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All Textbook Solutions for Calculus: Early Transcendental Functions

Finding Intercepts In Exercises 1-4, find any intercepts. y=5x8 Finding Intercepts In Exercises 1-4, find any intercepts. y=x28x+12 3RE 4RE 5RE 6RE 7RE 8RE 9RE 10RE 11RE 12RE 13RE 14RE 15RE 16RE 17RE 18RE 19RE 20RE 21RE 22RE 23RE Finding an Equation of a Line In Exercises 21-24, find an equation of the line that passes through the point and has the indicated slope. Then sketch the line. Point Slope (5, 4) m=0 25RE 26RE 27RE 28RE 29RE 30RE 31RE Finding an Equation of a Line In Exercises 31 and 32, find an equation of the line that passes through the points. Then sketch the line. (5,5), (10,1)Finding Equations of Lines Find equations of the lines passing through (3,5) and having the following characteristics. (a) Slope of 716 (b) Parallel lo the line 5x3y=3 (c) Perpendicular to the line 3x+4y=8 (d) Parallel to the y-axis 34RE 35RE 36RE 37RE 38RE 39RE 40RE 41RE 42RE 43RE Sketching a Graph of a Function In Exercises 43 and 44, sketch a graph of the function and find its domain and range. Use a graphing utility to verify your graph. g(x)=x+1 45RE 46RE 47RE 48RE 49RE Think About It What is the minimum degree of the polynomial function whose graph approximates the given graph? What sign must the leading coefficient have?Finding Composite Functions In Exercises 51 and 52, find (he composite functions fg and gf. find the domain of each composite function. Are the two composite functions equal? f(x)=3x+1 g(x)=x 52RE 53RE 54RE 55RE 56RE 57RE 58RE 59RE 60RE 61RE 62RE 63RE 64RE 65RE 66RE 67RE 68RE 69RE 70RE 71RE 72RE 73RE 74RE 75RE 76RE 77RE 78RE 79RE Solving a Trigonometric Equation In Exercises 75-80, solve the equation for , where 02. 2sec2+tan25=0 81RE 82RE 83RE 84RE 85RE 86RE 87RE 88RE 89RE 90RE 91RE 92RE 93RE 94RE 95RE Finding an Inverse Function In Exercises 95-100, (a) find the inverse function of f, (b) graph f and f1 on the same set of coordinate axes, (c) verify that f1(f(x))=x and f(f1(x))=x, and (d) state the domains and ranges of f and f1 f(x)=5x7 97RE 98RE 99RE 100RE 101RE 102RE 103RE 104RE 105RE 106RE 107RE 108RE 109RE 110RE Evaluating an Expression In Exercises 111 and 112, evaluate each expression without using a calculator. (Hint: Make a sketch of a right triangle.) (a) sin(arcsin12) (b) cos(arcsin12)112RE 113RE 114RE 115RE 116RE 117RE 118RE 119RE 120RE 121RE Matching In Exercises 121-124, match the function with its graph. [The graphs are labeled (a), (b), (c), and (d).] f(x)=ex 123RE 124RE 125RE 126RE 127RE 128RE 129RE Condensing a Logarithmic Expression In Exercises 129 and 130, write the expression as the logarithm of a single quantity. 3[lnx2ln(x2+1)]+2ln5 131RE 132RE Finding Tangent Lines Consider the circle x2+y26x8y=0 as shown in the figure. (a) Find the center and radius of the circle. (b) Find an equation of the tangent line to the circle at the point (0, 0). (c) Find an equation of die tangent line to the circle at the point (6, 0). (d) Where do die two tangent lines intersect? Figure for 1 2PS 3PS 4PS 5PS Maximum Area A rancher has 300 feet of fencing to enclose two adjacent pastures (see figure). (a) Write the total area A of the two pastures as a function of x. What is the domain of A? (b) Graph the area function and estimate the dimensions that yield the maximum amount of area for the pastures. (c) Find the dimensions that yield the maximum amount of area for the pastures by completing die square.Writing a Function You are in a boat 2 miles from the nearest point on the coast. You will travel to a point Q located 3 miles down the coast and 1 mile inland (see figure). You can row at 2 miles per hour and walk at 4 miles per hour. Write the total time T of the trip as a function of x.8PS Slope of a Tangent Line One of the fundamental themes of calculus is to find the slope of the tangent line to a curve at a point. To see how this can be done, consider the point (2.4) on the graph of f(x)=x2 (see figure). (a) Find the slope of the line joining (2, 4) and (3, 9). Is the slope of the tangent line at (2, 4) greater than or less than this number? (b) Find the slope of the line joining (2, 4) and (1, 1). Is the slope of the tangent line at (2, 4) greater than or less than this number? (c) Find the slope of the line joining (2, 4) and (2.1, 4.41). Is