Solved: Proof Generalizing the statement in Exercise 39,

Polynomial Curve Fitting In Exercises 112, (a) determine the polynomial function whose graph passes through the points, and (b) sketch the graph of the polynomial function, showing the points. (2, 5), (3, 2), (4, 5)

Polynomial Curve Fitting In Exercises 112, (a) determine the polynomial function whose graph passes through the points, and (b) sketch the graph of the polynomial function, showing the points. (0, 0), (2, 2), (4, 0)

Polynomial Curve Fitting In Exercises 112, (a) determine the polynomial function whose graph passes through the points, and (b) sketch the graph of the polynomial function, showing the points. (2, 4), (3, 6), (5, 10)

Polynomial Curve Fitting In Exercises 112, (a) determine the polynomial function whose graph passes through the points, and (b) sketch the graph of the polynomial function, showing the points. (2, 4), (3, 4), (4, 4)

Polynomial Curve Fitting In Exercises 112, (a) determine the polynomial function whose graph passes through the points, and (b) sketch the graph of the polynomial function, showing the points. (1, 3), (0, 0), (1, 1), (4, 58)

Polynomial Curve Fitting In Exercises 112, (a) determine the polynomial function whose graph passes through the points, and (b) sketch the graph of the polynomial function, showing the points. (0, 42), (1, 0), (2, 40), (3, 72)

Polynomial Curve Fitting In Exercises 112, (a) determine the polynomial function whose graph passes through the points, and (b) sketch the graph of the polynomial function, showing the points. (2, 28), (1, 0), (0, 6), (1, 8), (2, 0)

Polynomial Curve Fitting In Exercises 112, (a) determine the polynomial function whose graph passes through the points, and (b) sketch the graph of the polynomial function, showing the points. (4, 18), (0, 1), (4, 0), (6, 28), (8, 135)

Polynomial Curve Fitting In Exercises 112, (a) determine the polynomial function whose graph passes through the points, and (b) sketch the graph of the polynomial function, showing the points. (2013, 5), (2014, 7), (2015, 12)

Polynomial Curve Fitting In Exercises 112, (a) determine the polynomial function whose graph passes through the points, and (b) sketch the graph of the polynomial function, showing the points. (2012, 150), (2013, 180), (2014, 240), (2015, 360)

Polynomial Curve Fitting In Exercises 112, (a) determine the polynomial function whose graph passes through the points, and (b) sketch the graph of the polynomial function, showing the points. (0.072, 0.203), (0.120, 0.238), (0.148, 0.284)

Polynomial Curve Fitting In Exercises 112, (a) determine the polynomial function whose graph passes through the points, and (b) sketch the graph of the polynomial function, showing the points. (1, 1), (1.189, 1.587), (1.316, 2.080), (1.414, 2.520)

Use sin 0 = 0, sin 2 = 1, and sin = 0 to estimate sin 3 .

Use log2 1 = 0, log2 2 = 1, and log2 4 = 2 to estimate log2 3.

Equation of a Circle In Exercises 15 and 16, find an equation of the circle that passes through the points. (1, 3), (2, 6), (4, 2)

Equation of a Circle In Exercises 15 and 16, find an equation of the circle that passes through the points. (5, 1), (3, 2), (1, 1)

Population The U.S. census lists the population of the United States as 249 million in 1990, 282 million in 2000, and 309 million in 2010. Fit a second-degree polynomial passing through these three points and use it to predict the populations in 2020 and 2030. (Source: U.S. Census Bureau)

Population The table shows the U.S. populations for the years 1970, 1980, 1990, and 2000. (Source: U.S. Census Bureau) Year 1970 1980 1990 2000 Population (in millions) 205 227 249 282 (a) Find a cubic polynomial that fits the data and use it to estimate the population in 2010. (b) The actual population in 2010 was 309 million. How does your estimate compare?

