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All Textbook Solutions for Calculus: An Applied Approach (MindTap Course List)
4CPFind the slope-intercept form of the equation of the line that has a slope of 2 and passes through the point (1, 2).The sales per share for Amazon.com was $105.65 in 2011 and $134.40 in 2012. Using only this information, write a linear equation that gives the sales per share in terms of the year. Then estimate the sales per share in 2013. (Source: Amazon.com)7CP8CP1SWUIn Exercises 1 and 2, simplify the expression. 4(10)753SWU4SWU5SWU6SWU7SWU8SWU9SWU10SWU1E2E3E4EFinding the Slope and y-Intercept In Exercises 5-16, find the slope and y-intercept (if possible) of the equation of the line. y=x+7Finding the Slope and y-Intercept In Exercises 5-16, find the slope and y-intercept (if possible) of the equation of the line. y=4x+3Finding the Slope and y-Intercept In Exercises 5-16, find the slope and y-intercept (if possible) of the equation of the line. 5x+y=208E9EFinding the Slope and y-Intercept In Exercises 5-16, find the slope and y-intercept (if possible) of the equation of the line. 8x+3y=12Finding the Slope and y-Intercept In Exercises 5-16, find the slope and y-intercept (if possible) of the equation of the line. 3xy=1512EFinding the Slope and y-Intercept In Exercises 5-16, find the slope and y-intercept (if possible) of the equation of the line. x=4Finding the Slope and y-Intercept In Exercises 5-16, find the slope and y-intercept (if possible) of the equation of the line. x+5=0Finding the Slope and y-Intercept In Exercises 5-16, find the slope and y-intercept (if possible) of the equation of the line. y9=016EGraphing Linear Equations in Exercises 17-26, sketch the graph of the linear equation. Use a graphing utility to verify your result. See Example 1. y=218EGraphing Linear Equations in Exercises 17-26, sketch the graph of the linear equation. Use a graphing utility to verify your result. See Example 1. y=2x+1Graphing Linear Equations in Exercises 17-26, sketch the graph of the linear equation. Use a graphing utility to verify your result. See Example 1. y=3x2Graphing Linear Equations in Exercises 17-26, sketch the graph of the linear equation. Use a graphing utility to verify your result. See Example 1. 3x+2y=422EGraphing Linear Equations in Exercises 17-26, sketch the graph of the linear equation. Use a graphing utility to verify your result. See Example 1. 2xy3=024E25E26EFinding Slopes of Lines In Exercises 27-40, find the slope of the line passing through the pair of points. See Example 4. (0,2),(8,0).28EFinding Slopes of Lines In Exercises 27-40, find the slope of the line passing through the pair of points. See Example 4. (3,4),(5,2).30E31E32EFinding Slopes of Lines In Exercises 27-40, find the slope of the line passing through the pair of points. See Example 4. (8,3),(8,5)34EFinding Slopes of Lines In Exercises 27-40, find the slope of the line passing through the pair of points. See Example 4. (2,6),(1,6).36E37E38EFinding Slopes of Lines In Exercises 27-40, find the slope of the line passing through the pair of points. See Example 4. (23,52),(14,-56).Finding Slopes of Lines In Exercises 27-40, find the slope of the line passing through the pair of points. See Example 4. (78,34),(54,-14).41E42E43E44EFinding Points on a Line In Exercises 41-48, use the point on the line and the slope of the line to find three additional points through which the line passes. (There are many correct answers.) Point Slope (6,4)m=2346E47EFinding Points on a Line In Exercises 41-48, use the point on the line and the slope of the line to find three additional points through which the line passes. (There are many correct answers.) Point Slope (3,4) m is undefined.Using Slope In Exercises 49-52, use the concept of slope to determine whether the three points are collinear. (2,1),(1,0),(2,2)50E51E52EUsing the Point-Slope Form In Exercises 53-60, find an equation of the line that passes through the given point and has the given slope. Then sketch the line. See Example 5. Point Slope (0,3)m=3454EUsing the Point-Slope Form In Exercises 53-60, find an equation of the line that passes through the given point and has the given slope. Then sketch the line. See Example 5. Point Slope (2,7)m=056E57E58E59EUsing the Point-Slope Form In Exercises 5360, find an equation of the line that passes through the given point and has the given slope. Then sketch the line. See Example 5. PointSlope(32,0)m=1661E62E63E64E65E66E67E68E69E70E71E72EWriting an equation of a Line In Exercises 7174, find an equation of the line with the given characteristics. A line with a y-intercept at 10 and