Bartleby Sitemap - Textbook Solutions
All Textbook Solutions for Precalculus: Mathematics for Calculus (Standalone Book)
131EThe point that is 3 units to the right of the y-axis and 5 units below the x-axis has coordinates (_____, _____).2EThe point midway between (a, b) and (c, d) is __________. So the point midway between (1, 2) and (7, 10) is __________.If the point (2, 3) is on the graph of an equation in x and y, then the equation is satisfied when we replace x by __________ and y by __________. Is the point (2, 3) on the graph of the equation 2y = x + 1? Complete the table, and sketch a graph. x y (x, y) 2 1 0 1 25E6E7EThe graph of an equation is shown below. (a) The x-intercept(s) are __________, and the y-intercept(s) are _____. (b) The graph is symmetric about the _____ (x-axis/y-axis/origin).Yes or No? If No, give a reason. 9. If the graph of an equation is symmetric with respect to both the x- and y-axes, is it necessarily symmetric with respect to the origin?10E11E12EPoints in a Coordinate Plane Plot the given points in a coordinate plane. 13. (0, 5), (1, 0), (1, 2), (12,23)14E15E16E17E18ESketching Regions Sketch the region given by the set. 19. (a) {(x, y)| 2 x 2 and y 1} (b) {(x, y)| xy 0}20E21E22EDistance and Midpoint A pair of points is graphed. (a) Find the distance between them. (b) Find the midpoint of the segment that joins them.24E25E26E27E28E29E30E31E32EPlot the points A(1, 0), B(5, 0), C(4, 3), and D(2, 3) on a coordinate plane. Draw the segments AB, BC, CD, and DA. What kind of quadrilateral is ABCD, and what is its area?Plot the points P(5, 1), Q(0, 6), and R(5, 1) on a coordinate plane. Where must the point S be located so that the quadrilateral PQRS is a square? Find the area of this square.35E36E37E38EDistance Formula In these exercises we use the Distance Formula. 39. Show that the triangle with vertices A(0, 2), B(3, 1), and C(4, 3) is isosceles.40EPythagorean Theorem In these exercises we use the converse of the Pythagorean Theorem (Appendix A) to show that the given triangle is a right triangle. 41. Refer to triangle ABC in the figure below. (a) Show that triangle ABC is a right triangle by using the converse of the Pythagorean Theorem. (b) Find the area of triangle ABC.Show that the triangle with vertices A(6, 7), B(11, 3), and C(2, 2) is a right triangle by using the converse of the Pythagorean Theorem. Find the area of the triangle.43E44E45E46E47E48E49EThe point M in the figure is the midpoint of the line segment AB. Show that M is equidistant from the vertices of triangle ABC.51E52EPoints on a Graph? Determine whether the given points are on the graph of the equation. 53. x2 + xy + y2 = 4; (0, 2), (1, 2), (2,2)54E55E56E57E58E59E60E61E62EGraphing Equations Use a graphing calculator to graph the equation in the given viewing rectangle. 63. y=xx2+25; [50, 50] by [0.2, 0.2]64E65E66E67E68EGraphing Equations Make a table of values, and sketch the graph of the equation. Find the x- and y-intercepts, and test for symmetry. 69. (a) y=4x2 (b) x = y3 + 2y70E71E72EIntercepts Find the x- and y-intercepts of the graph of the equation. 73. (a) 9x2 4y2 = 36 (b) y 2xy + 4x = 174E75E76E77E78E79E80E81E82E83E84E85E86EGraphing Circles Find the center and radius of the circle, and sketch its graph. 87. (x + 3)2 + (y 4)2 = 2588E89E90E91E92EEquations of Circles Find an equation of the circle that satisfies the given conditions. 93. Endpoints of a diameter are P(1, 1) and Q(5, 9)Equations of Circles Find an equation of the circle that satisfies the given conditions. 94. Endpoints of a diameter are P(1, 3) and Q(7, 5)95E96EEquations of Circles Find the equation of the circle shown in the figure. 97.98E99E100E101E102E103E104E105E106E107E108E109E110E111E112E113E114E115E116EArea of a Region Find the area of the region that lies outside the circle x2 + y2 = 4 but inside the circle x2+y24y12=0Area of a Region Sketch the region in the coordinate plane that satisfies both the inequalities x2 + y2 9 and y | x |. What is the area of this region?Shifting the Coordinate Plane Suppose that each point in the coordinate plane is shifted 3 units to the right and 2 units upward. (a) The point (5, 3) is shifted to what new point? (b) The point (a, b) is shifted to what new point? (c) What point is shifted to (3, 4)? (d) Triangle ABC in the figure has been shifted to triangle