applied Calculus, 6e WileyPLUS + Loose-leaf

Ch. 4.1 - Prob. 11PCh. 4.1 - Prob. 12PCh. 4.1 - Prob. 13PCh. 4.1 - Prob. 14PCh. 4.1 - Prob. 15PCh. 4.1 - Prob. 16PCh. 4.1 - Prob. 17PCh. 4.1 - Prob. 18PCh. 4.1 - Prob. 19PCh. 4.1 - Prob. 20PCh. 4.1 - Prob. 21PCh. 4.1 - Prob. 22PCh. 4.1 - Prob. 23PCh. 4.1 - Prob. 24PCh. 4.1 - Prob. 25PCh. 4.1 - Prob. 26PCh. 4.1 - Prob. 27PCh. 4.1 - Prob. 28PCh. 4.1 - Prob. 29PCh. 4.1 - Prob. 30PCh. 4.1 - Prob. 31PCh. 4.1 - Prob. 32PCh. 4.1 - Prob. 33PCh. 4.1 - Prob. 34PCh. 4.1 - Prob. 35PCh. 4.1 - Prob. 36PCh. 4.1 - Prob. 37PCh. 4.1 - Prob. 38PCh. 4.1 - Prob. 39PCh. 4.1 - Prob. 40PCh. 4.1 - Prob. 41PCh. 4.1 - Prob. 42PCh. 4.1 - Prob. 43PCh. 4.1 - Prob. 44PCh. 4.1 - Prob. 45PCh. 4.1 - Prob. 46PCh. 4.1 - Prob. 47PCh. 4.1 - Prob. 48PCh. 4.1 - Prob. 49PCh. 4.1 - Prob. 50PCh. 4.1 - Prob. 51PCh. 4.1 - Prob. 52PCh. 4.1 - Prob. 53PCh. 4.1 - Prob. 54PCh. 4.1 - Prob. 55PCh. 4.1 - Prob. 56PCh. 4.1 - Prob. 57PCh. 4.1 - Prob. 58PCh. 4.2 - Prob. 1PCh. 4.2 - Prob. 2PCh. 4.2 - Prob. 3PCh. 4.2 - Prob. 4PCh. 4.2 - Prob. 5PCh. 4.2 - Prob. 6PCh. 4.2 - Prob. 7PCh. 4.2 - Prob. 8PCh. 4.2 - Prob. 9PCh. 4.2 - Prob. 10PCh. 4.2 - Prob. 11PCh. 4.2 - Prob. 12PCh. 4.2 - Prob. 13PCh. 4.2 - Prob. 14PCh. 4.2 - Prob. 15PCh. 4.2 - Prob. 16PCh. 4.2 - Prob. 17PCh. 4.2 - Prob. 18PCh. 4.2 - Prob. 19PCh. 4.2 - Prob. 20PCh. 4.2 - Prob. 21PCh. 4.2 - Prob. 22PCh. 4.2 - Prob. 23PCh. 4.2 - Prob. 24PCh. 4.2 - Prob. 25PCh. 4.2 - Prob. 26PCh. 4.2 - Prob. 27PCh. 4.2 - Prob. 28PCh. 4.2 - Prob. 29PCh. 4.2 - Prob. 30PCh. 4.2 - Prob. 31PCh. 4.2 - Prob. 32PCh. 4.2 - Prob. 33PCh. 4.2 - Prob. 34PCh. 4.2 - Prob. 35PCh. 4.2 - Prob. 36PCh. 4.2 - Prob. 37PCh. 4.2 - Prob. 38PCh. 4.2 - Prob. 39PCh. 4.2 - Prob. 40PCh. 4.2 - Prob. 41PCh. 4.3 - Prob. 1PCh. 4.3 - Prob. 2PCh. 4.3 - Prob. 3PCh. 4.3 - Prob. 4PCh. 4.3 - Prob. 5PCh. 4.3 - Prob. 6PCh. 4.3 - Prob. 7PCh. 4.3 - Prob. 8PCh. 4.3 - Prob. 9PCh. 4.3 - Prob. 10PCh. 4.3 - Prob. 11PCh. 4.3 - Prob. 12PCh. 4.3 - Prob. 13PCh. 4.3 - Prob. 14PCh. 4.3 - Prob. 15PCh. 4.3 - Prob. 16PCh. 4.3 - Prob. 17PCh. 4.3 - Prob. 18PCh. 4.3 - Prob. 19PCh. 4.3 - Prob. 20PCh. 4.3 - Prob. 21PCh. 4.3 - Prob. 22PCh. 4.3 - Prob. 23PCh. 4.3 - Prob. 24PCh. 4.3 - Prob. 25PCh. 4.3 - Prob. 26PCh. 4.3 - Prob. 27PCh. 4.3 - Prob. 28PCh. 4.3 - Prob. 29PCh. 4.3 - Prob. 30PCh. 4.3 - Prob. 31PCh. 4.3 - Prob. 32PCh. 4.3 - Prob. 33PCh. 4.3 - Prob. 34PCh. 4.3 - Prob. 35PCh. 4.3 - Prob. 36PCh. 4.3 - Prob. 37PCh. 4.3 - Prob. 38PCh. 4.3 - Prob. 39PCh. 4.3 - Prob. 40PCh. 4.3 - Prob. 41PCh. 4.3 - Prob. 42PCh. 4.3 - Prob. 43PCh. 4.3 - Prob. 44PCh. 4.3 - Prob. 45PCh. 4.3 - Prob. 46PCh. 4.3 - Prob. 47PCh. 4.3 - Prob. 48PCh. 4.3 - Prob. 49PCh. 4.3 - Prob. 50PCh. 4.3 - Prob. 51PCh. 4.3 - Prob. 52PCh. 4.3 - Prob. 53PCh. 4.3 - Prob. 54PCh. 4.3 - Prob. 55PCh. 4.3 - Prob. 56PCh. 4.3 - Prob. 57PCh. 4.3 - Prob. 58PCh. 4.3 - Prob. 59PCh. 4.3 - Prob. 60PCh. 4.4 - Prob. 1PCh. 4.4 - Prob. 2PCh. 4.4 - Prob. 3PCh. 4.4 - Prob. 4PCh. 4.4 - Prob. 5PCh. 4.4 - Prob. 6PCh. 4.4 - Prob. 7PCh. 4.4 - Prob. 8PCh. 4.4 - Prob. 9PCh. 4.4 - Prob. 10PCh. 4.4 - Prob. 11PCh. 4.4 - Prob. 12PCh. 4.4 - Prob. 13PCh. 4.4 - Prob. 14PCh. 4.4 - Prob. 15PCh. 4.4 - Prob. 16PCh. 4.4 - Prob. 17PCh. 4.4 - Prob. 18PCh. 4.4 - Prob. 19PCh. 4.4 - Prob. 20PCh. 4.4 - Prob. 21PCh. 4.4 - Prob. 22PCh. 4.4 - Prob. 23PCh. 4.4 - Prob. 24PCh. 4.4 - Prob. 25PCh. 4.4 - Prob. 26PCh. 4.4 - Prob. 27PCh. 4.4 - Prob. 28PCh. 4.4 - Prob. 29PCh. 4.4 - Prob. 30PCh. 4.4 - Prob. 31PCh. 4.4 - Prob. 32PCh. 4.4 - Prob. 33PCh. 4.4 - Prob. 34PCh. 4.4 - Prob. 35PCh. 4.5 - Prob. 1PCh. 4.5 - Prob. 2PCh. 4.5 - Prob. 3PCh. 4.5 - Prob. 4PCh. 4.5 - Prob. 5PCh. 4.5 - Prob. 6PCh. 4.5 - Prob. 7PCh. 4.5 - Prob. 8PCh. 4.5 - Prob. 9PCh. 4.5 - Prob. 10PCh. 4.5 - Prob. 11PCh. 4.5 - Prob. 12PCh. 4.5 - Prob. 13PCh. 4.5 - Prob. 14PCh. 4.5 - Prob. 15PCh. 4.5 - Prob. 16PCh. 4.5 - Prob. 17PCh. 4.6 - Prob. 1PCh. 4.6 - Prob. 2PCh. 4.6 - Prob. 3PCh. 4.6 - Prob. 4PCh. 4.6 - Prob. 5PCh. 4.6 - Prob. 6PCh. 4.6 - Prob. 7PCh. 4.6 - Prob. 8PCh. 4.6 - Prob. 9PCh. 4.6 - Prob. 10PCh. 4.6 - Prob. 11PCh. 4.6 - Prob. 12PCh. 4.6 - Prob. 13PCh. 4.6 - Prob. 14PCh. 4.6 - Prob. 15PCh. 4.6 - Prob. 16PCh. 4.6 - Prob. 17PCh. 4.6 - Prob. 18PCh. 4.6 - Prob. 19PCh. 4.6 - Prob. 20PCh. 4.6 - Prob. 21PCh. 4.6 - Prob. 22PCh. 4.6 - Prob. 23PCh. 4.6 - Prob. 24PCh. 4.6 - Prob. 25PCh. 4.6 - Prob. 26PCh. 4.6 - Prob. 27PCh. 4.6 - Prob. 28PCh. 4.6 - Prob. 29PCh. 4.6 - Prob. 30PCh. 4.6 - Prob. 31PCh. 4.7 - Prob. 1PCh. 4.7 - Prob. 