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All Textbook Solutions for Calculus (MindTap Course List)

Finding Intercepts In Exercises 1-4, find any intercepts. y=5x8 Finding Intercepts In Exercises 1-4, find any intercepts. y=x28x+12 Finding Intercepts In Exercises 1-4, find any intercepts. y=x3x4 4RE 5RE 6RE 7RE 8RE 9RE 10RE 11RE 12RE 13RE 14RE 15RE 16RE 17RE 18RE 19RE 20RE 21RE 22RE 23RE 24RE 25RE 26RE 27RE 28RE 29RE Sketching a Line in the Plane In Exercises 27-30, sketch the graph of the equation. 3x+2y=12 31RE 32RE Finding Equations of Lines Find equations of the lines passing through (-3,5) and having the following characteristics. (a) Slope of 716 (b) Parallel to the line 5x3y=3 (c) Perpendicular to the line 3x+4y=8 (d) Parallel to the y-axis 34RE Rate of Change The purchase price of a new machine is $12,500, and its value will decrease by $850 per year. Use this information to write a linear equation that gives the value V of the machine t years after it is purchased. Find its value at the end of 3 years.Break-Even Analysis A contractor purchases a piece of equipment for $36,500 that costs an average of $9.25 per hour for fuel and maintenance. The equipment operator is paid $13.50 per hour, and customers are charged $30 per hour. (a) Write a linear equation for the cost C of operating this equipment for t hours. (b) Write a linear equation for the revenue R derived from t hours of use. (c) Find the break-even point for this equipment by finding the time at which R=C.37RE 38RE Evaluating a Function In Exercises 37-40, evaluate the function at the given value(s) of the independent variable. Simplify the results. f(x)=4x2 f(x+x)f(x)x 40RE 41RE 42RE 43RE 44RE 45RE 46RE 47RE 48RE 49RE 50RE Transformations of Functions Use a graphing utility to graph f(x)=x23x2 Use the graph to write a formula for the function g shown in die figure.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?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 80RE 81RE 82RE 83RE 84RE 85RE 86RE 87RE 88RE 89RE 90RE 1PS Finding Tangent Lines There are two tangent lines from the point (0, 1) to the circle x2 + (y + l)2 = 1 (see figure). Find equations of these two lines by using the fact that each tangent line intersects the circle at exactly one point.Heaviside Function The Heaviside function H(x) is widely used in engineering applications. H(x)={ 1,x00,x0 Sketch the graph of the Heaviside function and the graphs of the following functions by hand. (a) H(x) 2 (b) H(x 2) (c) - H(x) (d) H(- x) (e) 12 H(x) (f) -H(x 2)+ 2 Sketching Transformations Consider the graph of the function f shown below. Use this graph to sketch the graphs of the following functions. To print an enlarged copy of the graph, go to MathGraphs.com. (a) f(x+1) (b) f(x)+1 (c) 2f(x) (d) f(x) (e) f(x) (f) | f(x) | (g) f(| x |)5PS 6PS 7PS 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.Slope of a Tangent Line Sketch the graph of the function f(x)=x and label the point (4, 2) on the graph. (a) Find the slope of the line joining (4, 2) and (9, 3). Is the slope of the tangent line at (4, 2) greater than or less than this number? (b) Find the slope of the line joining (4, 2) and (1, 1). Is the slope of the tangent line at (4, 2) greater than or less than this number? (c) Find the slope of the line joining (4, 2) and (4.41, 2.1). Is the slope of the tangent line at (4, 2) greater than or less than this number? (d) Find the slope of the line joining (4, 2) and (4 + h, f(4+h)) in terms of the nonzero number h. (e) What is the slope of the tangent line at (4, 2)? Explain how you arrived at your answer.11PS Graphing an Equation Explain how you would graph the equation y+| y |=x+| x |. Then sketch the graph.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 MathGraphs.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 listeners 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 the source. (a) Find the points on the x-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.Sound Intensity Suppose the speakers in Exercise 13 are 4 meters apart and the sound intensity of one speaker is k times 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.Lemniscate Let d1 and d2 be the distances from the point (x, y) to the points ( 1, 0) and (1, 0), respectively, as shown in the figure. Show that the equation of the graph of all points (x, y) satisfying d1d2 = 1 is (x2+y2)2=2(x2y2). This curve is called a lemniscate. Graph the lemniscate and identify three points on the graph.Finding Intercepts Describe how to find the x- and y-intercepts of the graph of an equation.CONCEPT CHECK 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 Matching In Exercises 3-6, match the equation with its graph. [The graphs are labeled (a), (b). (c), and (d).] y=9x2 Matching In Exercises 3-6, match the equation with its graph. [The graphs are labeled (a), (b). (c), and (d).] y=3x2 Matching In Exercises 3-6, match the equation with its graph. [The graphs are labeled (a), (b). (c), and (d).]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 716, 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 or each solution point accurate to two decimal places. y=5x (a) (2,y) (c) (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 or each solution point accurate to two decimal places. y=x55x (a) (0.5,y) (b) (x,4)Finding Intercepts In Exercises 19-28, find any intercepts. y=2x5 Finding Intercepts In Exercises 19-28, find any intercepts any intercepts. y=4x2+3 Finding Intercepts In Exercises 19-28, find any intercepts 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 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=x3+2 46E 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=xx+5 48E 49E 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 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=10x2+1 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 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. x2+4y2=4 Finding Points of Intersection In Exercises 57-62. find the points of intersection of the graphs of the equations. x+y=84xy=7 Finding Points of Intersection In Exercises 57-62. find the points of intersection of the graphs of the equations. 