Download Schaum's Outline of Beginning Calculus by Elliott Mendelson PDF
By Elliott Mendelson
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Example text
B) The line is not symmetric with respect to the y-axis, since (−1, 1) is on the line, but (1, 1), the reflection of (−1, 1) in the y-axis, is not on the line. (c) The line is symmetric with respect to the origin, by example (c) above. CHAP. 6] SYMMETRY 47 Fig. 6-7 Fig. 2 Determine whether the parabola x = y2 (see Fig. 6-8) is symmetric with respect to: (c) the origin. (a) the x-axis; (b) the y-axis; (a) The parabola is symmetric with respect to the x-axis, since x = y2 implies x = (−y)2 . (b) It is not symmetric with respect to the y-axis, since (1, 1) is on the parabola, but (−1, 1) is not.
The domain consists of all numbers x such that x ≤ 1. The range turns out to be all positive real numbers. This can be seen from Fig. 7-6, in which projection of the graph onto the y-axis produces all y such that y > 0. Note: In many treatments of the foundations of mathematics, a function is identified with its graph. In other words, a function is defined to be a set f of ordered pairs such that f does not contain two pairs (a, b) and (a, c) with b = c. ” However, this approach obscures the intuitive idea of a function.
GC Check on a graphing calculator. (d) Draw the graph of y = 1 . GC Check on a graphing calculator. 12 Sketch the graphs of the following equations. GC Check your answers on a graphing calculator. y2 x2 + =1 (b) 4x 2 + y2 = 4 (c) x 2 − y2 = 1 (a) 4 9 2 2 (y − 2) (x − 1) (d) y = x 3 + =1 (f ) xy = 2 (e) 4 9 [Hint: Parts (c) and (f ) are hyperbolas. 13 Find an equation whose graph consists of all points P(x, y) whose distance from the point F(0, p) is equal to its distance PQ from the horizontal line y = −p (p is a fixed positive number).