parallel to all horizontal lines74EFinding Parallel and Perpendicular Lines In Exercises 75-82, find equations of the lines that pass through the given point and are (a) parallel to the given line and (b) perpendicular to the given line. Then use a graphing utility to graph all three equations in the same viewing window using a square setting. See Example 7. Point Line (3,2)x+y=7Finding Parallel and Perpendicular Lines In Exercises 75-82, find equations of the lines that pass through the given point and are (a) parallel to the given line and (b) perpendicular to the given line. Then use a graphing utility to graph all three equations in the same viewing window using a square setting. See Example 7. Point Line (2,1)4x2y=377E78EFinding Parallel and Perpendicular Lines In Exercises 75-82, find equations of the lines that pass through the given point and are (a) parallel to the given line and (b) perpendicular to the given line. Then use a graphing utility to graph all three equations in the same viewing window using a square setting. See Example 7. Point Line (1,0)y=3=080E81E82EAverage Salary The graph shows the average salaries (in dollars) of postsecondary education administrators from 2008 through 2013. (Source: U.S. Bureau of Labor Statistics) (a) Determine the time periods in which the average salary increased the greatest and the least. (b) Find the slope of the line segment connecting the points for the years 2008 and 2013. (c) Interpret the meaning of the slope in part (b) in the context of the problem.Revenue The graph shows the revenue (in billions of dollars) for ATT for the years 2009 through 2013. (Source: ATT Inc.) (a) Determine the years in which the revenue increased the greatest and the least. (b) Find the slope of the line segment connecting the points for the years 2009 and 2013. (c) Interpret the meaning of the slope in part (b) in the context of the problem.85ETemperature Conversion Use the fact that water freezes at 0C (32F) and boils at 100C (212F). (a) Write a linear equation that expresses the relationship between the temperature in degrees Celsius C and the temperature in degrees Fahrenheit F. (b) A person has a temperature of 102.2F. What is this temperature on the Celsius scale? (c) The temperature in a room is 76F. What is this temperature on the Celsius scale?Population The resident population of Wisconsin (in thousands) was 5655 in 2009 and 5743 in 2013. Assume that the relationship between the population y and the year t is linear. Let t = 0 represent 2000. (Source: U.S. Census Bureau) (a) Write a linear model for the data. What is the slope and what does it tell you about the population? (b) Use the model to estimate the population in 2011. (c) Use your schools library, the Internet, or some other reference source to find the actual population in 2011. How close was your estimate? (d) Do you think your model could be used to predict the population in 2018? Explain.Personal Income Personal income (in billions of dollars) in the United States was 12,430 in 2008 and 14,167 in 2013. Assume that the relationship between the personal income y and the time t (in years) is linear. Let t = 0 represent 2000. (Source: U.S. Bureau of Economic Analysis) (a) Write a linear model for the data. (b) Estimate the personal incomes in 2011 and 2014. (c) Use your schools library, the Internet, or some other reference source to find the actual personal incomes in 2011 and 2014. How close were your estimates?Linear Depreciation A small business purchases a piece of equipment for $1025. After 5 years, the equipment will be outdated, having no value. (a) Write a linear equation giving the value y (in dollars) of the equipment in terms of the time t (in years), 0t5. (b) Use a graphing utility to graph the equation. (c) Move the cursor along the graph and estimate (to two-decimal-place accuracy) the value of the equipment after 3 years. (d) Move the cursor along the graph and estimate (to two-decimal-place accuracy) the time when the value of the equipment will be $600.Linear Depreciation A hospital purchases a $500,000 magnetic resonance imaging (MRI) machine that has a useful life of 9 years. The salvage value at the end of 9 years is $77,000. (a) Write a linear equation that describes the value y (in dollars) of the MRI machine in terms of the time t (in years), 0 t 9. (b) Find the value of the machine after 5 years. (c) Find the time when the value of the equipment will be $160,000.Choosing a Job As a salesperson, you receive a monthly salary of $2000, plus a commission of 7% 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 dollars) in terms of your monthly sales S (in