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All Textbook Solutions for Calculus (MindTap Course List)
If f(x)=x+2x and g(u)=u+2u, is it true that f=g?If f(x)=x2xx1 and g(x)=x is it true that f=g?The graph of a function f is given a State the value of f(1). b Estimate the value of f(1). c For what values of x is f(x)=1? d Estimate the value of x such that f(x)=0. e State the domain and range of f. f On what interval is f increasing?The graphs of f and g are given. a State the values of f(4) and g(3). b For what values of x is f(x)=g(x)? c Estimate the solution of the equation f(x)=1. d On what interval is f decreasing? e State the domain and range of f. f State the domain and range of g.Figure 1 was recorded by an instrument operated by the California Department of Mines and Geology at the University Hospital of the University of Southern California in Los Angeles. Use it to estimate the range of the vertical ground acceleration function at USC during the Northridge earthquake.In this section we discussed examples of ordinary, everyday functions: Population is a function of time, postage cost is a function of weight, water temperature is a function of time. Give three other examples of functions from everyday life that are described verbally. What can you say about the domain and range of each of your functions? If possible, sketch a rough graph of each function.Determine whether the curve is the graph of a function of x. If it is, state the domain and range of the function.Determine whether the curve is the graph of a function of x. If it is, state the domain and range of the function.Determine whether the curve is the graph of a function of x. If it is, state the domain and range of the function.Determine whether the curve is the graph of a function of x. If it is, state the domain and range of the function.Shown is a graph of the global average temperature T during the 20th century. Estimate the following. a The global average temperature in 1950 b The year when the average temperature was 14.2C c The year when the temperature was smallest; the year it was largest d The range of TTrees grow faster and form wider rings in warm years and grow more slowly and form narrower rings in cooler years. The figure shows ring widths of a Siberian pine from 1500 to 2000. a What is the range of the ring width function? b What does the graph tend to say about the temperature of the earth? Does the graph reflect the volcanic eruptions of the mid-19th century? Source: Adapted from G. Jacoby et al., Mongolian Tree Rings and 20th- Century Warming, Science 273 1996: 77173.You put some ice cubes in a glass, fill the glass with cold water, and then let the glass sit on a table. Describe how the temperature of the water changes as time passes. Then sketch a rough graph of the temperature of the water as a function of the elapsed time.Three runners compete in a 100-meter race. The graph depicts the distance run as a function of time for each runner. Describe in words what the graph tells you about this race. Who won the race? Did each runner finish the race?The graph shows the power consumption for a day in September in San Francisco. P is measured in megawatts; t is measured in hours starting at midnight. a What was the power consumption at 6 am? At 6 pm? b When was the power consumption the lowest? When was it the highest? Do these times seem reasonable? Pacific Gas & ElectricSketch a rough graph of the number of hours of daylight as a function of the time of year.Sketch a rough graph of the outdoor temperature as a function of time during a typical spring day.Sketch a rough graph of the market value of a new car as a function of time for a period of 20 years. Assume the car is well maintained.Sketch the graph of the amount of a particular brand of coffee sold by a store as a function of the price of the coffee.You place a frozen pie in an oven and bake it for an hour. Then you take it out and let it cool before eating it. Describe how the temperature of the pie changes as time passes. Then sketch a rough graph of the temperature of the pie as a function of time.A homeowner mows the lawn every Wednesday afternoon. Sketch a rough graph of the height of the grass as a function of time over the course of a four-week period.An airplane takes off from an airport and lands an hour later at another airport, 400 miles away. If t represents the time in minutes since the plane has left the terminal building, let x(t) be the horizontal distance traveled and y(t) be the altitude of the plane. a Sketch a possible graph of x(t). b Sketch a possible graph of y(t). c Sketch a possible graph of the ground speed. d Sketch a possible graph of the vertical velocity.Temperature readings Tin F were recorded every two hours from midnight to 2:00 pm in Atlanta on June 4, 2013. The time twas measured in hours from midnight. t 0 2 4 6 8 10 12 14 T 74 69 68 66 70 78 82 86 a Use the readings to sketch the graph of T as a function of t. b Use your graph to estimate the temperature at 9:00 am.Researchers measured the blood alcohol concentration BAC of eight adult male subjects after rapid consumption of 30 mL of ethanol corresponding to two standard alcoholic drinks. The table shows the data they obtained by averaging the BAC in g/mL of the eight men. a Use the readings to sketch the graph of the BAC as a function of t b Use your graph to describe how the effect of alcohol varies with time. t hours BAC 0 0 0.2 0.025 0.5 0.041 0.75 0.040 1.0 0.033 1.25 0.029 1.5 0.024 1.75 0.022 2.0 0.018 2.25 0.015 2.5 0.012 3.0 0.007 3.5 0.003 4.0 0.001 Source: Adapted from P. Wilkinson et.al., Pharmacokinetics of Ethanol after Oral Administration in the Fasting State, Journal