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SECTION 1.4 GRAPHING CALCULATORS AND COMPUTERS 37 1. f(x) = \sqrt{x^3} - 5x^2 (a) [-5, 5] by [-5, 5] (There is no graph shown.) 5 -5 5 -5 (b) [0, 10] by [0, 2] 2 (c) [0, 10] by [0, 10] 0 10 10 0 10 The most appropriate graph is produced in viewing rectangle (c). 2. f(x) = x^2 - 16x^2 + 20 (a) [-3, 3] by [-3, 3] 5 (c) [-50, 50] by [-50, 50] 3 -3 3 -3 50 50 -50 50 (b) [-10, 10] by [-10, 10] 10 -10 10 -10 (d) [-5, 5] by [-50, 50] 50 The most appropriate graph is produced in viewing rectangle (d). -5 5 50 -50 50 -50 3. Since the graph of f(x) = 5 + 20x - x^2 is a parabola opening downward, an appropriate viewing rectangle should include the maximum point. 30 4. An appropriate viewing rectangle for f(x) = x^3 + 30x^2 + 200x should include the high and low points. 150 -10 -50 30 500 -22 -500 38 CHAPTER 1 FUNCTIONS AND MODELS 5. f(x) = \sqrt{81 - x^4} is defined when 81 - x^4 >= 0 <=> x^4 <= 81 <=> |x| <= 3, so the domain of f is [-3, 3]. Also 0 <= \sqrt{81 - x^4} <= \sqrt{81} = 3, so the range is [0, 3]. 4 -4 4 4 -1 6. f(x) = \sqrt{0.1x + 20} is defined when 0.1x + 20 >= 0 <=> x >= -200, so the domain of f is [-200, ∞). 3500 50 -250 -1 7. The graph of f(x) = x^3 - 225x is symmetric with respect to the origin. Since f(x) = x^3 - 225x = x(x^2 - 225) = x(x + 15)(x - 15), there are x-intercepts at 0, -15, and 15. f(20) = 3500. 20 3500 8. The graph of f(x) = x/(x^2 + 100) is symmetric with respect to the origin. 0.1 50 -50 50 -0.1 9. The period of g(x) = \sin(1000\pi x) is 2\pi/1000 ≈ 0.0063 and its range is [-1, 1]. Since f(x) = \sin^2(1000\pi x) is the square of g, its range is [0, 1] and a viewing rectangle of [-0.01, 0.01] by [0, 1.1] seems appropriate. 0.01 -0.01 10. The period of f(x) = \cos(0.001x) is 2\pi/0.001 ≈ 6300 and its range is [-1, 1], so a viewing rectangle of [-10,000, 10,000] by [-1.5, 1.5] seems appropriate. -10,000 11. The domain of y = \sqrt{x} is x >= 0, so the domain of f(x) = \sin(\sqrt{x}) is [0, ∞) and the range is [-1, 1]. With a little trial-and-error experimentation, we find that an Xmax of 100 illustrates the general shape of f, so an appropriate viewing rectangle is [0, 100] by [-1.5, 1.5]. 100 -1.5 SECTION 1.4 GRAPHING CALCULATORS AND COMPUTERS 39 12. One period of y = \sec x occurs on the interval (-π/2, π/2)∪(π/2, 3π/2). -π/2 < 2πnx < π/2, 3π/2 < 3π/2 or equivalently, -0.025 < x < 0.075 0.025 5 -5 -0.025 13. The first term, 10 \sin x, has period 2π and range [-10, 10]. It will be the dominant term in any "large" graph of y = 10 \sin x + \sin 100πx, as shown in the first figure. The second term, \sin 100πx, has period 2π/100π = 1/50 and range [-1, 1]. It causes the bumps in the first figure and will be the dominant term in any "small" graph, as shown in the view near the origin in the second figure. 2π/100π 31 π -2π π/25 25 2 -2 0.025 14. y = x^2 + 0.02 \sin(50x) 2 -0.5 0.5 4 -2 2 0.25 -0.02 0.02 15. We must solve the given equation for y to obtain equations for the upper and lower halves of the ellipse. 