Net Profit The table shows the net profits (in millions of dollars) for Microsoft from 2007 through 2014. (Source: Microsoft Corp.) Year 2007 2008 2009 2010 Net Profit 14,065 17,681 14,569 18,760 Year 2011 2012 2013 2014 Net Profit 23,150 23,171 22,453 22,074 (a) Set up a system of equations to fit the data for the years 2007, 2008, 2009, and 2010 to a cubic model. (b) Solve the system. Does the solution produce a reasonable model for determining net profits after 2010? Explain.

Sales The table shows the sales (in billions of dollars) for Wal-Mart stores from 2006 through 2013. (Source: Wal-Mart Stores, Inc.) Year 2006 2007 2008 2009 Sales 348.7 378.8 405.6 408.2 Year 2010 2011 2012 2013 Sales 421.8 447.0 469.2 476.2 (a) Set up a system of equations to fit the data for the years 2006, 2007, 2008, 2009, and 2010 to a quartic model. (b) Solve the system. Does the solution produce a reasonable model for determining sales after 2010? Explain.

Network Analysis The figure shows the flow of traffic (in vehicles per hour) through a network of streets. (a) Solve this system for xi , i = 1, 2, . . . , 5. (b) Find the traffic flow when x3 = 0 and x5 = 100. (c) Find the traffic flow when x3 = x5 = 100.

Network Analysis The figure shows the flow of traffic (in vehicles per hour) through a network of streets. (a) Solve this system for xi, i = 1, 2, . . . , 5. (b) Find the traffic flow when x2 = 200 and x3 = 50. (c) Find the traffic flow when x2 = 150 and x3 = 0.

Network Analysis The figure shows the flow of traffic (in vehicles per hour) through a network of streets. (a) Solve this system for xi, i = 1, 2, 3, 4. (b) Find the traffic flow when x4 = 0. (c) Find the traffic flow when x4 = 100. (d) Find the traffic flow when x1 = 2x2.

Network Analysis Water is flowing through a network of pipes (in thousands of cubic meters per hour), as shown in the figure. (a) Solve this system for the water flow represented by xi , i = 1, 2, . . . , 7. (b) Find the water flow when x1 = x2 = 100. (c) Find the water flow when x6 = x7 = 0. (d) Find the water flow when x5 = 1000 and x6 = 0.

Network Analysis Determine the currents I1, I2, and I3 for the electrical network shown in the figure.

Network Analysis Determine the currents I1, I2, I3, I4, I5, and I6 for the electrical network shown in the figure.

Network Analysis (a) Determine the currents I1, I2, and I3 for the electrical network shown in the figure. (b) How is the result affected when A is changed to 2 volts and B is changed to 6 volts?

CAPSTONE (a) Explain how to use systems of linear equations for polynomial curve fitting. (b) Explain how to use systems of linear equations to perform network analysis.

Temperature In Exercises 29 and 30, the figure shows the boundary temperatures (in degrees Celsius) of an insulated thin metal plate. The steady-state temperature at an interior junction is approximately equal to the mean of the temperatures at the four surrounding junctions. Use a system of linear equations to approximate the interior temperatures T1, T2, T3, and T4. 1

Temperature In Exercises 29 and 30, the figure shows the boundary temperatures (in degrees Celsius) of an insulated thin metal plate. The steady-state temperature at an interior junction is approximately equal to the mean of the temperatures at the four surrounding junctions. Use a system of linear equations to approximate the interior temperatures T1, T2, T3, and T4. 2

Partial Fraction Decomposition In Exercises 3134, use a system of equations to find the partial fraction decomposition of the rational expression. Solve the system using matrices. 4x2 (x + 1)2(x 1) = A x 1 + B x + 1 + C (x + 1)2

Partial Fraction Decomposition In Exercises 3134, use a system of equations to find the partial fraction decomposition of the rational expression. Solve the system using matrices. 3x2 7x 12 (x + 4)(x 4)2 = A x + 4 + B x 4 + C (x 4)2

Partial Fraction Decomposition In Exercises 3134, use a system of equations to find the partial fraction decomposition of the rational expression. Solve the system using matrices. x2 3x 2 (x + 2)(x 2)2 = A x + 2 + B x 2 + C (x 2)2

