Explanation Given information: The equation that relates federal deficit of the government (in trillions of dollars) with time (in year) is given by, D ( t ) = − 0.038898 t 3 + 0.30858 t 2 − 0.31849 t + 0.22 ( 0 ≤ t ≤ 6 ) (1) Where, t is the time in years. The time t = 0 corresponds to year 2006. Definition used: Second derivative test: Put the point value of variable obtained by First derivative test in the second derivative of the equation, if, (a) The numeric value of second derivative comes positive, then it is a point of minima . (b) The numeric value of second derivative comes negative, then it is a point of maxima . Formula used : Power rule of differentiation , d d x ( x n ) = n x n − 1 . Calculation : First derivative test: Equate the first derivative of an equation to zero; it gives the point of extrema (either maxima or minima). Use power rule to differentiate equation (1) with respect to t . D ′ ( t ) = d d t ( − 0.038898 t 3 + 0.30858 t 2 − 0.31849 t + 0.22 ) = d ( − 0.038898 t 3 ) d t + d ( 0.30858 t 2 ) d t + d ( − 0.31849 t ) d t + d ( 0.22 ) d t = − 0.116694 t 2 + 0.61716 t − 0.31849 + 0 = − 0.116694 t 2 + 0.61716 t − 0.31849 First derivative of federal deficit gives the rate of change of federal deficit. D ′ ( t ) = − 0.116694 t 2 + 0.61716 t − 0.31849 . (2) Equate above equation to 0. − 0.116694 t 2 + 0.61716 t − 0.31849 = 0 0.116694 t 2 − 0.61716 t + 0.31849 = 0 The general form of a quadratic equation is, a x 2 + b x + c = 0 The solution of above equation is given by, x = − b ± b 2 − 4 a c 2 a (3) Compare given equation with the general form to obtain the values of coefficients, The value of a is 0.116694 , b is − 0.61716 and c is 0.31849 . Substitute 0.116694 for a , − 0.61716 for b and 0.31849 for c in equation (3) to get the roots of the equation. x = 0.61716 ± ( − 0.61716 ) 2 − 4 ( 0.116694 ) ( 0.31849 ) 2 ( 0

Applied Calculus for the Managerial, Life, and Social Sciences (MindTap Course List)

10th Edition

ISBN: 9781305657861

Author: Soo T. Tan

Publisher: Cengage Learning

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Chapter 4.4, Problem 64E

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The largest federal deficit between years 2006 to 2010.

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Chapter 4.4, Problem 63E

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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.

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Chapter 4 Solutions

Applied Calculus for the Managerial, Life, and Social Sciences (MindTap Course List)

Ch. 4.1 - Explain each of the following: a. f is increasing...Ch. 4.1 - Prob. 2CQ Ch. 4.1 - Prob. 3CQ Ch. 4.1 - Prob. 4CQ Ch. 4.1 - Prob. 5CQ Ch. 4.1 - Prob. 1E Ch. 4.1 - Prob. 2E Ch. 4.1 - Prob. 3E Ch. 4.1 - Prob. 4E Ch. 4.1 - Prob. 5E

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The largest federal deficit between years 2006 to 2010.

Solution Summary: The author explains that the largest federal deficit between years 2006 to 2010 is 1.50.

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