960 CHAPTER 14 A Preview of Calculus: The Limit, Derivative, and Integral of a Function In Problems 77–82, determine where each rational function R is undefined. Determine whether an asymptote or a hole appears in the graph of R at such numbers. 77. R x x x x x x x 1 2 2 3 2 4 3 ( ) = − + − − + − 78. R x x x x x x x 3 3 2 2 3 2 4 3 ( ) = + + + + + + 79. R x x x x x x 2 4 8 6 3 2 2 ( ) = − + − + − 80. R x x x x x x 3 3 3 4 3 2 2 ( ) = − + − + − 81. R x x x x x x x 2 2 2 3 2 4 3 ( ) = + + + + + 82. R x x x x x x x 3 4 12 3 3 3 2 4 3 ( ) = − + − − + − For Problems 83–88, use a graphing utility to graph the functions R given in Problems 77–82. Do the graphs support the solutions found for Problems 77 – 82 ? Explaining Concepts 89. Name three functions that are continuous at every real number. 90. Create a function that is not continuous at the number 5. Retain Your Knowledge Problems 91–94 are based on previously learned material. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for a final exam, or subsequent courses such as calculus. 91. Find any vertical or horizontal asymptotes for the graph of f x x x 3 4 4 . ( ) = − − 92. Evaluate the permutation P 5,3 ( ) . 93. Write x y z 5 ln 2 ln 4 ln + − as a single natural logarithm. 94. Write the augmented matrix for the following system: x y z x z y z 3 2 4 2 5 3 2 + + = + = − = − ⎧ ⎨ ⎪⎪ ⎪ ⎩ ⎪⎪ ⎪ ‘Are You Prepared?’ Answers 1. f f 0 0; 2 3 ( ) ( ) = = 2. Domain: x x 0 ; { } > range: { } −∞< <∞ y y 3. True 4. Secant, cosecant, tangent, cotangent 5. True 6. False 14.4 The Tangent Problem; The Derivative Now Work the ‘Are You Prepared?’ problems on page 966. • Point-Slope Form of a Line (Section 1.5, p. 36) • Average Rate of Change (Section 2.3, pp. 93–95) PREPARING FOR THIS SECTION Before getting started, review the following: OBJECTIVES 1 Find an Equation of the Tangent Line to the Graph of a Function (p. 961) 2 Find the Derivative of a Function (p. 962) 3 Find Instantaneous Rates of Change (p. 963) 4 Find the Instantaneous Velocity of an Object (p. 964) The Tangent Problem One question that motivated the development of calculus was a geometry problem, the tangent problem .This problem asks,“What is the slope of the tangent line to the graph of a function ( ) = y f x at a point P on its graph?” See Figure 13 on the next page. We first need to define what is meant by a tangent line. In plane geometry, the tangent line to a circle at a point is defined as the line that intersects the circle at exactly that one point. Look at Figure 14 on the next page. Notice that the tangent line just touches the graph of the circle.
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