Partial Fraction Decomposition In Exercises 3134, use a system of equations to find the partial fraction decomposition of the rational expression. Solve the system using matrices. + 2)(x 2)2 = A x + 2 + B x 2 + C (x 2)2

Calculus In Exercises 35 and 36, find the values of x, y, and that satisfy the system of equations. Such systems arise in certain problems of calculus, and is called the Lagrange multiplier. 2x x + 2y y + + 4 = = = 0 0 0

Calculus In Exercises 35 and 36, find the values of x, y, and that satisfy the system of equations. Such systems arise in certain problems of calculus, and is called the Lagrange multiplier. 2x 2x + 2y y + + 2 + + 2 1 100 = = = 0 0 0

Calculus The graph of a parabola passes through the points (0, 1) and ( 1 2, 1 2) and has a horizontal tangent line at ( 1 2, 1 2). Find an equation for the parabola and sketch its graph.

Calculus The graph of a cubic polynomial function has horizontal tangent lines at (1, 2) and (1, 2). Find an equation for the function and sketch its graph.

Guided Proof Prove that if a polynomial function p(x) = a0 + a1x + a2x2 is zero for x = 1, x = 0, and x = 1, then a0 = a1 = a2 = 0. Getting Started: Write a system of linear equations and solve the system for a0, a1, and a2. (i) Substitute x = 1, 0, and 1 into p(x). (ii) Set each result equal to 0. (iii) Solve the resulting system of linear equations in the variables a0, a1, and a2.

Proof Generalizing the statement in Exercise 39, if a polynomial function p(x) = a0 + a1x + . . . + an1xn1 is zero for more than n 1 x-values, then a0 = a1 = . . . = an1 = 0. Use this result to prove that there is at most one polynomial function of degree n 1 (or less) whose graph passes through n points in the plane with distinct x-coordinates.

(a) The graph of a function f passes through the points (0, 1), (2, 1 3), and (4, 1 5). Find a quadratic function whose graph passes through these points. (b) Find a polynomial function p of degree 2 or less that passes through the points (0, 1), (2, 3), and (4, 5). Then sketch the graph of y = 1'p(x) and compare this graph with the graph of the polynomial function found in part (a).

Writing Try to find a polynomial to fit the data shown in the table. What happens, and why? x 12334 y 11234

Polynomial Curve Fitting In Exercises 1-12, (a) determine the polynomial function whose graph passes through the points, and (b) sketch the graph of the polynomial function, showing the points. (2, 5), (3, 2), (4, 5)

Polynomial Curve Fitting In Exercises 1-12, (a) determine the polynomial function whose graph passes through the points, and (b) sketch the graph of the polynomial function, showing the points. (0, 0), (2, ?2), (4, 0)

Polynomial Curve Fitting In Exercises 1-12, (a) determine the polynomial function whose graph passes through the points, and (b) sketch the graph of the polynomial function, showing the points. (2, 4), (3, 6), (5, 10)

Polynomial Curve Fitting In Exercises 1-12, (a) determine the polynomial function whose graph passes through the points, and (b) sketch the graph of the polynomial function, showing the points. (2, 4), (3, 4), (4, 4)

Polynomial Curve Fitting In Exercises 1-12, (a) determine the polynomial function whose graph passes through the points, and (b) sketch the graph of the polynomial function, showing the points. (?1, 3), (0, 0), (1, 1), (4, 58)

Polynomial Curve Fitting In Exercises 1-12, (a) determine the polynomial function whose graph passes through the points, and (b) sketch the graph of the polynomial function, showing the points. (0, 42), (1, 0), (2, ?40), (3, ?72)

Polynomial Curve Fitting In Exercises 1-12, (a) determine the polynomial function whose graph passes through the points, and (b) sketch the graph of the polynomial function, showing the points. (?2, 28), (?1, 0), (0, ?6), (1, ?8), (2, 0)

Polynomial Curve Fitting In Exercises 1-12, (a) determine the polynomial function whose graph passes through the points, and (b) sketch the graph of the polynomial function, showing the points. (?4, 18), (0, 1), (4, 0), (6, 28), (8, 135)

