Calculus of a Single Variable
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point on the graph. This means that the graph is unchanged by a rotation of 180Њ about the origin. y-axis symmetry y (x, y) x (x, −y) x-axis symmetry TESTS FOR SYMMETRY 1. The graph of an equation in x and y is symmetric with respect to the y-axis if replacing x by Ϫx yields an equivalent equation. 2. The graph of an equation in x and y is symmetric with respect to the x-axis if replacing y by Ϫy yields an equivalent equation. 3. The graph of an equation in x and y is symmetric with respect
equation Because replacing both x by Ϫx and y by Ϫy yields an equivalent equation, you can conclude that the graph of y ϭ 2x3 Ϫ x is symmetric with respect to the origin, as shown in Figure P.8. ■ 6 Chapter P Preparation for Calculus EXAMPLE 4 Using Intercepts and Symmetry to Sketch a Graph Sketch the graph of x Ϫ y 2 ϭ 1. Solution The graph is symmetric with respect to the x-axis because replacing y by Ϫy yields an equivalent equation. y x − y =1 2 (5, 2) 2 (2, 1) 1 (1, 0) x 2 3 4
Welcome to the Ninth Edition of Calculus of a Single Variable! We are proud to offer you a new and revised version of our textbook. Much has changed since we wrote the first edition over 35 years ago. With each edition we have listened to you, our users, and have incorporated many of your suggestions for improvement. 6th 7th 9th 8th Throughout the years, our objective has always been to write in a precise, readable manner with the fundamental concepts and rules of calculus clearly defined and
Exercises 27–30, discuss the continuity of each function. 47. f ͑x͒ ϭ xϩ2 x 2 Ϫ 3x Ϫ 10 48. f ͑x͒ ϭ xϪ1 x2 ϩ x Ϫ 2 x→4 24. limϩ͑2x Ϫ ͠x͒͡ x→2 25. lim ͑2 Ϫ ͠Ϫx͡ ͒ x→3 x→1 Γ 2xΔ 27. f ͑x͒ ϭ x2 1 Ϫ4 28. f ͑x͒ ϭ x2 Ϫ 1 xϩ1 y 3 2 1 3 2 1 x −3 −1 −2 −3 49. f ͑x͒ ϭ Խx ϩ 7Խ 50. f ͑x͒ ϭ Խx Ϫ 8Խ y 1 3 x −3 −2 −1 −3 1 2 3 xϩ7 xϪ8 Άx,x , xx >Յ 11 Ϫ2x ϩ 3, x < 1 52. f ͑x͒ ϭ Ά x , x Ն 1 51. f ͑x͒ ϭ 2 2 44. f ͑x͒ ϭ 1 x2 ϩ 1 x 2 x x2 Ϫ 1 80 Chapter 1 53. f ͑x͒ ϭ Ά
2.2 you learned that the derivative of the sum of two functions is simply the sum of their derivatives. The rules for the derivatives of the product and quotient of two functions are not as simple. THEOREM 2.7 THE PRODUCT RULE NOTE A version of the Product Rule that some people prefer is d ͓ f ͑x͒g ͑x͔͒ ϭ fЈ ͑x͒g͑x͒ ϩ f ͑x͒gЈ͑x͒. dx The advantage of this form is that it generalizes easily to products of three or more factors. The product of two differentiable functions f and g is itself