dollars) for your current job and for your job offer. (b) Use a graphing utility to graph each equation and find the point of intersection. What does the point of intersection signify? (c) You think you can sell $20,000 worth of a product per month. Should you change jobs? Explain.HOW DO YOU SEE IT? Match the description of the situation with its graph. Then write the equation of the line. [The graphs are labeled (i), (ii), (iii), and (iv).] (a) You are paying $10 per week to repay a $100 loan. (b) An employee is paid $12.50 per hour plus $1.50 for each unit produced per hour. (c) A sales representative receives $50 per day for food plus $0.58 for each mile traveled. (d) A computer that was purchased for $600 depreciates $100 per year.1QY2QY3QY4QY5QY6QY7QY8QY9QY10QY11QYA business manufactures a product at a cost of $4.55 per unit and sells the product for $7.19 per unit. The companys initial investment to produce the product was $12,500. How many units must the company sell to break even?13QY14QY15QY16QY17QY18QY19QYA company had sales of $1,330,000 in 2011 and $1,800,000 in 2015. The companys sales can be modeled by a linear equation. Predict the sales in 2019 and 2022.21QY22QYDecide whether each equation defines y as a function of x. xy=1 b. x2+y2=4 c. y2+x=2 d. x2y=0Find the domain and range of each function. y=x+1 b. y={ x2,x0x,x0Find the values of f(x)=x25x+1 when x is 0, 1, and 4. Is f one-to-one?Given f(x)=x2+3, evaluate each expression. f(x+x) b. f(x+x)f(x)x5CPFind the inverse function of each function informally. a. f(x)=15x b. f(x)=x67CP8CPIn Exercises 1-6, simplify the expression. 5(1)26(1)+92SWU3SWU4SWU5SWU6SWU7SWU8SWU9SWU10SWU11SWU12SWUDeciding Whether Equations Are Functions In Exercises 1-8, decide whether the equation defines y as a function of x. See Example1. x2+y2=16Deciding Whether Equations Are Functions In Exercises 1-8, decide whether the equation defines y as a function of x. See Example 1. x+y2=4Deciding Whether Equations Are Functions In Exercises 1-8, decide whether the equation defines y as a function of x. See Example 1. 12x6y=3Deciding Whether Equations Are Functions In Exercises 1-8, decide whether the equation defines y as a function of x. See Example 1. 3x2y+5=0Deciding Whether Equations Are Functions In Exercises 1-8, decide whether the equation defines y as a function of x. See Example 1. x2+y=46EDeciding Whether Equations Are Functions In Exercises 1-8, decide whether the equation defines y as a function of x. See Example 1. y=| x+2 |Deciding Whether Equations Are Functions In Exercises 1-8, decide whether the equation defines y as a function of x. See Example 1. x2y23x2+4y2=0Vertical Line Test In Exercises 9-12, use the Vertical Line Test to determine whether y is a function of x. x2+y2=9Vertical Line Test In Exercises 9-12, use the Vertical Line Test to determine whether y is a function of x. xxy+y+1=0Vertical Line Test In Exercises 9-12, use the Vertical Line Test to determine whether y is a function of x. x2=xy1Vertical Line Test In Exercises 9-12, use the Vertical Line Test to determine whether y is a function of x. x=| y |Finding the Domain and Range of a Function In Exercises 13-16, find the domain and range of the function. Use interval notaion to write your result. See Example 2. f(x)=x3Finding the Domain and Range of a Function In Exercises 13-16, find the domain and range of the function. Use interval notaion to write your result. See Example 2. f(x)=2x3Finding the Domain and Range of a Function In Exercises 13-16, find the domain and range of the function. Use interval notaion to write your result. See Example 2. f(x)=4x2Finding the Domain and range of a Function In Exercises 1316, find the domain and range of the function. Use interval notation to write your result. See Example 2. f(x)=| x2 |Finding the Domain and Range of a Function In Exercises 17-24, find the domain and range of the function. Use a graphing utility to verify your results. See Example 2. f(x)=x22x+3Finding the Domain and Range of a Function In Exercises 17-24, find the domain and range of the function. Use a graphing utility to verify your results. See Example 2. f(x)=5x3+6x21Finding the Domain and range of a Function In Exercises 1724, find the domain and range of the function. Use a graphing utility to verify your results. See Example 2. f(x)=| x |xFinding the Domain and range of a Function In Exercises 1724, find the domain and range of the function. Use a graphing utility to verify your results. See Example 2. f(x)=xx9Finding the Domain and Range of a Function In Exercises 17-24, find the domain and range of the function. Use a graphing utility to verify your