Schaum's Outline of Calculus (6th Edition) (Schaum's by Elliott Mendelson, Frank Ayres

By Elliott Mendelson, Frank Ayres

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C) Describe the line having the equation y = −x. 12. (a) Find the slopes and y intercepts of the lines that have the following equations: (i) y = 3x − 2; (ii) 2x − 5y = 3; y (iii) y = 4x − 3; (iv) y = −3; (v) + x = 1. 2 3 (b) Find the coordinates of a point other than (0, b) on each of the lines of part (a). Ans. (a) (i) m = 3, b = −2; (ii) m = 25 b = − 35 (iii) m = 4, b = −3; (iv) m = 0, b = −3; (v) m = − 23 ; b = 2; (b) (i) (1, 1); (ii) (−6, −3); (iii) (1, 1); (iv) (1, −3); (v) (3, 0) 13. If the point (3, k) lies on the line with slope m = −2 passing through the point (2, 5), find k.

Sketch the graph of the function defined as follows: f ( x ) = 5 when 0 < x ≤ 1 f ( x ) = 10 when 1 < x ≤ 2 f ( x ) = 15 when 2 < x ≤ 3 f ( x ) = 20 when 3 < x ≤ 4 etc. Determine the domain and range of the function. The graph is shown in Fig. 6-3. The domain is the set of all positive real numbers, and the range is the set of integers, 5, 10, 15, 20, . . Fig. 6-3 5. A rectangular plot requires 2000 ft of fencing to enclose it. If one of its dimensions is x (in feet), express its area y (in square feet) as a function of x, and determine the domain of the function.

Let F and F′ be two distinct points at a distance 2c from each other. Show that the set of all points P(x, y) such that PF + PF ′ = 2a, a > c is an ellipse. CHAPTER 5 Equations and Their Graphs 42 Construct a coordinate system such that the x axis passes through F and F′, the origin is the midpoint of the segment FF′, and F lies on the positive x axis. Then the coordinates of F and F′ are (c, 0) and (−c, 0). Fig. 5-11 (See Fig. ) Thus, the condition PF + PF ′ = 2a is equivalent to ( x − c)2 + y 2 + ( x + c)2 + y 2 = 2a.

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