Bartleby Sitemap - Textbook Solutions
All Textbook Solutions for Calculus
Matching In Exercises 3-6, match the equation with its graph. [The graphs are labeled (a), (b). (c), and (d).] y=32x+3Matching In Exercises 3-6, match the equation with its graph. [The graphs are labeled (a), (b). (c), and (d).] y=9x2Matching In Exercises 3-6, match the equation with its graph. [The graphs are labeled (a), (b). (c), and (d).] y=3x2Matching In Exercises 3-6, match the equation with its graph. [The graphs are labeled (a), (b). (c), and (d).]5ESketching 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=4x28ESketching a Graph by Point Plotting In Exercises 514, sketch the graph of the equation by point plotting. y=| x+2 |Sketching a Graph by Point Plotting In Exercises 716, 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 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=2x5Finding Intercepts In Exercises 19-28, find any intercepts any intercepts. y=4x2+3Finding Intercepts In Exercises 19-28, find any intercepts 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+127E28E29E30E31E32E33E34E35E36E37E38E39E40E41E42EUsing 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+244EUsing 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+546E47E48EUsing 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=8xUsing 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+151EUsing 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|53EUsing 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=4Using Intercepts and Symmetry to Sketch a Graph In Exercises 3956, find any intercepts and test for symmetry. Then sketch the graph of the equation. x+3y2=656EFinding Points of Intersection In Exercises 57-62. find the points of intersection of the graphs of the equations. x+y=84xy=7Finding Points of Intersection In Exercises 57-62. find the points of intersection of the graphs of the equations. 3x2y=44x+2y=1059EFinding Points of Intersection In Exercises 57-62, find the points of intersection of the graphs of the equations. x=3y2y=x161EFinding Points of Intersection In Exercises 57-62. find the points of intersection of the graphs of the equations. x2+y2=16x+2y=4Finding 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+3x164E65EFinding 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=6xModeling Data The table shows the Gross Domestic Product, or GDP (in trillions of dollars), for selected years. (Source: U.S. Bureau of Economic Analysis) Year 1980 1985 1990 1995 GDP 2.8 4.2 5.8 7.4 Year 2000 2005 2010 GDP 10.0 12.6 14.5 (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 GDP (in trillions of dollars) and t represents the year, with t = 0 corresponding to 1980. (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 2020.Modeling 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 is R = 3.29x.Copper Wire The resistance y in ohms of 1000 feet of solid copper wire at 77F can be approximated by the model y=10,770x20.37,5x100 where x is the diameter of the wire in mils (0.001 in.). Use a graphing utility to graph the model. By about what factor is the resistance changed when the diameter of the wire is doubled?Using Solution Points For what values of k does the graph of y = kx3 pass through the point? (a) (1,4) (b) (2, 1) (c) (0, 0) (d) (1, 1)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)WRITING ABOUT CONCEPTS Writing Equations In Exercises 73 and 74, write an equation whose graph has the indicated property. (There may be more than one correct answer.) The graph has intercepts at x = 4, x = 3, and x = 8.EXPLORING CONCEPTS Using Intercepts Write an equation whose graph has intercepts at x=32,x=4andx=52. (There is more than one correct answer.)75EHOW 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.78ETrue 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.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.2E3E4E5E6EFinding 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) UndefinedSketching 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) 013EFinding 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.15EFinding 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=2Finding 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=3418E19E20E21E22E23EModeling Data The table shows the populations y (in millions) of the United States 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 2020.25EFinding the Slope and y-Intercept In Exercises 2934, find the slope and the y-intercept (if possible) of the line. x+y=127E28E29E30ESketching a Line in the Plane In Exercises 35-42, sketch the graph of the equation. y=332E33E34E35E36E37E38E39E40E41E42E43E44E45EFinding an Equation of a Line In Exercises 3946, find an equation of the line that passes through the points. Then sketch the line. (78,34),(54,14)Find an equation of the vertical line with x-intercept at 3.48EWriting 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)50E51E52EWriting 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)54EFinding 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=1Finding 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=3Finding 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=7Finding 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=2Finding 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=0Finding 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=8Finding Parallel and Perpendicular Lines In Exercises 5562, write the general forms of the equations of the lines through the point (a) parallel to the given line and (b) perpendicular to the given line. Point Line (34,78) 5x3y=062E63E64E65E66E67E68E69E70E71E72EAnalyzing 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).82E74E75EReimbursed Expenses A company reimburses its sales representatives $200 per day for lodging and meals plus $0.51 per mile driven. Write a linear equation giving the daily cost C to the company in terms of x, the number of miles driven. How much does it cost the company if a sales representative drives 137 miles on a given day?77EStraight-Line Depreciation A small business purchases a piece of equipment for $875. After 5 years, the equipment will be outdated, having no value. (a) Write a linear equation giving the value y of the equipment in terms of the time x (in years), 0 x 5. (b) Find the value of the equipment when x = 2. (c) Estimate (to two-decimal-place accuracy) the time when the value of the equipment is $200.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.80E83E84E85E86E87E88E89E90E91E92E93E94ETrue 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.96EEvaluating a Function In Exercises 110, evaluate the function at the given value(s) of the independent variable. Simplify the results. f(x)=7x4 (a) f(0) (b) f(3) (c) f(b) (d) f(x 1)2E3E4EEvaluating a Function In Exercises 1-10, evaluate the function at the given value(s) of the independent variable. Simplify the results.