Polynomial Curve Fitting In Exercises 1-12, (a) determine the polynomial function whose graph passes through the points, and (b) sketch the graph of the polynomial function, showing the points. (2013, 5), (2014, 7), (2015, 12)

Polynomial Curve Fitting In Exercises 1-12, (a) determine the polynomial function whose graph passes through the points, and (b) sketch the graph of the polynomial function, showing the points. (2012, 150), (2013, 180), (2014, 240), (2015, 360)

Polynomial Curve Fitting In Exercises 1-12, (a) determine the polynomial function whose graph passes through the points, and (b) sketch the graph of the polynomial function, showing the points. (0.072, 0.203), (0.120, 0.238), (0.148, 0.284)

Polynomial Curve Fitting In Exercises 1-12, (a) determine the polynomial function whose graph passes through the points, and (b) sketch the graph of the polynomial function, showing the points. (1, 1), (1.189, 1.587), (1.316, 2.080), (1.414, 2.520)

Use \(\sin 0=0, \sin \frac{\pi}{2}=1\), and \(\sin \pi=0\) to estimate \(\sin \frac{\pi}{3}\). Text Transcription: sin 0=0,sin pi/2=1 sin pi=0 sin pi/3

Use \(\log _{2} 1=0, \log _{2} 2=1\), and \(\log _{2} 4=2\) to estimate \(\log _{2} 3\). Text Transcription: Log_2 1=0, log_2 2=1 Log_2 4=2 Log_2 3

Equation of a Circle In Exercises 15 and 16, find an equation of the circle that passes through the points. (1, 3), (?2, 6), (4, 2)

Equation of a Circle In Exercises 15 and 16, find an equation of the circle that passes through the points. (?5, 1), (?3, 2), (?1, 1)

Population The U.S. census lists the population of the United States as 249 million in 1990, 282 million in 2000, and 309 million in 2010. Fit a second-degree polynomial passing through these three points and use it to predict the populations in 2020 and 2030. (Source: U.S. Census Bureau)

Population The table shows the U.S. populations for the years 1970, 1980, 1990, and 2000. (Source: U.S. Census Bureau) (a) Find a cubic polynomial that fits the data and use it to estimate the population in 2010. (b) The actual population in 2010 was 309 million. How does your estimate compare?

Net Profit The table shows the net profits (in millions of dollars) for Microsoft from 2007 through 2014. (Source: Microsoft Corp.) (a) Set up a system of equations to fit the data for the years 2007, 2008, 2009, and 2010 to a cubic model. (b) Solve the system. Does the solution produce a reasonable model for determining net profits after 2010? Explain.

Sales The table shows the sales (in billions of dollars) for Wal-Mart stores from 2006 through 2013. (Source: Wal-Mart Stores, Inc.) (a) Set up a system of equations to fit the data for the years 2006, 2007, 2008, 2009, and 2010 to a quartic model. (b) Solve the system. Does the solution produce a reasonable model for determining sales after 2010? Explain.

Network Analysis The figure shows the flow of traffic (in vehicles per hour) through a network of streets. (a) Solve this system for \(x_{i}, i=1,2, \ldots, 5\). (b) Find the traffic flow when \(x_{3}=0 \text { and } x_{5}=100\). (c) Find the traffic flow when \(x_{3}=x_{5}=100\). Text Transcription: x_i, i=1,2, …, 5 x_3=0 and x_5=100 x_3=x_5=100

Network Analysis The figure shows the flow of traffic (in vehicles per hour) through a network of streets. (a) Solve this system for \(x_{i}, i=1,2, \ldots, 5\). (b) Find the traffic flow when \(x_{2}=200 \text { and } x_{3}=50\). (c) Find the traffic flow when \(x_{2}=150 \text { and } x_{3}=0\). Text Transcription: x_{i}, i=1,2, …, 5 x_{2}=200 and x_{3}=50 x_{2}=150 and x_{3}=0