results. See Example 2. f(x)={ x5,x03x5,x0Finding the Domain and Range of a Function In Exercises 17-24, find the domain and range of the function. Use a graphing utility to verify your results. See Example 2. f(x)={ 3x+2,x02x,x0Finding the Domain and Range of a Function In Exercises 17-24, find the domain and range of the function. Use a graphing utility to verify your results. See Example 2. f(x)=x2x+4Finding the Domain and range of a Function In Exercises 1724, find the domain and range of the function. Use a graphing utility to verify your results. See Example 2. f(x)=x21xEvaluating a Function In Exercises 25-28, evaluate the function at the specified values of the independent variable. Simplify the results. See Example 3. f(x)=3x2 (a) f(0) (b) f(5) (c) f(x1)Evaluating a Function In Exercises 25-28, evaluate the function at the specified values of the independent variable. Simplify the results. See Example 3. f(x)=x24x+1 (a) f(1) (b) f(12) (c) f(c+2)Evaluating a Function In Exercises 25-28, evaluate the function at the specified values of the independent variable. Simplify the results. See Example 3. g(x)=1/x (a) g(15) (b) g(0.6) (c) g(x+4)Evaluating a Function In Exercises 25-28, evaluate the function at the specified values of the independent variable. Simplify the results. See Example 3. f(x)=| x |+4 (a) f(3) (b) f(0.8) (c) f(x+2)29E30EEvaluating a Function In Exercises 29-34, evaluate the difference quotient and simplify the result. See Example 4. g(x)=x+1g(4+x)g(4)xEvaluating a Function In Exercises 29-34, evaluate the difference quotient and simplify the result. See Example 4. f(x)=1xf(x)f(2)x2Evaluating a Function In Exercises 29-34, evaluate the difference quotient and simplify the result. See Example 4. f(x)=1x2f(x+x)f(x)x34ECombinations of Functions In Exercises 35-38, find (a) f(x)+g(x), (b) f(x)g(x), (c) f(x)g(x), (d) f(x)/g(x), (e) f(g(x)), and (f) g(f(x)), if defined. See Example 5. f(x)=2x5g(x)=43x36E37E38E39E40E41E42E43E44E45E46E47E48E49E50E51E52EFinding an Inverse Function In Exercises 45-56, find the inverse function of f. See Example 7. f(x)=9x2,0x354EFinding an Inverse Function In Exercises 45-56, find the inverse function of f. See Example 7. f(x)=x2/3,x056E57EDetermining Whether a Function is one-to-one In Exercises 5762, use a graphing utility to graph the function. Then use the Horizontal Line Test to determine whether the function is one-to-one. If it is, find its inverse function. f(x)=x259E60E61E62E63E64EWriting a Function Use the graph of f (x) = x2 to write an equation for each function whose graph is shown.Writing a Function Use the graph of f(x) = x3 to write an equation for each function whose graph is shown.67EHow do You See it? An electronically controlled thermostat in a home is programmed to lower the temperature automatically during the night. The temperature in the house T (in degrees Fahrenheit) is given in terms of t, the time in hours on a 24-hour clock (see figure). (a) Explain why T is a function of t. (b) Approximate T(4) and T(15). (c) The thermostat is reprogrammed to produce a temperature H for which H(t)=T(t1). How does this change the temperature? (d) The thermostat is reprogrammed to produce a temperature H for which H(t)=T(t)1. How does this change the temperature?69EBirths and Deaths From 2008 to 2012, the total numbers of births B (in thousands) and deaths D (in thousands) in the United States can be approximated by the models B(t)=4.917t3124.71t2+925.9t+2308 and D(t)=7.083t3222.64t2+2281.8t+10,104 where t represents the year, with t = 8 corresponding to 2008. Find B(t) D(t) and interpret this function. (Source: U.S. National Center for Health Statistics)Cost The inventor of a new game believes that the variable cost for producing the game is $2.89 per unit. The fixed cost is $8000. (a) Express the total cost C as a function of x, the number of games sold. (b) Find a formula for the average cost per unit C=Cx. (c) The selling price for each game is $6.89. How many units must be sold before the average cost per unit falls below the selling price?Demand The demand function for a commodity is p=14.751+0.01x,x0 where p is the price per unit and x is the number of units sold. (a) Find x as a function of p. (b) Find the number of units sold when the price is $10.Cost The weekly cost C of producing x units in a manufacturing process is given by C(x) = 70x + 500. The number of units x produced in t hours is given by x(t) = 40t. (a) Find and interpret C(x(t)). (b) Find the cost of 4 hours of production. (c) After how much time does the cost of production reach $18,000?Revenue For groups of 80 or more people, a charter bus company