5. f(x) = cos 2x
6E7E8E9E10E11E12E13EFinding the Domain and Range of a Function In Exercises 1322, 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)=16x218E19EFinding the Domain and Range of a Function In Exercises 1122, find the domain and range of the function. h(t)=cott21E22E23E24E25E26E27E28E29E30E31E32E33E34E35E36E37E38E39E40E41ESketching a Graph A student who commutes 27 miles to attend college remembers, after driving a few minutes, that a term paper that is due has been forgotten. Driving faster than usual, the student returns home, picks up the paper, and once again starts toward school. Sketch a possible graph of the students distance from home as a function of time.43EUsing 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=045E46E47E48E49E50E51E52E53E54E55EMatching In Exercises 51-56, use the graph of y=f(x) to match the function with its graph. y=f(x)557E58E59E60ESketching 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 MathGraphs.com. (a) f(x+3) (b) f(x1) (c) f(x)+2 (d) f(x)4 (e) 3f(x) (f) 14f(x) (g) f(x) (h) f(x)Sketching 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 MtilhGraphx.com. (a) f(x4) (b) f(x+2) (c) f(x)+4 (d) f(x)1 (e) 2f(x) (f) 12f(x) (g) f(x) (h) f(x)63E64E65E66EFinding Composite Functions In Exercises 63-66, find the composite functions fg and gf . Find the domain of each composite function. Are the two composite functions equal? f(x)=x2 g(x)=x68E69E70EEvaluating 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).73E74EThink About It In Exercises 71 and 72, find the coordinates or a second point on the graph of a function f when the given point is on the graph and the function is (a) even and (b) odd. (32,4)76EEver, and Odd Functions The graphs of f, g, and h are shown in the figure. Decide whether each function is even, odd, or neither.78E79E80E81E82E83E84E85E86E87E88E89E90EDomain 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.Graphical Reasoning An electronically controlled thermostat is programmed to lower the temperature during the night automatically (see figure). The temperature T in degrees Celsius is given in terms of t, the time in hours on a 24-hour clock. (a) Approximate T(4) and T(15). (b) The thermostat is reprogrammed to produce a temperature H(t)=T(t1). How does this change the temperature? Explain. (c) The thermostat is reprogrammed to produce a temperature H(t)=T(t1). How does this change the temperature? Explain.HOW DO YOU SEE IT? Water runs into a vase of height 30 centimeters at a constant rate. The vase is full after 5 seconds. Use this information and the shape of the vase shown to answer the questions when d is the depth of the water in centimeters and t is the time in seconds (see figure). (a) Explain why d is a function of t. (b) Determine the domain and range of the function. (c) Sketch a possible graph of the function (d) Use the graph in part (c) to approximate d(4). What does this represents?96E95E97E98EProof Prove that the function is odd f(x)=a2n+1x2n+1++a3x3+a1xProof Prove that the function is even. f(x)=a2nx2n+a2n2x2n2++a2x2+a0101E102ELength A right triangle is formed in the first quadrant by the x- and y-axes and a line through the point (3, 2) (see figure). Write the length L of the hypotenuse as a function of x.Volume 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 comer 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.105E106E107E108E109E110E111E112E1E2EHooke's Law Hookes Law states that the force F required to compress or stretch a spring (within its clastic limits) is proportional to the distance d that the spring is compressed or stretched from its original length. That is, F = kd, where k is a measure of the stiffness of the spring and is called the spring constant. The table shows the elongation d in centimeters of a spring when a force of F newtons is applied. F 20 40 60 80 100 d 1.4 2.5 4.0 5.3 6.6 (a) Use the regression capabilities of a graphing utility to find a linear model for the data. (b) Use a graphing utility to plot the data and graph the model. How well does the model fit the data? Explain. (c) Use the model to estimate the elongation of the spring when a force of 55 newtons is applied.4E5E6EBeam Strength Students in a lab measured the breaking strength S (in pounds) of wood 2 inches thick, x inches high, and 12 inches long. The results are shown in the table. x 4 6 8 10 12 S 2370 5460 10,310 16,250 23,860 (a) Use the regression capabilities of a graphing utility to find a quadratic model for the data. (b) Use a graphing utility to plot the data and graph the model. (c) Use the model to approximate the breaking strength when x = 2. (d) How many times greater is the breaking strength for a 4inch-high board than for a 2inch-high board? (e) How many times greater is the breaking strength for a 12inch-high board than for a 6inch-high board? When the height of a board increases by a factor, does the breaking strength increase by the same factor? Explain.Car Performance The time t (in seconds) required to attain a speed of s miles per hour from a standing start for a Volkswagen Passat is shown in the table. (Source: Car Driver) s 30 40 50 60 70 80 90 t 2.7 3.8 4.9 6.3 8.0 9.9 12.2 (a) Use the regression capabilities of a graphing utility to find a quadratic model for the data. (b) Use a graphing utility to plot the data and graph the model. (c) Use the graph in part (b) to state why the model is not appropriate for determining the times required to attain speeds of less than 20 miles per hour. (d) Because the test began from a standing start, add the point (0, 0) to the data. Fit a quadratic model to the revised data and graph the new model. (e) Does the quadratic model in part (d) more accurately model the behavior of the car? Explain.Engine Performance A 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.Boiling Temperature The table shows the temperatures T (in degrees Fahrenheit) at which water boils at selected pressures p (in pounds per square inch). (Source: Standard Handbook for Mechanical Engineers) p 5 10 14.696 (1 atmosphere) 20 T 162.24 193.21 212.00 227.96 p 30 40 60 80 100 T 250.33 267.25 292.71 312.03 327.81 (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 graph to estimate the pressure required for the boiling point of water to exceed 300F. (d) Explain why the model would not be accurate for pressures exceeding 100 pounds per square inch.11E12E