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All Textbook Solutions for Calculus: Early Transcendental Functions (MindTap Course List)
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=x3xMatching 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=x3xMatching 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=x3xMatching 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=x3x5ESketching a Graph by Point Plotting In Exercises 7-16, sketch the graph of the equation by point plotting. y=52xSketching a Graph by Point Plotting In Exercises 7-16, sketch the graph of the equation by point plotting. y=4x28E9ESketching a Graph by Point Plotting In Exercises 7-16, sketch the graph of the equation by point plotting. y = |x| - 1Sketching a Graph by Point Plotting In Exercises 7-16, sketch the graph of the equation by point plotting. y=x612E13E14EApproximating 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=2x5Finding Intercepts In Exercises 19-28, find any intercepts. y=4x2+3Finding Intercepts In Exercises 19-28, find any intercepts. y=x2+x220EFinding Intercepts In Exercises 19-28, find any intercepts. y=x16x2Finding Intercepts In Exercises 19-28, find any intercepts. y=(x1)x2+1Finding Intercepts In Exercises 19-28, find any intercepts. y=2x5x+1Finding Intercepts In Exercises 19-28, find any intercepts. y=x2+3x(3x+1)2Finding Intercepts In Exercises 19-28, find any intercepts. x2yx2+4y=0Finding Intercepts In Exercises 19-28, find any intercepts. y=2xx2+127E28E29E30E31E32E33E34E35E36E37E38E39E40E41E42E43EUsing 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=x34x45E46EUsing 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=y348EUsing 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=8x50E51EUsing 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=6x53E54E55E56E57E58E59EFinding Points of Intersection In Exercises 57-62, find the points of intersection of the graphs of the equations. x=3y2 y=x1Finding Points of Intersection In Exercises 57-62, find the points of intersection of the graphs of the equations. x2+y2=5 xy=162EFinding 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+3x164E65E66E67EModeling Data The table shows the numbers of cellular phone subscribers (in millions) in the United States for selected years. (Source: CTIA-The Wireless) Year 1995 1998 2001 2004 2007 2010 Number 34 69 128 182 255 303 (a) Use the 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 = 5 corresponding to 1995. (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 cellular phone subscribers in the United States in the year 2020.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.70E71EUsing 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)73E74E75E76ETrue 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.79ETrue 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.1E2E3E4EFinding 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)6E7E8E9EFinding 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)11E12EFinding 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=014E15E16E17E18E19E20EFinding 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=322EConveyor 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 Stales for 2004 through 2009. The variable t represents the time in years, with t = 4 corresponding to 2004. (Source: U.S. Census Bureau) t 4 5 6 7 8 9 y 293.0 295.8 298.6 301.6 304.4 307.0 (a) Plot the data by hand and connect adjacent points with a line segment. (b) Use the slope of each line segment to determine the year when the population increased least rapidly. (c) Find the average rate of change of the population of the United States from 2004 through 2009. (d) Use the average rate of change of the population to predict the population of the United States in 2020.25E26E27E28E29E30E31E32E33E34E35ESketching a Line in the Plane In Exercises 35-42, sketch the graph of the equation. y1=3(x+4)37E38E39EFinding 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)42E43EFinding 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)45E46E47EUsing Intercepts Show that the line with intercepts {a, 0) and (0, b) has the following equation. xa+yb=1,a0,b0Writing 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)50E51EWriting an Equation in General Form In Exercises 4954, use the result of Exercise 48 to write an equation of the line in general form. Point on line: (3, 4) x-intercept: (a, 0) y-intercept: (0, a) (a0)53E54E55E56E58EFinding 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=259E61E60E62ERate 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 year64ERate 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 year66ECollinear 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)68E69E70EWRITING ABOUT CONCEPTS Finding Points of Intersection In Exercises 6971, find the coordinates of the point of intersection of the given segments. Explain your reasoning.72E73E74E75E76EChoosing 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.78E79E80E87EDistance Write the distance d between the point (3, 1) and the line y=mx+4 in terms of m. Use a graphing utility to graph the equation. When is the distance 0? Explain the result geometrically.81E82E83E84E85E86EProof Prove that the diagonals of a rhombus intersect at right angles. (A rhombus is a quadrilateral with sides of equal lengths.)90EProof Prove that if the points (x1,y1) and (x2,y2) lie on the same line as (x1*,y1*) and (x2*,y2*) then y2y1*x2x1*=y2y1x2x1. Assume x1x2 and x1x2*92E93E94ETrue or False? In Exercises 85 and 86, determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. If a line contains points in both the first and third quadrants, then its slope must be positive.96E1EEvaluating a Function In Exercises 5-12, evaluate the function at the given value(s) of the independent variable. Simplify the results. f(x)=x+5 (a) f(4) (b) f (11) (c) f (4) (d) f(x+x)Evaluating a Function In Exercises 5-12, evaluate the function at the given value(s) of the independent variable. Simplify the results. g(x)=5x2 (a) g (0) (b) g(5) (c) g(2) (d) g(t1)4E5E6E7E8E9E10EFinding the Domain and Range of a Function In Exercises 13-22, find the domain and range of the function. f(x)=4x212E13EFinding the Domain and Range of a Function In Exercises 13-22, find the domain and range of the function. h(x)=4x215EFinding the Domain and Range of a Function In Exercises 13-22, find the domain and range of the function. h(x)=x+3Finding the Domain and Range of a Function In Exercises 13-22, find the domain and range of the function. f(x)=16x218E19E20EFinding the Domain and Range of a Function In Exercises 13-22, find the domain and range of the function. f(x)=3xFinding the Domain and Range of a Function In Exercises 13-22, find the domain and range of the function. f(x)=x2x+423EFinding the Domain of a Function In Exercises 23-26, find the domain of the function. f(x)=x23x+225E26E27EFinding the Domain of a Function In Exercises 23-26, find the domain of the function. g(x)=1x24Finding the Domain and Range of a Piecewise Function In Exercises 27-30, evaluate the function at the given value(s) of the independent variable. Then find the domain and range. f(x)={2x+1,x02x+2,x0 (a) f(1) (b) f (0) (c) f (2) (d) f(t2+1)Finding the Domain and Range of a Piecewise Function In Exercises 27-30, evaluate the function at the given value(s) of the independent variable. Then find the domain and range. f(x)={x2+2,x12x2+2,x1 (a) f(2) (b) f (0) (c) f (1) (d) f(s2+2)31EFinding the Domain and Range of a Piecewise Function In Exercises 27-30, evaluate the function at the given value(s) of the independent variable. Then find the domain and range. f(x)={x+4,x5(x5)2,x5 (a) f(3) (b) f(0) (c) f(5) (d) f(10)Sketching a Graph of a Function In Exercises 31-38, sketch a graph of the function and find its domain and range. Use a graphing utility to verify your graph. f(x)=4x34E35E36E37E38E39E40E41E42E43EUsing the Vertical Line Test In Exercises 39-42, use the Vertical Line Test to determine whether y is a function of x. To print an enlarged copy of the graph, go to MathGraphs.com. x24y=0Using the Vertical Line Test In Exercises 39-42, use the Vertical Line Test to determine whether y is a function of x. To print an enlarged copy of the graph, go to MathGraphs.com. y={x+1,x0x+2,x046E47EDeciding Whether an Equation Is a Function In Exercises 43-46, determine whether y is a function of x. x2+y=1649E50E51E52E53E54EMatching In Exercises 51-56, use the graph of y=f(x) to match the function with its graph. y=f(x+5)Matching In Exercises 51-56, use the graph of y=f(x) to match the function with its graph. y=f(x)557E58E59EMatching In Exercises 51-56, use the graph of y=f(x) to match the function with its graph. y=f(x1)+361ESketching Transformations Use the graph of f shown in the figure to sketch the graph of" each function. To print an enlarged copy of the graph, go to MathGrafths.com. (a) f(x4) (b) f(x+2) (c) f(x)+4 (d) f(x)1 (e) f(x) (f) 12f(x) (g) f(x) (h) f(x)63E64E65E66E67E68E69E70EEvaluating Composite Functions Use the graphs of f and g to evaluate each expression. If the result is undefined, explain why. (a) (fg)(3) (b) g(f(2)) (c) g(f(5)) (d) (fg)(3) (e) (gf)(1) (f) f(g(1))Ripples A pebble is dropped into a calm pond, causing ripples in the form of concentric circles. The radius (in feet) of the outer ripple is given by r(t)=0.6t, where t is the time in seconds after the pebble strikes the water. The area of the circle is given by the function A(r)=r2. Find and interpret (Ar)(t).73E74E75E76E77E78E79EEven and Odd Functions and Zeros of Functions In Exercises 75-78, determine whether the function is even, odd, or neither. Then find the zeros of the function. Use a graphing utility to verify your result. f(x)=x381E82E83EWriting Functions In Exercises 79-82, write an equation for a function that has the given graph. Line segment connecting (3, 1) and (5, 8)Writing FunctionsIn Exercises 79-82, write an equation for a function that has the given graph. The bottom half of the parabola x+y2=086E87E88E89E90EDomain Find the value of c such that the domain of f(x)=cx2 is [5,5]Domain Find all values of c such that the domain of f(x)=x+3x2+3cx+6 is the set of all real numbers.93E94E95E96E97E98E99E100E101E102E103EVolume An open box of maximum volume is to be made from a square piece of material 24 centimeters on a side by cutting equal squares from the comers and turning up the sides (see figure). (a) Write the volume V as a function of x, the length of the corner squares. What is the domain of the function? (b) Use a graphing utility to graph the volume function and approximate the dimensions of the box that yield a maximum volume. (c) Use the table feature of a graphing utility to verify your answer in part (b). (The first two rows of the table are shown.) Height, x Length and Width Volume, V 1 24 2(1) 1[24 2(1)]2 = 484 2 24 2(2) 2[24 2(2)]2 = 800105E106E107ETrue or False?In Exercises 103-108, determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. If f is a function, then f(ax)=af(x).109E110E111E112E1EQuiz ScoresThe ordered pairs represent the scores on two consecutive 15point quizzes for a class of 15 students. (7, 13), (9, 7), (14, 14), (15, 15), (10, 15), (9, 7), (11, 14), (7, 14), (14, 11). (14, 15), (8, 10), (15, 9), (10, 11), (9, 10), (11, 10) (a) Plot the data. From the graph, does the relationship between consecutive scores appear to be approximately linear? (b) If the data appear to be approximately linear, find a linear model for the data. If not, give some possible explanations.3E4E5E6E7E8EEngine PerformanceA V8 car engine is coupled to a dynamometer, and the horsepower y is measured at different engine speeds x (in thousands of revolutions per minute). The results are shown in the table. X 1 2 3 4 5 6 y 40 85 140 200 225 245 (a) Use the regression capabilities of a graphing utility to find a cubic model for the data. (b) Use a graphing utility to plot the data and graph the model. (c) Use the model to approximate the horsepower when the engine is running at 4500 revolutions per minute.10E11E12E