3x2y=44x+2y=10 Finding Points of Intersection In Exercises 57-62. find the points of intersection of the graphs of the equations. x2+y=153x+y=11 Finding Points of Intersection In Exercises 57-62, find the points of intersection of the graphs of the equations. x=3y2y=x1 61E Finding Points of Intersection In Exercises 57-62. find the points of intersection of the graphs of the equations. x2+y2=16x+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+x1y=x2+3x1 64E 65E Finding Points of Intersection Using Technology In Exercises 6366, use a graphing utility to find the points of intersection of the graphs of the equations. Check your results analytically. y=2x3+6 y=6x 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 the GDP (in trillions of dollars) and t represents the year, with t = 9 corresponding to 2009. (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 GDP in the year 2024.68E 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 is 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)EXPLORING CONCEPTS Using Intercepts Write an equation whose graph has intercepts at x=32,x=4andx=52. (There is more than one correct answer.)EXPLORING CONCEPTS 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.73E HOW DO YOU SEE IT? Use the graphs of the two equations to answer the questions below (a) What are the intercepts for each equation? (b) Determine the symmetry for each equation. (c) Determine the point of intersection of the two equations.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 x-axis, then (4,5) is also a point on the graph.76E 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. b24ac0 and a0. then the graph of y=ax2+bx+c has two x-intercepts.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 then a0, the graph of y=ax2+bx+c has only one x-intercept.Slope-Intercept Form In the form y = mx + b, what does m represent? What does b represent?Perpendicular Lines Is it possible for two lines with positive slopes to be perpendicular? Why or why not?Estimating Slope In Exercises 36, estimate the slope of the line from its graph. To print an enlarged copy of the graph, go to MathGraphs.com.4E 5E 6E 7E 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. (0,0), (-2, 3)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. (4.6), (4,1)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. (3,5),(5,5)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. (12,23),(34,16)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)Sketching Lines In Exercises 13 and 14. sketch the lines through the point with the indicated slopes. Make the sketches on the same set of coordinate axes. Point Slopes (3,4) (a) 1(b) -2(c) 32 (d) Undefined Sketching Lines In Exercises 13 and 14, sketch the lines through the point with the indicated slopes. Make the sketches on the same set of coordinate axes. Point Slopes (2, 5)(a) 3(b) 3(c) 13 (d) 0 15E Finding Points on a Line In Exercises 1518, 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.) PointSlope(4,3)misundefined.17E Finding Points on a Line In Exercises 1518, 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.) PointSlope(2,2)m=2 Finding an Equation of a Line In Exercises 19-24, find an equation of the line that passes through the point and has the indicated slope. Then sketch the line. PointSlope(0,3)m=34 20E 21E 22E 23E 24E 25E 26E 27E 28E 29E Finding the Slope and y-Intercept In Exercises 2934, find the slope and the y-intercept (if possible) of the line. x+y=1 31E 32E 33E 34E Sketching a Line in the Plane In Exercises 35-42, sketch the graph of the equation. y=3 36E 37E 38E 39E 40E 41E 42E 43E 44E 45E 46E 47E 48E 49E 50E 51E 52E 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 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. Point on line:(9,-2) x-intercept: (2a,0) y-intercept: (0,a) (a0)56E 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 (h) perpendicular to the given line. PointLine(7,2)x=1 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 (h) perpendicular to the given line. PointLine(1,0)y=3 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 (h) perpendicular to the given line. (3,2)x+y=7 Finding Parallel and Perpendicular Lines In Exercises 5762, write the general forms of the equations of the lines that pass through the point and are (a) parallel to the given line and (b) perpendicular to the given line. Point Slope (2,5) xy=2 Finding Parallel and Perpendicular Lines In Exercises 5762, write the general forms of the equations of the lines that pass through the point and are (a) parallel to the given line and (b) perpendicular to the given line. Point Slope (34,78) 5x3y=0 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 (h) perpendicular to the given line. Point Slope (56,12)7x+4y=8 63E 64E 65E 66E 67E Analyzing a Line A line is represented by the equation ax+by=4. (a) When is the line parallel to the x-axis? (b) When is the line parallel to the y-axis? (c) Give values for a and b such that the line has a slope of 58. (d) Give values for a and b such that the line is perpendicular to y=25x+3. (e) Give values for a and b such that the line coincides with the graph of 5x+6y=8.Tangent Line Find an equation of the line tangent to the circle x2+y2=169 at the point (5, 12).70E 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 74E Apartment Rental A real estate office manages an apartment complex with 50 units. When the rent is $780 per month, all 50 units are occupied. However, when the rent is $825, the average number of occupied units drops to 47. Assume that the relationship between the monthly rent p and the demand x is linear. (Note: The term demand refers to the number of occupied units.) (a) Write a linear equation giving die demand x in terms of the rent p. (b) Linear extrapolation Use a graphing utility to graph the demand equation and use the trace feature to predict the number of units occupied when the rent is raised to $855. (c) Linear interpolation Predict the number of units occupied when the rent is lowered to $795. Verify graphically.76E 77E 78E 79E 80E 81E 82E 83E 84E 85E True 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.Writing Describe how a relation and a function are different.2E 3E 4E 5E 6E Evaluating a Function In Exercises 5-12, evaluate the function at the given value(s) of the independent variable. Simplify the results. f(x)=x2+4 (a) f(2) (b) f(3) (c) f(2) (d) f(x+bx)8E 9E 10E 11E 12E 13E 14E 15E Finding the Domain and Range of a Function In Exercises 1322, find the domain and range of the function. h(x)=4x2 17E Finding the Domain and Range of a Function In Exercises 13-22, find the domain and range of the function. h(x)=x+3 Finding the Domain and Range of a Function In Exercises 13-22, find the domain and range of the function. f(x)=16x2 20E 21E 22E 23E 24E 25E 26E 27E 28E 29E 30E 31E

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