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Unformatted text preview: Chapter 1
the diﬀerential equation dy/dx = x + sin y . Thus d dx ⇒ dy dx = d (x + sin y ); dx dy (chain rule) = 1 + (cos y ) dx = 1 + (cos y )(x + sin y ) = 1 + x cos y + sin y cos y ; 1 = 1 + x cos y + sin 2y. 2 d2 y dx2 d2 y dx2 ⇒ (d) Relative minima occur when the ﬁrst derivative, dy/dx, is equal to zero and the second derivative, d2 y/dx2 , is greater than zero. The value of the ﬁrst derivative at the point (0, 0) is given by dy = 0 + sin 0 = 0. dx This tells us that the solution has a critical point at the point (0, 0). Using the second derivative found in part (c) we have d2 y 1 = 1 + 0 · cos 0 + sin 0 = 1. 2 dx 2 This tells us the point (0, 0) is a relative minimum. 7. (a) The graph of the directional ﬁeld is shown in Figure B.5 in the answers section of the text. (b) The direction ﬁeld indicates that all solution curves with p(0) > 1 will approach the horizontal line (asymptote) p = 2 as t → +∞. Thus limt→+∞ p(t) = 2 when p(0) = 3. (c) The direction ﬁeld shows that a population between 1000 and 2000 (that is 1 < p(0) < 2) will approach the horizontal line p = 2 as t → +∞. (d) The direction ﬁeld shows that an initial population less than 1000 (that is 0 ≤ p(0) < 1) will approach zero as t → +∞. (e) As noted in part (d), the line p = 1 is an asymptote. The direction ﬁeld indicates that a population of 900 (p(0) = 0.9) steadily decreases with time and therefore cannot increase to 1100. 12 ...
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