By Robert M. Miura
Lawsuits of the NSF examine Workshop on touch modifications, Held in Nashville, Tennessee, 1974
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Extra info for Baecklund transformations, the inverse scattering method, solitons, and their applications
Specifically, nonvertical lines with the same slope are parallel and nonvertical lines whose slopes are negative reciprocals are perpendicular. 19 STUDY TIP In mathematics, the phrase “if and only if” is a way of stating two implications in one statement. For instance, the first statement at the right could be rewritten as the following two implications. a. If two distinct nonvertical lines are parallel, then their slopes are equal. b. If two distinct nonvertical lines have equal slopes, then they are parallel.
3 (a) Use the regression capabilities of a graphing utility to find a mathematical model of the form y ϭ at 2 ϩ bt ϩ c for the data. In the model, y represents the CPI and t represents the year, with t ϭ 5 corresponding to 1975. (b) Use a graphing utility to plot the data and graph the model. Compare the data with the model. (c) Use the model to predict the CPI for the year 2010. Number 1990 1993 1996 1999 2002 2005 5 16 44 86 141 208 (a) Use the regression capabilities of a graphing utility to find a mathematical model of the form y ϭ at 2 ϩ bt ϩ c for the data.
Y Ϫ 4 ϭ Ϫ1͑x ϩ 1͒ Linear Models and Rates of Change (x1, y1) (x1*, y1*) c. y Ϫ 4 ϭ Ϫ 12͑x ϩ 1͒ x y * − y1* y2 − y1 m= 2 = x2* − x1* x2 − x1 d. y Ϫ 4 ϭ 0͑x ϩ 1͒ e. y Ϫ 4 ϭ 12͑x ϩ 1͒ Any two points on a nonvertical line can be used to determine its slope. f. y Ϫ 4 ϭ 1͑x ϩ 1͒ g. 14 Use your results to write an equation of a line passing through ͑Ϫ1, 4͒ with a slope of m. You can write an equation of a nonvertical line if you know the slope of the line and the coordinates of one point on the line. Suppose the slope is m and the point is ͑x1, y1͒.
Baecklund transformations, the inverse scattering method, solitons, and their applications by Robert M. Miura