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Extra info for Cram101 textbook outlines to accompany: calculus early transcendentals
90 SOLUTION (a) The mass is initially 24 mg and is halved during each 25-year period, so m͑0͒ 24 m͑25͒ 1 ͑24͒ 2 m͑50͒ 1 1 1 и ͑24͒ 2 ͑24͒ 2 2 2 m͑75͒ 1 1 1 и 2 ͑24͒ 3 ͑24͒ 2 2 2 m͑100͒ 1 1 1 и 3 ͑24͒ 4 ͑24͒ 2 2 2 From this pattern, it appears that the mass remaining after t years is m͑t͒ 1 2 t͞25 ͑24͒ 24 и 2 Ϫt͞25 This is an exponential function with base a 2Ϫ1͞25 1͞21͞25. 9 mg (c) We use a graphing calculator or computer to graph the function m͑t͒ 24 и 2 Ϫt͞25 in Figure 10.
Write a formula for V͑t͒ in terms of H͑t͒. (c) Sketch the graph of the voltage V͑t͒ in a circuit if the switch is turned on at time t 5 seconds and 240 volts are applied instantaneously to the circuit. Write a formula for V͑t͒ in terms of H͑t͒. ) 60. The Heaviside function defined in Exercise 59 can also be used f 57. A stone is dropped into a lake, creating a circular ripple that travels outward at a speed of 60 cm͞s. (a) Express the radius r of this circle as a function of the time t (in seconds).
1Ϫx SOLUTION Figure 10(a) shows the graph produced by a graphing calculator with viewing rectangle ͓Ϫ9, 9͔ by ͓Ϫ9, 9͔. In connecting successive points on the graph, the calculator produced a steep line segment from the top to the bottom of the screen. That line segment is not truly part of the graph. Notice that the domain of the function y 1͑͞1 Ϫ x͒ is ͕x x 1͖. We can eliminate the extraneous near-vertical line by experimenting with a change of scale. 7͔ on this particular calculator, we obtain the much better graph in Figure 10(b).