Chapter Review 749 Chapter Review Things to Know Equations Parabola (pp. 682–687) See Tables 1 and 2 (pp. 684 and 686). Ellipse (pp. 692–699) See Table 3 (p. 697). Hyperbola (pp. 705–714) See Table 4 (p. 712). General equation of a conic (p. 726) Ax Bxy Cy Dx Ey F 0 2 2 + + + + + = Parabola if B AC 4 0 2 − = Ellipse (or circle) if B AC 4 0 2 − < Hyperbola if B AC 4 0 2 − > Polar equations of a conic with focus at the pole (pp. 728–733) See Table 5 (p. 731). Parametric equations of a plane curve (p. 736) x x t y y t t , , ( ) ( ) = = is the parameter Definitions Parabola (p. 682) Set of points P in a plane for which d F P d P D , , , ( ) ( ) = where F is the focus and D is the directrix Ellipse (p. 692) Set of points P in a plane the sum of whose distances from two fixed points (the foci) is a constant Hyperbola (p. 705) Set of points P in a plane the difference of whose distances from two fixed points (the foci) is a constant Conic in polar coordinates (p. 729) The collection of points P for which d F P d D P e , , ( ) ( ) = Parabola if e 1 = Ellipse if e 1 < Hyperbola if e 1 > Formulas Rotation formulas (p. 722) x x y y x y cos sin sin cos θ θ θ θ = ′ − ′ = ′ + ′ Angle θ of rotation that eliminates the x y-term ′ ′ (p. 723) A C B cot 2 0 90 θ θ ( ) = − ° < < ° Retain Your Knowledge Problems 67–75 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 subsequent sections, a final exam, or later courses such as calculus. 67. Graph the equation x y 3 4 8 − = on the xy-plane. 68. Graph y x x 2cos 2 sin 2( ) ( ) = + on the xy-plane. 69. The International Space Station (ISS) orbits Earth at a height of approximately 248 miles above the surface. What is the distance, in miles, on the surface of Earth that can be observed from the ISS? Assume that Earth’s radius is 3960 miles. Source: nasa.gov 70. The displacement d (in meters) of an object at time t (in seconds) is given by d t t 2 cos 4 . ( ) ( ) = (a) Describe the motion of the object. (b) What is the maximum displacement of the object from its rest position? (c) What is the time required for 1 oscillation? (d) What is the frequency? 71. Find the oblique asymptote of R x x x x 4 9 7 2 1 2 ( ) = − + + 72. Find the difference quotient of f x x 1 3 ( ) = + as h 0. → 73. Find the exact value of cos285 .° 74. Solve x x x log 7 log3 5 log24. 5 5 5 ( ) ( ) ( ) − + + = 75. If f x x 1 4 1 3 ( ) = + and g x x 3 4 ,2 ( ) = find all numbers c in the interval 0, 2 [ ] where g c( ) equals the average rate of change of f over the interval. ‘Are You Prepared?’ Answers 1. 3; 2 π
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