11th Edition

ISBN: 9780136845881

Author: Sullivan

Publisher: VST

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Chapter 2.3, Problem 82AYU

National Debt The size of the total debt owed by the United States federal government continues to grow. In fact, according to the Department of the Treasury, the debt per person living in the United States is approximately $ 63 , 720 (or over $ 172 , 000 per U.S. household). The following data represent the U.S. debt for the years 2007 − 2017 . Since the debt D depends on the year y , and each input corresponds to exactly one output, the debt is a function of the year. So D ( y ) represents the debt for each year y .

Year Debt(billions of dollars) 2007 9008 2008 10 , 025 2009 11 , 910 2010 13 , 562 2011 14 , 790 2012 16 , 066 2013 16 , 738 2014 17 , 824 2015 18 , 151 2016 19 , 573 2017 20 , 245

Plot the points ( 2007 , 9008 ) , ( 2008 , 10 025 ) , and so on.

Draw a line segment from the point ( 2007 , 9008 ) to ( 2012 , 16 066 ) . What does the slope of this line segment represent?

Find the average rate of change of the debt from 2008 to 2010 .

Find the average rate of change of the debt from 2011 to 2013 .

Find the average rate of change of the debt from 2014 to 2016 .

What appears to be happening to the average rate of change as time passes?

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Chapter 2.3, Problem 81AYU

Chapter 2.3, Problem 83AYU

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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.

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Solution Summary: The author plots the U.S debt for the years 2007- 2017, and the debt is a function D(y) for each year y.

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