Network Analysis The figure shows the flow of traffic (in vehicles per hour) through a network of streets. (a) Solve this system for \(x_{i}, i=1,2,3,4\). (b) Find the traffic flow when \(x_{4}=0\). (c) Find the traffic flow when \(x_{4}=100\). (d) Find the traffic flow when \(x_{1}=2 x_{2}\). Text Transcription: x_{i}, i=1,2,3,4 x_{4}=0 x_{4}=100 x_{1}=2 x_{2}

Network Analysis Water is flowing through a network of pipes (in thousands of cubic meters per hour), as shown in the figure. (a) Solve this system for the water flow represented by \(x_{i}, i=1,2, \ldots .7\). (b) Find the water flow when \(x_{1}=x_{2}=100\). (c) Find the water flow when \(x_{6}=x_{7}=0\). (d) Find the water flow when \(x_{5}=1000 \text { and } x_{6}=0\). Text Transcription: x_{i}, i=1,2, \ldots .7 x_{1}=x_{2}=100 x_{6}=x_{7}=0 x_{5}=1000 \text { and } x_{6}=0

Network Analysis Determine the currents \(I_{1}, I_{2}, \text { and } I_{3}\) for the electrical network shown in the figure. Text Transcription: I_{1}, I_{2}, and I_{3}

Network Analysis Determine the currents \(I_{1}, I_{2}, I_{3},I_{4}, I_{5} \text { and } I_{6}\) for the electrical network shown in the figure. Text Transcription: I_{1}, I_{2}, I_{3},I_{4}, I_{5} and I_{6}

Network Analysis (a) Determine the currents \(I_{1}, I_{2}, \text { and } I_{3}\) for the electrical network shown in the figure. (b) How is the result affected when A is changed to 2 volts and B is changed to 6 volts? Text Transcription: I_{1}, I_{2}, and I_{3}

CAPSTONE (a) Explain how to use systems of linear equations for polynomial curve fitting. (b) Explain how to use systems of linear equations to perform network analysis.

Temperature In Exercises 29 and 30, the figure shows the boundary temperatures (in degrees Celsius) of an insulated thin metal plate. The steady-state temperature at an interior junction is approximately equal to the mean of the temperatures at the four surrounding junctions. Use a system of linear equations to approximate the interior temperatures \(T_{1}, T_{2}, T_{3}, \text { and } T_{4}\). Text Transcription: T_{1}, T_{2}, T_{3}, and T_{4}

Temperature In Exercises 29 and 30, the figure shows the boundary temperatures (in degrees Celsius) of an insulated thin metal plate. The steady-state temperature at an interior junction is approximately equal to the mean of the temperatures at the four surrounding junctions. Use a system of linear equations to approximate the interior temperatures \(T_{1}, T_{2}, T_{3}, \text { and } T_{4}\). Text Transcription: T_{1}, T_{2}, T_{3}, and T_{4}

Partial Fraction Decomposition In Exercises 31-34, use a system of equations to find the partial fraction decomposition of the rational expression. Solve the system using matrices. \(\frac{4 x^{2}}{(x+1)^{2}(x-1)}=\frac{A}{x-1}+\frac{B}{x+1}+\frac{C}{(x+1)^{2}}\) Text Transcription: 4x^2/(x+1)^2(x-1)=A/x-1+B/x+1+C/(x+1)^2

Partial Fraction Decomposition In Exercises 31-34, use a system of equations to find the partial fraction decomposition of the rational expression. Solve the system using matrices. \(\frac{3 x^{2}-7 x-12}{(x+4)(x-4)^{2}}=\frac{A}{x+4}+\frac{B}{x-4}+\frac{C}{(x-4)^{2}}\) Text Transcription: 3x^2-7x-12/(x+4)(x-4)^2=A/x+4+B/x-4+C/(x-4)^2

