An Introduction to Analytic Geometry and Calculus by A. C. Burdette (Auth.)

By A. C. Burdette (Auth.)

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Choose a suitable set of coordinate axes and determine an equation for the parabola of which this arch is a segment. EXERCISES 3-3 41 6. A suspension bridge is 200 ft long with towers 45 ft high. If the roadway is 5 ft below the lowest point on the cable, how high is the cable above the roadway at a point 25 ft from the end of the bridge ? 7. Water squirts from the end of a horizontal pipe 18 ft above the ground. At a point 6 ft below the pipe, the stream of water is 8 ft beyond the end of the pipe.

3-12 Example 3-17. Determine an equation of the parabola with vertex at the origin, with axis on y = 0, and passing through the point (1,3). Equation (3-6) takes the form kx = y2 in this case. If the parabola passes through (1, 3), these coordinates satisfy this equation and we have /c(l) = (3) 2 . Thus the required equation is 9x = y2. 3-7. THE PARABOLA 39 Example 3-18. Determine an equation of the parabola with its axis parallel to the j-axis and passing through the three points (0, 1), (1, 0), ( — 2, — 1).

Discuss the function c where c(x) = cos x. cos x is defined for all real x and its values range from 1 to —1. There­ fore the domain is — oo < x < oo and the range is — 1 ^ c(x) ^ 1. 48 4. FUNCTIONS AND LIMITS In all preceding examples the correspondence between x and the function values was defined by a single formula. This need not be true. The functions in the following two examples illustrate this fact. Example 4-5. Discuss the function/where f(x) = h x<0; f(x) = 1 + x, x ^ 0. This function has the set of all real numbers as its domain since the two statements provide a rule for calculating a real/(x) when any real x is specified, the first for negative values of x, the second for nonnegative values.

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