determines the rate r (in dollars per person) according to the formula r=150.05(n80),n80 where n is the number of people. (a) Express the revenue R for the bus company as a function of n. (b) Complete the table. n 100 125 150 175 200 225 250 R (c) Is the formula for the rate a good one for the company? Explain your reasoning.Cost, Revenue, and Profit A company invests $98,000 for equipment to produce a new product. Each unit of the product costs $12.30 and is sold for $17.98. Let x be the number of units produced and sold. (a) Write the total cost C as a function of x. (b) Write the revenue R as a function of x. (c) Write the profit P as a function of x.Profit A manufacturer charges $90 per unit for units that cost $60 to produce. To encourage large orders from distributors, the manufacturer will reduce the price by $0.01 per unit for orders in excess of 100 units. (For example, an order of 101 units would have a price of $89.99 per unit, and an order of 102 units would have a price of $89.98 per unit.) This price reduction is discontinued when the price per unit drops to $75. (a) Express the price per unit p as a function of the order size x. (b) Express the profit P as a function of the order size x.77E78E79E80E81E82E83E84E85E1CPFind the limit. limx2f(x) a. f(x)=x24x2 b. f(x)=| x2 |x2 c. f(x)={ x2,x20,x=23CP4CPFind the limit: limx2x38x2.Find the limit: limx3x2+x12x3Find the limit: limx0x+42x.8CPFind the limit of f(x) as x approaches 0. f(x)={ x2+1,x02x+1,x010CP11CPIn Exercises 1-4, simplify the expression by factoring. 2x3+x26x2SWU3SWU4SWU5SWU6SWU7SWU8SWU9SWUIn Exercises 9-12, find the domain and range of the function and sketch its graph. g(x)=25x2In Exercises 9-12, find the domain and range of the function and sketch its graph. f(x)=| x3 |12SWU13SWU14SWUFinding Limits Graphically In Exercises 1-4, use the graph to find the limit. See Examples 1 and 2. (a)limx2f(x)(b)limx1f(x)Finding Limits Graphically In Exercises 1-4, use the graph to find the limit. See Examples 1 and 2. (a)limx1f(x)(b)limx3f(x)Finding Limits Graphically In Exercises 1-4, use the graph to find the limit. See Examples 1 and 2. (a)limx0g(x)(b)limx1g(x)Finding Limits Graphically In Exercises 1-4, use the graph to find the limit. See Examples 1 and 2. (a)limx2h(x)(b)limx0h(x)Finding Limits Numerically In Exercises 5-12, complete the table and use the result to estimate the limit. Use a graphing utility to graph the function to confirm your result. See Examples 1 and 2. limx62x+35 x 5.9 5.99 5.999 6 6.001 6.01 6.1 f(x) ?Finding Limits Numerically In Exercises 5-12, complete the table and use the result to estimate the limit. Use a graphing utility to graph the function to confirm your result. See Examples 1 and 2. limx1(x24x1) x 0.9 0.99 0.999 1 1.001 1.01 1.1 f(x) ?Finding Limits Numerically In Exercises 5-12, complete the table and use the result to estimate the limit. Use a graphing utility to graph the function to confirm your result. See Examples 1 and 2. limx4x4x25x+4 x 3.9 3.99 3.999 4 4.001 4.01 4.1 f(x) ?Finding Limits Numerically In Exercises 5-12, complete the table and use the result to estimate the limit. Use a graphing utility to graph the function to confirm your result. See Examples 1 and 2. limx2x2x24 x 1.9 1.99 1.999 2 2.001 2.01 2.1 f(x) ?Finding Limits Numerically In Exercises 5-12, complete the table and use the result to estimate the limit. Use a graphing utility to graph the function to confirm your result. See Examples 1 and 2. limx0x+16-44 x -0.1 -0.01 -0.001 0 0.001 0.01 0.1 f(x) ?Finding Limits Numerically In Exercises 5-12, complete the table and use the result to estimate the limit. Use a graphing utility to graph the function to confirm your result. See Examples 1 and 2. limx0x+2-2x x -0.1 -0.01 -0.001 0 0.001 0.01 0.1 f(x) ?Finding Limits Numerically In Exercises 5-12, complete the table and use the result to estimate the limit. Use a graphing utility to graph the function to confirm your result. See Examples 1 and 2. limx41x+414x x -4.1 -4.01 -4.001 -4 -3.999 -3.99 -3.9 f(x) ?Finding Limits Numerically In Exercises 5-12, complete the table and use the result to estimate the limit. Use a graphing utility to graph the function to confirm your result. See Examples 1 and 2. limx212-1x+22x x -2.1 -2.01 -2.001 -2 -1.999 -1.99 -1.9 f(x) ?Evaluating Basic Limits In Exercises 13-20, find the limit. See Example 3. limx3614E15E16E17E18E19EEvaluating Basic Limits In Exercises 13-20, find the limit. See Example 3. limx1x321E22E23E24E25E26E27E28E29E30E31E32E33E34E35E36E37EFinding Limits In Exercises 37-58, find the limit (if it exists). See Examples 5, 6, 7, 9, and 11. limx12x2-x-3x+139E40E41E