Partial Fraction Decomposition In Exercises 31-34, use a system of equations to find the partial fraction decomposition of the rational expression. Solve the system using matrices. \(\frac{3 x^{2}-3 x-2}{(x+2)(x-2)^{2}}=\frac{A}{x+2}+\frac{B}{x-2}+\frac{C}{(x-2)^{2}}\) Text Transcription: 3x^2-3x-2/(x+2)(x-2)^2=A/x+2+B/x-2+C/(x-2)^2

Partial Fraction Decomposition In Exercises 31-34, use a system of equations to find the partial fraction decomposition of the rational expression. Solve the system using matrices. \(\frac{20-x^{2}}{(x+2)(x-2)^{2}}=\frac{A}{x+2}+\frac{B}{x-2}+\frac{C}{(x-2)^{2}}\) Text Transcription: 20-x^2/(x+2)(x-2)^2=A/x+2+B/x-2+C/(x-2)^2

Calculus In Exercises 35 and 36, find the values of x, y, and \(\lambda\) that satisfy the system of equations. Such systems arise in certain problems of calculus, and \(\lambda\) is called the Lagrange multiplier. \(\begin{aligned} 2 x+\lambda &=0 \\ 2 y+\lambda &=0 \\ x+y-4 &=0 \end{aligned} \) Text Transcription: Lambda 2 x+lambda =0 2 y+lambda =0 x+y-4 &=0

Calculus In Exercises 35 and 36, find the values of x, y, and \(\lambda\) that satisfy the system of equations. Such systems arise in certain problems of calculus, and \(\lambda\) is called the Lagrange multiplier. \(\begin{aligned} 2 y+2 \lambda+2 &=0 \\ 2 x+\lambda+1 &=0 \\ 2 x+y-100 &=0 \end{aligned} \) Text Transcription: Lambda 2 y+2 lambda+2 =0 2 x+lambda+1 =0 2 x+y-100 =0

Calculus The graph of a parabola passes through the points (0, 1) and \(\left(\frac{1}{2}, \frac{1}{2}\right)\) and has a horizontal tangent line at \(\left(\frac{1}{2}, \frac{1}{2}\right)\). Find an equation for the parabola and sketch its graph. Text Transcription: (1/2, 1/2)

Calculus The graph of a cubic polynomial function has horizontal tangent lines at (1, -2) and (-1, 2). Find an equation for the function and sketch its graph.

Guided Proof Prove that if a polynomial function \(p(x)=a_{0}+a_{1} x+a_{2} x^{2}\) is zero for x = -1, x = 0, and x =1, then \(a_{0}=a_{1}=a_{2}=0\). Getting Started: Write a system of linear equations and solve the system for \(a_{0}, a_{1}, \text { and } a_{2}\). (i) Substitute x = - 1, 0, and 1 into p(x). (ii) Set each result equal to 0. (iiii) Solve the resulting system of linear equations in the variables \(a_{0}, a_{1}, \text { and } a_{2}\). Text Transcription: p(x)=a_0+a_1 x+a_2 x^2 a_0=a_1=a_2=0 a_0, a_1, and a_2

Proof Generalizing the statement in Exercise 39, if a polynomial function \(p(x)=a_{0}+a_{1} x+\cdots+a_{n-1} x^{n-1}\) is zero for more than n - 1 x-values, then \(a_{0}=a_{1}=\cdots=a_{n-1}=0\). Use this result to prove that there is at most one polynomial function of degree n - 1 (or less) whose graph passes through n points in the plane with distinct x-coordinates. Text Transcription: p(x)=a_0+a_1 x+...+a_n-1 x^n-1 a_0=a_1=...=a_n-1=0.

(a) The graph of a function f passes through the points (0, 1), \(\left(2, \frac{1}{3}\right)\), and \(\left(4, \frac{1}{5}\right)\). Find a quadratic function whose graph passes through these points. (b) Find a polynomial function p of degree 2 or less that passes through the points (0, 1), (2, 3), and (4, 5). Then sketch the graph of y = 1/p(x) and compare this graph with the graph of the polynomial function found in part (a). Text Transcription: (2, 1/3) (4,1/5)

Writing Try to find a polynomial to fit the data shown in the table. What happens, and why? x 12334 y 11234