of Pharmacokinetics and Biopharmaceutics 51977: 20724.If f(x)=3x2x+2, find f(2),f(2),f(a),f(a),f(a+1),2f(a),f(2a),f(a2),[f(a)]2, and f(a+h).A spherical balloon with radius r inches has volume V(r)=43r3. Find a function that represents the amount of air required to inflate the balloon from a radius of r inches to a radius of r+1 inches.Evaluate the difference quotient for the given function. Simplify your answer. f(x)=4+3xx2, f(3+h)f3h28EEvaluate the difference quotient for the given function. Simplify your answer. f(x)=1x, f(x)f(a)xa30EFind the domain of the function. f(x)=x+4x29Find the domain of the function. f(x)=2x35x2+x633E34E35E36EFind the domain of the function. F(p)=2pFind the domain and range and sketch the graph of the function h(x)=4x2Find the domain and sketch the graph of the function. f(x)=1.6x2.4Find the domain and sketch the graph of the function. g(t)=t21t+1Evaluate f(3),f(0), and f(2) for the piecewise defined function. Then sketch the graph of the function. f(x)={x+2ifx01xifx0Evaluate f(3),f(0), and f(2) for the piecewise defined function. Then sketch the graph of the function. f(x)={312xifx22x5ifx243EEvaluate f(3),f(0), and f(2) for the piecewise defined function. Then sketch the graph of the function. f(x)={1ifx172xifx1Sketch the graph of the function. f(x)=x+|x|Sketch the graph of the function. f(x)=|x+2|Sketch the graph of the function. g(t)=|13t|Sketch the graph of the function. h(t)=|t|+|t+1|Sketch the graph of the function. f(x)={|x|if|x|11if|x|150EFind an expression for the function whose graph is the given curve. The line segment joining the points (1,3)and(5,7)52EFind an expression for the function whose graph is the given curve. The bottom half of the parabola x+(y1)2=054EFind an expression for the function whose graph is the given curve.Find an expression for the function whose graph is the given curve.57EFind a formula for the described function and state its domain. A rectangle has area 16m2. Express the perimeter of the rectangle as a function of the length of one of its sides.Find a formula for the described function and state its domain. Express the area of an equilateral triangle as a function of the length of a side.Find a formula for the described function and state its domain. A closed rectangular box with volume 8ft3 has length twice the width. Express the height of the box as a function of the width.61EA Norman window has the shape of a rectangle surmounted by a semicircle. If the perimeter of the window is 30ft, express the area A of the window as a function of the width x of the window.A box with an open top is to be constructed from a rectangular piece of cardboard with dimensions 12 in. by 20 in. by cutting out equal squares of side x at each comer and then folding up the sides as in the figure. Express the volume V of the box as a function of x.A cell phone plan has a basic charge of 35 a month. The plan includes 400 free minutes and charges 10 cents for each additional minute of usage. Write the monthly cost C as a function of the number x of minutes used and graph C as a function of x for 0x600.In a certain state the maximum speed permitted on freeways is 65 mi/h and the minimum speed is 40 mi/h. The fine for violating these limits is 15 for every mile per hour above the maximum speed or below the minimum speed. Express the amount of the fine F as a function of the driving speed x and graph F(x) for 0x100.An electricity company charges its customers a base rate of 10 a month, plus 6 cents per kilowatt-hour kWh for the first 1200 kWh and 7 cents per kWh for all usage over 1200 kWh. Express the monthly cost E as a function of the amount x of electricity used. Then graph the function E for 0x2000.In a certain country, income tax is assessed as follows. There is no tax on income up to 10, 000. Any income over 10, 000 is taxed at a rate of 10, up to an income of 20, 000. Any income over 20, 000 is taxed at 15. a Sketch the graph of the tax rate R as a function of the income I. b How much tax is assessed on an income of 14, 000? On 26, 000? c Sketch the graph of the total assessed tax T as a function of the income I.68EGraphs of f and g are shown. Decide whether each function is even, odd, or neither. Explain your reasoning.Graphs of f and g are shown. Decide whether each function is even, odd, or neither. Explain your reasoning.a If the point (5,3) is on the graph of an even function, what other point must also be on the graph? b If the point (5,3) is on the graph of an odd function, what other point must also be on the graph?A function f has domain [5,5] and a portion of its graph is shown. a Complete the graph of f if it is known that f is even. b Complete the graph of f if it is known that f is odd.73EDetermine whether f is even, odd, or neither. If you have a graphing calculator, use it to check your answer visually. f(x)=x2x4+1Determine whether f is even, odd, or neither. If you have a graphing calculator, use it to check your answer visually. f(x)=xx+176E77EDetermine whether f is even, odd, or neither. If you have a graphing calculator, use it to check your answer visually. f(x)=1+3x3x579EIf f and g are both even functions, is the product fg even? If f and g are both odd functions, is fg odd? What if f is even and g is odd? Justify your answers.Classify each function as a power function, root function, polynomial state its degree, rational function, algebraic function, trigonometric function, exponential function, or logarithmic function. a f(x)=log2x b g(x)=x4 c h(x)=2x31x2 d u(t)=11.1t+2.54t2 e v(t)=5t f w()=sincos2Classify each function as a power function, root function, polynomial state its degree, rational function, algebraic function, trigonometric