By H. Jerome Keisler
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Extra info for Elementary Calculus: An Infinitesimal Approach, 2nd Edition
3 Solutions to Self-Check Exercises 1. a. The points are plotted in the following figure: The distance between A and C is d1A, C2 ϭ 21Ϫ3 Ϫ 4 2 2 ϩ 3 1 Ϫ 1Ϫ22 4 2 y ϭ 21Ϫ72 2 ϩ 32 ϭ 249 ϩ 9 ϭ 258 5 B(2, 3) c. We will show that [d(A, C)]2 ϭ [d(A, B)]2 ϩ [d(B, C)]2 C(– 3, 1) x –5 5 A(4, –2) –5 From part (b), we see that [d(A, B)]2 ϭ 29, [d(B, C)]2 ϭ 29, and [d(A, C)]2 ϭ 58, and the desired result follows. 2. The distance between city A and city B is d1A, B2 ϭ 22002 ϩ 502 Ϸ 206 or 206 mi. The distance between city B and city C is b.
75. 3 THE CARTESIAN COORDINATE SYSTEM Explore & Discuss In the Cartesian coordinate system, the two axes are perpendicular to each other. Consider a coordinate system in which the x- and y-axes are not collinear and are not perpendicular to each other (see the accompanying figure). y x O 1. Describe how a point is represented in this coordinate system by an ordered pair (x, y) of real numbers. Conversely, show how an ordered pair (x, y) of real numbers uniquely determines a point in the plane. 2.
97. 4x Ϫ 12x Ϫ 6x 5 4 3 131. 1 2 x Ϫxϩ1ϭ0 4 132. 2 PRECALCULUS REVIEW II In Exercises 133–138, solve the equation by using the quadratic formula. 133. 4x2 ϩ 5x Ϫ 6 ϭ 0 134. 3x2 Ϫ 4x ϩ 1 ϭ 0 135. 8x2 Ϫ 8x Ϫ 3 ϭ 0 136. x2 Ϫ 6x ϩ 6 ϭ 0 137. 2x2 ϩ 4x Ϫ 3 ϭ 0 138. 2x2 ϩ 7x Ϫ 15 ϭ 0 15 In Exercises 140–142, determine whether the statement is true or false. If it is true, explain why it is true. If it is false, give an example to show why it is false. 139. 5, where y is the number of families with an income of x or more dollars.