ABC. Find the coordinates of the points A, B and C.120E121EDistances in a City A city has streets that run north and south and avenues that run east and west, all equally spaced. Streets and avenues are numbered sequentially, as shown in the figure. The walking distance between points A and B is 7 blocksthat is, 3 blocks east and 4 blocks north. To find the straight-line distance d, we must use the Distance Formula. (a) Find the straight-line distance (in blocks) between A and B. (b) Find the walking distance and the straight-line distance between the corner of 4th St. and 2nd Ave. and the corner of 11th St. and 26th Ave. (c) What must be true about the points P and Q if the walking distance between P and Q equals the straight-line distance between P and Q?Halfway Point Two friends live in the city described in Exercise 122, one at the corner of 3rd St. and 7th Ave., the other at the corner of 27th St. and 17th Ave. They frequently meet at a coffee shop halfway between their homes. (a) At what intersection is the coffee shop located? (b) How far must each of them walk to get to the coffee shop? 122. Distances in a City A city has streets that run north and south and avenues that run east and west, all equally spaced. Streets and avenues are numbered sequentially, as shown in the figure. The walking distance between points A and B is 7 blocksthat is, 3 blocks east and 4 blocks north. To find the straight-line distance d, we must use the Distance Formula. (a) Find the straight-line distance (in blocks) between A and B. (b) Find the walking distance and the straight-line distance between the corner of 4th St. and 2nd Ave. and the corner of 11th St. and 26th Ave. (c) What must be true about the points P and Q if the walking distance between P and Q equals the straight-line distance between P and Q?Orbit of a Satellite A satellite is in orbit around the moon. A coordinate plane containing the orbit is set up with the center of the moon at the origin, as shown in the graph, with distances measured in megameters (Mm). The equation of the satellites orbit is (x3)225+y216=1 (a) From the graph, determine the closest and the farthest that the satellite gets to the center of the moon. (b) There are two points in the orbit with y-coordinates 2. Find the x-coordinates of these points, and determine their distances to the center of the moon.125E126E127EWe find the steepness, or slope, of a line passing through two points by dividing the difference in the _____-coordinates of these points by the difference in the _____-coordinates. So the line passing through the points (0, 1) and (2, 5) has slope _____.2E3E4E5E6E7E8E9E10E11E12E13E14E15ESlope Find the slope of the line through P and Q. 16. P(3, 2), Q(6, 2)Slope Find the slopes of the lines l1, l2, l3, and l4 in the figure below.Slope (a) Sketch lines through (0, 0) with slopes 1,0,12,2, and 1. (b) Sketch lines through (0, 0) with slopes 13,12,13, and 3.Equations of Lines Find an equation for the line whose graph is sketched. 19.20E21E22E23E24EFinding Equations of Lines Find an equation of the line that satisfies the given conditions. 25. Through (2, 3); slope 526E27E28E29EFinding Equations of Lines Find an equation of the line that satisfies the given conditions. 30. Through (1, 2) and (4, 3)Finding Equations of Lines Find an equation of the line that satisfies the given conditions. 31. Through (2, 5) and (1, 3)32EFinding Equations of Lines Find an equation of the line that satisfies the given conditions. 33. x-intercept 1; y-intercept 334E35E36E37E38EFinding Equations of Lines Find an equation of the line that satisfies the given conditions. 39. Through (1, 2); parallel to the line y = 3x 540E41E42E43EFinding Equations of Lines Find an equation of the line that satisfies the given conditions. 44. y-intercept 6; parallel to the line 2x + 3y + 4 = 045E46E47E48E49E50E51E52E53EFamilies of Lines Use a graphing device to graph the given family of lines in the same viewing rectangle. What do the lines have in common? 54. y = mx 3 for m = 0, 0.25, 0.75, 1.5Families of Lines Use a graphing device to graph the given family of lines in the same viewing rectangle. What do the lines have in common? 55. y = m(x 3) for m = 0, 0.25, 0.75, 1.556EUsing Slopes and y-Intercepts to Graph Lines Find the slope and y-intercept of the line, and draw its graph. 