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39. 5422(1.018) 19 < 7609.7 million 40. (a) [ 2 5, 5] by [ 2 2, 10] In this window, it appears they cross twice, although a third crossing off-screen appears likely. (b) It happens by the time x 5 4. (c) Solving graphically, x < 2 0.7667, x 5 2, x 5 4. (d) The solution set is approximately ( 2 0.7667, 2) < (4, ). 41. Since f (1) 5 4.5 we have ka 5 4.5, and since f ( 2 1) 5 0.5 we have ka 2 1 5 0.5. Dividing, we have } ka ka 2 1 } 5 } 4 0 . . 5 5 } a 2 5 9 a 5 6 3 Since f ( x ) 5 k ? a x is an exponential function, we require a . 0, so a 5 3. Then ka 5 4.5 gives 3 k 5 4.5, so k 5 1.5. The values are a 5 3 and k 5 1.5. 42. Since f (1) 5 1.5 we have ka 5 1.5, and since f ( 2 1) 5 6 we have ka 2 1 5 6. Dividing, we have } ka ka 2 1 } 5 } 1 6 .5 } a 2 5 0.25 a 5 6 0.5 Since f ( x ) 5 k ? a x is an exponential function, we require a . 0, so a 5 0.5. Then ka 5 1.5 gives 0.5 k 5 1.5, so k 5 3. The values are a 5 0.5 and k 5 3. Section 1.4 Parametric Equations (pp. 26–31) Exploration 1 Parametrizing Circles 1. Each is a circle with radius ) a ) . As ) a ) increases, the radius of the circle increases. [ 2 4.7, 4.7] by [ 2 3.1, 3.1] 2. 0 # t # } p 2 } : [ 2 4.7, 4.7] by [ 2 3.1, 3.1] 0 # t # p : [ 2 4.7, 4.7] by [ 2 3.1, 3.1] 0 # t # } 3 2 p } : [ 2 4.7, 4.7] by [ 2 3.1, 3.1] 2 p # t # 4 p : [ 2 4.7, 4.7] by [ 2 3.1, 3.1] 0 # t # 4 p : [ 2 4.7, 4.7] by [ 2 3.1, 3.1] Let d be the length of the parametric interval. If d , 2 p , you get } 2 d p } of a complete circle. If d 5 2 p , you get the complete circle. If d . 2 p , you get the complete circle but portions of the circle will be traced out more than once. For example, if d 5 4 p the entire circle is traced twice. 16 Section 1.4 x change in Y 1 change in Y 2 1 3 2 2 5 4 3 7 8 4

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3. } p 2 } # t # } 3 2 p } initial point: (0, 3) terminal point: (0, 2 3) p # t # 2 p initial point: ( 2 3, 0) terminal point: (3, 0) } 3 2 p } # t # 3 p initial point: (0, 2 3) terminal point: ( 2 3, 0) p # t # 5 p initial point: ( 2 3, 0) terminal point: ( 2 3, 0) 4. For 0 # t # 2 p the complete circle is traced once clock- wise beginning and ending at (2, 0). For p # t # 3 p the complete circle is traced once clock- wise beginning and ending at ( 2 2, 0). For } p 2 } # t # } 3 2 p } the half circle below is traced clockwise starting at (0, 2 2) and ending at (0, 2). Exploration 2 Parametrizing Ellipses 1. a 5 2, b 5 3: [ 2 12, 12] by [ 2 8, 8] a 5 2, b 5 4: [ 2 12, 12] by [ 2 8, 8] a 5 2, b 5 5: [ 2 12, 12] by [ 2 8, 8] a 5 2, b 5 6: [ 2 12, 12] by [ 2 8, 8] 2. a 5 3, b 5 4: [ 2 9, 9] by [ 2 6, 6] a 5 5, b 5 4: [ 2 9, 9] by [ 2 6, 6] a 5 6, b 5 4: [ 2 9, 9] by [ 2 6, 6] a 5 7, b 5 4: [ 2 9, 9] by [ 2 6, 6] 3. If ) a ) . ) b ) , then the major axis is on the x -axis and the minor on the y -axis. If ) a ) , ) b ) , then the major axis is on the y
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