4x^2 + 2y^2 = 1 <=> 2y^2 = 1 - 4x^2 1 - 4x^2 y^2 = \frac{1 -}{2} <=> y = \pm\sqrt{1 + 9x^2} 1 -1 4 -4 2 16. y^2 - 9x^2 = 1 <=> y^2 = 1 + 9x^2 17. From the graph of y = 3x^2 - 6x + 1 and y = 0.23x - 2.25 in the viewing rectangle [-1, 3] by [-2.5, 1.5], it is difficult to see if the graphs intersect. If we zoom in on the fourth quadrant, we see the graphs do not intersect. 1.5 3 1.8 0.8 -2.5 -2.2 1.5 1.2 x SECTION 1.5 EXPONENTIAL FUNCTIONS 43 35. The graphing window is 95 pixels wide and we want to start with x = 0 and end with x = 2π. Since there are 94 "gaps" between pixels, the distance between pixels is . Thus, the x-values that the calculator actually plots are x = n , where n = 0, 1, 2, ..., 93, 94. For y = sin 2x, the actual points plotted by the calculator are for n = 0, 1,...,94. For y = sin 96x, the points plotted are for n = 0, 1,...,94. But . n) = sin(96 . )n = sin(94 . n) = sin(94 . n) = sin(2π + )n) = sin(2π + n) = sin(94 . n) [by periodicity of sine], n = 0,1,...,94 sin So the y-values, and hence the points, plotted for y = sin 96x are identical to those plotted for y = sin 2x. Note: Try graphing y = sin 94x. Can you see why all the y-values are zero? 36. As in Exercise 35, we know that the points being plotted for y = sin 45x are and for n = 0, 1,...,94. But sin(45 . n) = sin(47 . n) = sin(45 . n) = sin(77 . n) = sin(2π . n) = sin(πn) (cos . n) - cos(πn) sin(2 . n) [Subtraction formula for the sine] = 0 . cos(2 . n) So the och ) - (1) sin(2 . ) = sin(n) )n) , n = 0, 1,...,94 So the y-values, and hence the points, plotted for y = sin 45x lie on either y = sin 2x or y = sin 2x.
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Texto de pré-visualização
SECTION 1.4 GRAPHING CALCULATORS AND COMPUTERS 37 1. f(x) = \sqrt{x^3} - 5x^2 (a) [-5, 5] by [-5, 5] (There is no graph shown.) 5 -5 5 -5 (b) [0, 10] by [0, 2] 2 (c) [0, 10] by [0, 10] 0 10 10 0 10 The most appropriate graph is produced in viewing rectangle (c). 2. f(x) = x^2 - 16x^2 + 20 (a) [-3, 3] by [-3, 3] 5 (c) [-50, 50] by [-50, 50] 3 -3 3 -3 50 50 -50 50 (b) [-10, 10] by [-10, 10] 10 -10 10 -10 (d) [-5, 5] by [-50, 50] 50 The most appropriate graph is produced in viewing rectangle (d). -5 5 50 -50 50 -50 3. Since the graph of f(x) = 5 + 20x - x^2 is a parabola opening downward, an appropriate viewing rectangle should include the maximum point. 30 4. An appropriate viewing rectangle for f(x) = x^3 + 30x^2 + 200x should include the high and low points. 150 -10 -50 30 500 -22 -500 38 CHAPTER 1 FUNCTIONS AND MODELS 5. f(x) = \sqrt{81 - x^4} is defined when 81 - x^4 >= 0 <=> x^4 <= 81 <=> |x| <= 3, so the domain of f is [-3, 3]. Also 0 <= \sqrt{81 - x^4} <= \sqrt{81} = 3, so the range is [0, 3]. 4 -4 4 4 -1 6. f(x) = \sqrt{0.1x + 20} is defined when 0.1x + 20 >= 0 <=> x >= -200, so the domain of f is [-200, ∞). 3500 50 -250 -1 7. The graph of f(x) = x^3 - 225x is symmetric with respect to the origin. Since f(x) = x^3 - 225x = x(x^2 - 225) = x(x + 15)(x - 15), there are x-intercepts at 0, -15, and 15. f(20) = 3500. 20 3500 8. The graph of f(x) = x/(x^2 + 100) is symmetric with respect to the origin. 