the slope of the tangent line at (2, 4) greater than or less than this number? (d) Find the slope of the line joining (2, 4) and (2+h,f(2+h)) in terms of the nonzero number h. Verify that h=1,1, and 0, 1 yield the solutions to parts (a)-(c) above. (e) What is the slope of the tangent line at (2.4)? Explain how you arrived at your answer.10PS 11PS 12PS Sound Intensity A Large room contains two speakers that are 3 meters apart. The sound intensity I of one speaker is twice that of the other, as shown in the figure. (To print an enlarged copy of the graph, go to MaihGraphs.com.) Suppose the listener is free to move about the room to find those positions that receive equal amounts of sound from both speakers. Such a location satisfies two conditions: (1) the sound intensity at the listener's position is directly proportional to the sound level of a source, and (2) the sound intensity is inversely proportional to the square of the distance from die source. (a) Find the points on the a-axis that receive equal amounts of sound from both speakers. (b) Find and graph the equation of all locations (x, y) where one could stand and receive equal amounts of sound from both speakers. Figure for 13 Sound Intensity Suppose the speakers in Exercise 13 are 4 meters apart and ihe sound intensity of one speaker is k limes that of the other, as shown in the figure. To print an enlarged copy of the graph, go to MathGraphs.com. (a) Find the equation of all locations (x, y) where one could stand and receive equal amounts of sound from both speakers. (b) Graph the equation for the case k=3. (c) Describe the set of locations of equal sound as k becomes very large. Figure for 14 15PS Finding Intercepts Describe how to find the x and y intercepts of the graph of an equation.Verifying Points of Intersection How can you check that an ordered pair is a point of intersection of two graphs?Matching In Exercises 3-6, match the equation with its graph. [The graphs are labeled (a), (b), (c), and (d).] y=32x+3 y=9x2 y=3x2 y=x3x Matching In Exercises 3-6, match the equation with its graph. [The graphs are labeled (a), (b). (c), and (d).] y=32x+3 y=9x2 y=3x2 y=x3x Matching In Exercises 3-6, match the equation with its graph. [The graphs are labeled (a), (b). (c), and (d).] y=32x+3 y=9x2 y=3x2 y=x3x Matching In Exercises 3-6, match the equation with its graph. [The graphs are labeled (a), (b). (c), and (d).] y=32x+3 y=9x2 y=3x2 y=x3x 7E Sketching a Graph by Point Plotting In Exercises 7-16, sketch the graph of the equation by point plotting. y=52x Sketching a Graph by Point Plotting In Exercises 7-16, sketch the graph of the equation by point plotting. y=4x2 10E Sketching a Graph by Point Plotting In Exercises 7-16, sketch the graph of the equation by point plotting. y=x+1 Sketching a Graph by Point Plotting In Exercises 7-16, sketch the graph of the equation by point plotting. y = |x| - 1 Sketching a Graph by Point Plotting In Exercises 7-16, sketch the graph of the equation by point plotting. y=x6 14E 15E 16E Approximating Solution Points Using Technology In Exercises 17 and 18, use a graphing utility to graph the equation. Move the cursor along the curve to approximate the unknown coordinate of each solution point accurate to two decimal places. y=5x (a) (2, y) (b) (x, 3)Approximating Solution Points Using Technology In Exercises 17 and 18, use a graphing utility to graph the equation. Move the cursor along the curve to approximate the unknown coordinate of each solution point accurate to two decimal places. y=x55x (a) (0.5,y) (b) (x,4)Finding InterceptsIn Exercises 19-28, find any intercepts. y=2x5 Finding Intercepts In Exercises 19-28, find any intercepts. y=4x2+3 Finding Intercepts In Exercises 19-28, find any intercepts. y=x2+x2 22E Finding Intercepts In Exercises 19-28, find any intercepts. y=x16x2 Finding Intercepts In Exercises 19-28, find any intercepts. y=(x1)x2+1 Finding Intercepts In Exercises 19-28, find any intercepts. y=2x5x+1 Finding Intercepts In Exercises 19-28, find any intercepts. y=x2+3x(3x+1)2 Finding Intercepts In Exercises 19-28, find any intercepts. x2yx2+4y=0 Finding Intercepts In Exercises 19-28, find any intercepts. y=2xx2+1 29E 30E 31E 32E 33E 34E 35E 36E 37E 38E 39E 40E 41E 42E 43E 44E 45E Using Intercepts and Symmetry to Sketch a Graph In Exercises 41-56, find any intercepts and test for