57. y = 3 x58EUsing Slopes and y-Intercepts to Graph Lines Find the slope and y-intercept of the line, and draw its graph. 59. 2x + y = 760E61E62E63E64E65EUsing Slopes and y-Intercepts to Graph Lines Find the slope and y-intercept of the line, and draw its graph. 66. y = 267E68E69E70E71E72E73E74E75E76E77E78E79E80E81E82E83E84E85E86E87EDrug Dosages If the recommended adult dosage for a drug is D (in mg), then to determine the appropriate dosage c for a child of age a, pharmacists use the equation c=0.0417D(a+1) Suppose the dosage for an adult is 200 mg. (a) Find the slope. What does it represent? (b) What is the dosage for a newborn?Flea Market The manager of a weekend flea market knows from past experience that if she charges x dollars for a rental space at the flea market, then the number y of spaces she can rent is given by the equation y = 200 4x. (a) Sketch a graph of this linear equation. (Remember that the rental charge per space and the number of spaces rented must both be nonnegative quantities.) (b) What do the slope, the y-intercept, and the x-intercept of die graph represent?90E91E92EDepreciation A small business buys a computer for 4000. After 4 years the value of the computer is expected to be 200. For accounting purposes the business uses linear depreciation to assess the value of the computer at a given time. This means that if V is the value of the computer at time t, then a linear equation is used to relate V and t. (a) Find a linear equation that relates V and t. (b) Sketch a graph of this linear equation. (c) What do the slope and V-intercept of the graph represent? (d) Find the depreciated value of the computer 3 years from the date of purchase.94E95EDISCUSS: Collinear Points Suppose that you are given the coordinates of three points in the plane and you want to see whether they lie on the same line. How can you do this using slopes? Using the Distance Formula? Can you think of another method?The solutions of the equation x2 2x 3 = 0 are the _____-intercepts of the graph of y = x2 2x 3.2EThe figure shows a graph of y = x4 3x3 x2 + 3x. Use the graph to do the following. (a) Find the solutions of the equation x4 3x3 x2 + 3x = 0. (b) Find the solutions of the inequality x4 3x3 x2 + 3x 0.The figure shows the graphs of y = 5x x2 and y = 4. Use the graphs to do the following. (a) Find the solutions of the equation 5x x2 = 4. (b) Find the solutions of the inequality 5x x2 4.5E6E7EEquations Solve the equation both algebraically and graphically. 8. 4x+262x=52x+49E10E11E12E13E14E15E16E17EEquations Solve the equation graphically in the given interval. State each answer rounded to two decimals. 18. x2 0.75x + 0.125 = 0; [2, 2]19E20E21E22E23E24E25E26E27E28E29E30E31E32E33EInequalities Find the solutions of the inequality by drawing appropriate graphs. State each answer rounded to two decimals. 34. 0.5x2 + 0.875x 0.2535E36E37E38E39E40E41E42E43E44E45E46E47E48EWRITE: Algebraic and Graphical Solution Methods Write a short essay comparing the algebraic and graphical methods for solving equations. Make up your own examples to illustrate the advantages and disadvantages of each method.50EIf the quantities x and y are related by the equation y = 3x, then we say that y is _______________ _______________ to x and the constant of _______________ is 3.2E3E4E5E6E7EEquations of Proportionality Write an equation that expresses the statement. 8. P is directly proportional to w.Equations of Proportionality Write an equation that expresses the statement. 9. v is inversely proportional to z.Equations of Proportionality Write an equation that expresses the statement. 10. w is proportional to the product of m and n.11E12E13E14E15EEquations of Proportionality Write an equation that expresses the statement. 16. S is proportional to the product of the squares of r and .Equations of Proportionality Write an equation that expresses the statement. 17. R is proportional to the product of the squares of P and t and inversely proportional to the cube of b.Equations of Proportionality Write an equation that expresses the statement. 18. A is jointly proportional to the square roots of x and y.19E20E21EConstants of Proportionality Express the statement as an equation. Use the given information to find the constant of proportionality. 22. P is directly proportional to T. If T = 300, then P = 20.23E24E25E26E