0.1 50 -50 50 -0.1 9. The period of g(x) = \sin(1000\pi x) is 2\pi/1000 ≈ 0.0063 and its range is [-1, 1]. Since f(x) = \sin^2(1000\pi x) is the square of g, its range is [0, 1] and a viewing rectangle of [-0.01, 0.01] by [0, 1.1] seems appropriate. 0.01 -0.01 10. The period of f(x) = \cos(0.001x) is 2\pi/0.001 ≈ 6300 and its range is [-1, 1], so a viewing rectangle of [-10,000, 10,000] by [-1.5, 1.5] seems appropriate. -10,000 11. The domain of y = \sqrt{x} is x >= 0, so the domain of f(x) = \sin(\sqrt{x}) is [0, ∞) and the range is [-1, 1]. With a little trial-and-error experimentation, we find that an Xmax of 100 illustrates the general shape of f, so an appropriate viewing rectangle is [0, 100] by [-1.5, 1.5]. 100 -1.5 SECTION 1.4 GRAPHING CALCULATORS AND COMPUTERS 39 12. One period of y = \sec x occurs on the interval (-π/2, π/2)∪(π/2, 3π/2). -π/2 < 2πnx < π/2, 3π/2 < 3π/2 or equivalently, -0.025 < x < 0.075 0.025 5 -5 -0.025 13. The first term, 10 \sin x, has period 2π and range [-10, 10]. It will be the dominant term in any "large" graph of y = 10 \sin x + \sin 100πx, as shown in the first figure. The second term, \sin 100πx, has period 2π/100π = 1/50 and range [-1, 1]. It causes the bumps in the first figure and will be the dominant term in any "small" graph, as shown in the view near the origin in the second figure. 2π/100π 31 π -2π π/25 25 2 -2 0.025 14. y = x^2 + 0.02 \sin(50x) 2 -0.5 0.5 4 -2 2 0.25 -0.02 0.02 15. We must solve the given equation for y to obtain equations for the upper and lower halves of the ellipse. 4x^2 + 2y^2 = 1 <=> 2y^2 = 1 - 4x^2 1 - 4x^2 y^2 = \frac{1 -}{2} <=> y = \pm\sqrt{1 + 9x^2} 1 -1 4 -4 2 16. y^2 - 9x^2 = 1 <=> y^2 = 1 + 9x^2 17. From the graph of y = 3x^2 - 6x + 1 and y = 0.23x - 2.25 in the viewing rectangle [-1, 3] by [-2.5, 1.5], it is difficult to see if the graphs intersect. If we zoom in on the fourth quadrant, we see the graphs do not intersect. 1.5 3 1.8 0.8 -2.5 -2.2 1.5 1.2 x SECTION 1.5 EXPONENTIAL FUNCTIONS 43 35. The graphing window is 95 pixels wide and we want to start with x = 0 and end with x = 2π. Since there are 94 "gaps" between pixels, the distance between pixels is . Thus, the x-values that the calculator actually plots are x = n , where n = 0, 1, 2, ..., 93, 94. For y = sin 2x, the actual points plotted by the calculator are for n = 0, 1,...,94. For y = sin 96x, the points plotted are for n = 0, 1,...,94. But . n) = sin(96 . )n = sin(94 . n) = sin(94 . n) = sin(2π + )n) = sin(2π + n) = sin(94 . n) [by periodicity of sine], n = 0,1,...,94 sin So the y-values, and hence the points, plotted for y = sin 96x are identical to those plotted for y = sin 2x. Note: Try graphing y = sin 94x. Can you see why all the y-values are zero? 36. As in Exercise 35, we know that the points being plotted for y = sin 45x are and for n = 0, 1,...,94. But sin(45 . n) = sin(47 . n) = sin(45 . n) = sin(77 . n) = sin(2π . n) = sin(πn) (cos . n) - cos(πn) sin(2 . n) [Subtraction formula for the sine] = 0 . cos(2 . n) So the och ) - (1) sin(2 . ) = sin(n) )n) , n = 0, 1,...,94 So the y-values, and hence the points, plotted for y = sin 45x lie on either y = sin 2x or y = sin 2x.