symmetry. Then sketch the graph of the equation. y=x34x 47E 48E Using Intercepts and Symmetry to Sketch a Graph In Exercises 41-56, find any intercepts and test for symmetry. Then sketch the graph of the equation. y=y3 50E Using Intercepts and Symmetry to Sketch a Graph In Exercises 41-56, find any intercepts and test for symmetry. Then sketch the graph of the equation. y=8x 52E 53E Using Intercepts and Symmetry to Sketch a Graph In Exercises 41-56, find any intercepts and test for symmetry. Then sketch the graph of the equation. y=6x 55E 56E 57E 58E Finding Points of Intersection In Exercises 57-62, find the points of intersection of the graphs of the equations. x2+y=15 3x+y=11 Finding Points of Intersection In Exercises 57-62, find the points of intersection of the graphs of the equations. x=3y2 y=x1 Finding Points of Intersection In Exercises 57-62, find the points of intersection of the graphs of the equations. x2+y2=5 xy=1 Finding Points of Intersection In Exercises 57-62, find the points of intersection of the graphs of the equations. x2+y2=16 x+2y=4 Finding Points of Intersection Using Technology In Exercises 63-66, use a graphing utility to find the points of intersection of the graphs of the equations. Check your results analytically. y=x32x2+x1 y=x2+3x1 64E 65E 66E Modeling Data The table shows the Gross Domestic Product, or GDP (in trillions of dollars), for 2009 through 2014, (Source: U.S. Bureau of Economic Analysis) Year 2009 2010 2011 2012 2013 2014 GDP 14.4 15.0 15.5 16.2 16.7 17.3 (a) Use the regression capabilities of a graphing utility to find a mathematical model of the form y=at+b for the data. In the model, y represents die GDP (in trillions of dollars) and t represents the year, with t=9 corresponding to 2009. (b) Use a graphing utility lo plot the data and graph the model. Compare the data with the model. (c) Use the model to predict the GDP in the year 2024.Modeling Data The table shows [he numbers of cell phone subscribers (in millions) in the United States for selected years. (Source: CTIA-The Wireless Association) Year 2000 2002 2004 2006 Number 109 141 182 233 Year 2008 2010 2012 2014 Number 270 296 326 355 (a) Use die regression capabilities of a graphing utility to find a mathematical model of the form y=at2+bt+c for the data. In the model, y represents the number of subscribers (in millions) and t represents the year, with t=0 corresponding to 2000. (b) Use a graphing utility to plot the data and graph the model. Compare the data with the model. (c) Use the model to predict the number of cell phone subscribers in the United Sates in the year 2024.Break-Even Point Find the sales necessary to break even (R=C) when the cost C of producing x units is C=2.04x+5600 and the revenue R from selling x units us R=3.29x.Using Solution Points For what values of k does the graph of y2=4kx pass through the point. (a) (1, 1) (b) (2, 4) (c) (0, 0) (d) (3, 3)Using Intercepts Write an equation whose graph has intercepts at x=32,x=4. and x=52 (There is more than one correct answer.)Symmetry A graph is symmetric with respect to the x-axis and to the y-axis. Is the graph also symmetric with respect to the origin? Explain.Symmetry A graph is symmetric with respect to one axis and to the origin. Is the graph also symmetric with respect to the other axis? Explain.74E True or False? In Exercises 75-78, determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. If (4,5) is a point on a graph that is symmetric with respect to the r-axis, then (4,5) is also a point on the graph.True or False? In Exercises 75-78, determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. If (4,5) is a point on a graph that is symmetric with respect to the y-axis. then (4,5) is also a point on the graph.77E True or False? In Exercises 75-78, determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. If b24ac=0 and a0, then the graph of y=ax2+bx+c has only one x-intercept.Slope-Intercept Form la the form y=mx+b2 what does m represent? What does h represent?2E 3E 4E 5E 6E Finding the Slope of a Line In Exercises 7-12, plot the pair of points and find the slope of the line passing through them. (3, -4), (5, 2)8E 9E 10E 11E Finding the Slope of a Line In Exercises 7-12, plot the pair of points and find the slope of the line passing through them. (78,34),(54,14)13E 14E Finding Points on a Line In Exercises 