2PCh. 4.7 - Prob. 3PCh. 4.7 - Prob. 4PCh. 4.7 - Prob. 5PCh. 4.7 - Prob. 6PCh. 4.7 - Prob. 7PCh. 4.7 - Prob. 8PCh. 4.7 - Prob. 9PCh. 4.7 - Prob. 10PCh. 4.7 - Prob. 11PCh. 4.7 - Prob. 12PCh. 4.7 - Prob. 13PCh. 4.7 - Prob. 14PCh. 4.7 - Prob. 15PCh. 4.7 - Prob. 16PCh. 4.7 - Prob. 17PCh. 4.7 - Prob. 18PCh. 4.7 - Prob. 19PCh. 4.7 - Prob. 20PCh. 4.7 - Prob. 21PCh. 4.7 - Prob. 22PCh. 4.7 - Prob. 23PCh. 4.7 - Prob. 24PCh. 4.8 - Prob. 1PCh. 4.8 - Prob. 2PCh. 4.8 - Prob. 3PCh. 4.8 - Prob. 4PCh. 4.8 - Prob. 5PCh. 4.8 - Prob. 6PCh. 4.8 - Prob. 7PCh. 4.8 - Prob. 8PCh. 4.8 - Prob. 9PCh. 4.8 - Prob. 10PCh. 4.8 - Prob. 11PCh. 4.8 - Prob. 12PCh. 4.8 - Prob. 13PCh. 4 - Prob. 1SYUCh. 4 - Prob. 2SYUCh. 4 - Prob. 3SYUCh. 4 - Prob. 4SYUCh. 4 - Prob. 5SYUCh. 4 - Prob. 6SYUCh. 4 - Prob. 7SYUCh. 4 - Prob. 8SYUCh. 4 - Prob. 9SYUCh. 4 - Prob. 10SYUCh. 4 - Prob. 11SYUCh. 4 - Prob. 12SYUCh. 4 - Prob. 13SYUCh. 4 - Prob. 14SYUCh. 4 - Prob. 15SYUCh. 4 - Prob. 16SYUCh. 4 - Prob. 17SYUCh. 4 - Prob. 18SYUCh. 4 - Prob. 19SYUCh. 4 - Prob. 20SYUCh. 4 - Prob. 21SYUCh. 4 - Prob. 22SYUCh. 4 - Prob. 23SYUCh. 4 - Prob. 24SYUCh. 4 - Prob. 25SYUCh. 4 - Prob. 26SYUCh. 4 - Prob. 27SYUCh. 4 - Prob. 28SYUCh. 4 - Prob. 29SYUCh. 4 - Prob. 30SYUCh. 4 - Prob. 31SYUCh. 4 - Prob. 32SYUCh. 4 - Prob. 33SYUCh. 4 - Prob. 34SYUCh. 4 - Prob. 35SYUCh. 4 - Prob. 36SYUCh. 4 - Prob. 37SYUCh. 4 - Prob. 38SYUCh. 4 - Prob. 39SYUCh. 4 - Prob. 40SYUCh. 4 - Prob. 41SYUCh. 4 - Prob. 42SYUCh. 4 - Prob. 43SYUCh. 4 - Prob. 44SYUCh. 4 - Prob. 45SYUCh. 4 - Prob. 46SYUCh. 4 - Prob. 47SYUCh. 4 - Prob. 48SYUCh. 4 - Prob. 49SYUCh. 4 - Prob. 50SYUCh. 4 - Prob. 51SYUCh. 4 - Prob. 52SYUCh. 4 - Prob. 53SYUCh. 4 - Prob. 54SYUCh. 4 - Prob. 55SYUCh. 4 - Prob. 56SYUCh. 4 - Prob. 57SYUCh. 4 - Prob. 58SYUCh. 4 - Prob. 59SYUCh. 4 - Prob. 60SYUCh. 4 - Prob. 61SYUCh. 4 - Prob. 62SYUCh. 4 - Prob. 63SYUCh. 4 - Prob. 64SYUCh. 4 - Prob. 65SYUCh. 4 - Prob. 66SYUCh. 4 - Prob. 67SYUCh. 4 - Prob. 68SYUCh. 4 - Prob. 69SYUCh. 4 - Prob. 70SYUCh. 4 - Prob. 71SYUCh. 4 - Prob. 72SYUCh. 4 - Prob. 73SYUCh. 4 - Prob. 74SYUCh. 4 - Prob. 75SYUCh. 4 - Prob. 76SYUCh. 4 - Prob. 77SYUCh. 4 - Prob. 78SYUCh. 4 - Prob. 79SYUCh. 4 - Prob. 80SYU
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