By Ross L. Finney, Franklin D. Demana, Bert K. Waits, Daniel Kennedy

The major objective of this 3rd variation is to realign with the adjustments within the complex Placement (AP ) calculus syllabus and the recent form of AP examination questions. we've got additionally extra conscientiously aligned examples and workouts and up-to-date the knowledge utilized in examples and workouts. Cumulative quickly Quizzes at the moment are supplied or thrice in each one chapter.

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**Extra resources for Calculus: Graphical, Numerical, Algebraic**

**Sample text**

F (x) ϭ 3Ϫx 29. f (x) ϭ ln x 30. f (x) ϭ log x 31. f (x) ϭ sinϪ1 x 32. f (x) ϭ tanϪ1 x 3 15. f Ϫ1(x) ϭ (x ϩ 1)1/3 or ͙ෆ xϩ1 16. 60 Source: Statistical Abstract of the United States, 2004–2005. (b) Estimate the number of cubic feet of natural gas produced by Canada in 2002. 63 trillion cubic feet in 2002. 16 trillion cubic feet (c) Predict when Canadian natural gas production will reach 7 trillion cubic feet. sometime during 2003 x Ϫ1 43. 5 Functions and Logarithms 45 False. Consider f (x) ϭ x2, g(x) ϭ ͙xෆ.

B) Find a Cartesian equation for a curve that contains the parametrized curve. What portion of the graph of the Cartesian equation is traced by the parametrized curve? 45. x ϭ Ϫsec t, y ϭ tan t, Ϫp͞2 Ͻ t Ͻ p͞2 (b) x2 – y2 ϭ 1; left branch (or x ϭ Ϫ͙ෆ y2 ϩ 1; all) 46. x ϭ tan t, y ϭ Ϫ2 sec t, Ϫp͞2 Ͻ t Ͻ p͞2 ϩ 1; all) Ϫ x ϭ 1; lower branch (or y ϭ Ϫ2͙xෆ y (b) ᎏᎏ 2 2 2 2 Extending the Ideas 47. The Witch of Agnesi The bell-shaped witch of Agnesi can be constructed as follows. Start with the circle of radius 1, centered at the point (0, 1) as shown in the figure.

37 The graphs of y ϭ 2x (x1 ϭ t, y1 ϭ 2t), its inverse y ϭ log2 x (x2 ϭ 2t, y2 ϭ t), and y ϭ x (x3 ϭ t, y3 ϭ t). log10 x ϭ log x. The function y ϭ ln x is called the natural logarithm function and y ϭ log x is often called the common logarithm function. Properties of Logarithms Because a x and loga x are inverses of each other, composing them in either order gives the identity function. This gives two useful properties. Inverse Properties for ax and loga x 1. Base a: aloga x ϭ x, 2. Base e: eln x ϭ x, loga a x ϭ x, ln ex ϭ x, a Ͼ 1, x Ͼ 0 xϾ0 These properties help us with the solution of equations that contain logarithms and exponential functions.