15-18, use the point on the line and the slope of the line to find three additional points that the line passes through. (There is more than one correct answer.) (6, 2) m=0 16E 17E 18E 19E 20E 21E 22E Finding an Equation of a Line In Exercises 19-24, find an equation or the line that passes through the point and has the indicated slope. Then sketch the line. (3,2) m=3 24E 25E Conveyor Design.......... A moving conveyor is built to rise I meter for each 3 meters of horizontal change (a) Find the slope of the conveyor. (b) Suppose the conveyor runs between two floors in a factory. Find the length of the conveyor when the vertical distance between floors is 10 feet.Modeling Data The table shows the populations y (in millions) of the United States for 2009 through 2014. The variable t represents the time in years, with t=9 corresponding to 2009. (Source: US. Census Bureau) t 9 10 11 12 13 14 y 307.0 309.3 311.7 314.1 316.5 318.9 (a) Plot the data by hand and connect adjacent points with a line segment. Use the slope of each line segment to determine the year when the population increased least rapidly. (b) Find the average rate of change of the population of the United States from 2009 through 2014. (c) Use the average rate of change of the population to predict the population of the United States in 2025.28E 29E 30E 31E 32E 33E 34E 35E 36E 37E 38E 39E Sketching a Line in the Plane In Exercises 35-42, sketch the graph of the equation. y1=3(x+4)41E 42E 43E Finding an Equation of a Line In Exercises 43-50, find an equation of the line that passes through the points. Then sketch the line. (2,2), (1, 7)Finding an Equation of a Line In Exercises 43-50, find an equation of the line that passes through the points. Then sketch the line. (2, 8), (5, 0)46E 47E Finding an Equation of a Line In Exercises 43-50, find an equation of the line that passes through the points. Then sketch the line. (1,2),(3,2)49E 50E 51E Using Intercepts Show that the line with intercepts {a, 0) and (0, b) has the following equation. xa+yb=1,a0,b0 Writing an Equation in General Form In Exercises 53-56, use the result of Exercise 52 to write an equation of the line with the given characteristics in general form. x-Intercept: (2, 0) y-intercept: (0, 3)54E 55E 56E 57E 58E 59E Finding Parallel and Perpendicular Lines In Exercises 57-62, write the general forms of the equations of the lines that pass through the point and are (a) parallel to the given line and lb) perpendicular to the given line. (2,5) xy=2 61E 62E Rate of Change In Exercises 63 and 64, you are given the dollar value of a product in 2016 and the rate at which the value of the product is expected to change during the next 5 years. Write a linear equation that gives the dollar value V of the product in terms of the year r. (Let t=0 represent 2010.) 2016 Value Rate $1850 $250 increase per year Rate of Change In Exercises 63 and 64, you are given the dollar value of a product in 2016 and the rate at which the value of the product is expected to change during the next 5 years. Write a linear equation that gives the dollar value V of the product in terms of the year r. (Let t=0 represent 2010.) 2016 Value Rate $1850 $250 increase per year $17.200 $1600 decrease per year Collinear Points In Exercises 65 and 66, determine whether the points are collinear. (Three points are collinear if they lie on the same line.) (2,1),(1,0),(2,2)66E 67E 68E Tangent Line Find an equation of the line tangent to the Circle x2+y2=169 at the point (5, 12).Tangent Line Find an equation of the line tangent to the circle (x1)2+(y1)2=25 at die point (4,3).Finding Points of Intersection Find the coordinates of the point of intersection of the given segments. Explain your reasoning (a) Perpendicular bisectors (b) Medians 72E 73E Choosing a Job As a salesperson, you receive a monthly salary of $2000, plus a commission of 1% of sales. You are offered a new job at $2300 per month, plus a commission of 5% of sales. (a) Write linear equations for your monthly wage W in terms of your monthly sales s for your current job and your job offer. (b) Use a graphing utility to graph each equation and find the point of intersection. What does it signify? (c) You think you can sell 520,000 worth of a product per month. Should you change jobs? Explain.75E

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