function, exponential function, or logarithmic function. a y=x b y=x c y=x2(2x3) d y=tantcost e y=s1+s f y=x311+x33EMatch each equation with its graph. Explain your choices. Dont use a computer or graphing calculator. a y=3x b y=3x c y=x3 d y=x3Find the domain of the function. f(x)=cosx1sinx6Ea Find an equation for the family of linear functions with slope 2 and sketch several members of the family. b Find an equation for the family of linear functions such that f(2)=1 and sketch several members of the family. c Which function belongs to both families?What do all members of the family of linear functions f(x)=1+m(x+3) have in common? Sketch several members of the family.What do all members of the family of linear functions f(x)=cx have in common? Sketch several members of the family.Find expressions for the quadratic functions whose graphs are shown.11ERecent studies indicate that the average surface temperature of the earth has been rising steadily. Some scientists have modeled the temperature by the linear function T=0.02t+8.50, where T is temperature in C and t represents years since 1900. a What do the slope and T-intercept represent? b Use the equation to predict the average global surface temperature in 2100.If the recommended adult dosage for a drug is D in mg, then to determine the appropriate dosage c for a child of age a, pharmacists use the equation c=0.0417D(a+1). Suppose the dosage for an adult is 200 mg. a Find the slope of the graph of c. What does it represent? b What is the dosage for a newborn?The manager of a weekend flea market knows from past experience that if he charges x dollars for a rental space at the market, then the number y of spaces he can rent is given by the equation y=2004x. a Sketch a graph of this linear function. Remember that the rental charge per space and the number of spaces rented cant be negative quantities. b What do the slope, the y-intercept, and the x-intercept of the graph represent?The relationship between the Fahrenheit F and Celsius C temperature scales is given by the linear function F=95C+32. a Sketch a graph of this function. b What is the slope of the graph and what does it represent? What is the F-intercept and what does it represent?16E17EThe manager of a furniture factory finds that it costs 2200 to manufacture 100 chairs in one day and 4800 to produce 300 chairs in one day. a Express the cost as a function of the number of chairs produced, assuming that it is linear. Then sketch the graph. b What is the slope of the graph and what does it represent? c What is the y-intercept of the graph and what does it represent?At the surface of the ocean, the water pressure is the same as the air pressure above the water, 15lb/in2. Below the surface, the water pressure increases by 4.34lb/in2 for every 10 ft of descent. a Express the water pressure as a function of the depth below the ocean surface. b At what depth is the pressure 100lb/in2?20EFor each scatter plot, decide whay type of function you might choose as a model for the data. Explain your choices.For each scatter plot, decide whay type of function you might choose as a model for the data. Explain your choices.The table shows lifetime peptic ulcer rates per 100 population for various family incomes as reported by the National Health Interview Survey. Income Ulcer rate per 100 population 4, 000 6, 000 8, 000 12, 000 16, 000 20, 000 30, 000 45, 000 60, 000 14.1 13.0 13.4 12.5 12.0 12.4 10.5 9.4 8.2 a Make a scatter plot of these data and decide whether a linear model is appropriate. b Find and graph a linear model using the first and last data points. c Find and graph the least squares regression line. d Use the linear model in part c to estimate the ulcer rate for an income of 25, 000. e According to the model, how likely is someone with an income of 80, 000 to suffer from peptic ulcers? f Do you think it would be reasonable to apply the model to someone with an income of 200, 000?Biologists have observed that the chirping rate of crickets of a certain species appears to be related to temperature. The table shows the chirping rates for various temperatures. a Make a scatter plot of the data. b Find and graph the regression line. c Use the linear model in part b to estimate the chirping rate at 100 F. Temperature (F) Chirping rate chirps/min 50 55 60 65 70 75 80 85 90 20 46 79 91 113 140 173 198 211Anthropologists use a linear model that relates human femur thighbone length to height. The model allows an anthropologist to determine the height of an individual when only a partial skeleton including the femur is found. Here we find the model by analyzing the data on femur length and height for the eight males given in the following table. a Make a scatter plot of the data. b Find and graph the regression line that models the data. c An anthropologist finds a human femur of length 53 cm. How tall was the person? Femur length cm Height cm 50.1 48.3 45.2 44.7 44.5 42.7 39.5 38.0 178.5 173.6 164.8 163.7 168.3 165.0 155.4 155.8When laboratory rats are exposed to asbestos fibers, some of them develop lung tumors. The table lists the results of several experiments by different scientists. a Find the regression line for the data. b Make a scatter plot and graph the regression line. Does the regression line appear to be a suitable model for the data? c What does the y-intercept of the regression line represent? Asbestos Exposure fibers/mL Percent of mice that develop lung tumors 50 400 500 900 1100 1600 1800 2000 3000 2 6 5 10 26 42 37 38 50The table shows world average daily oil consumption from 1985 to 2010 measured in thousands of barrels per day. a Make a scatter plot and decide whether a linear model is appropriate. b Find and graph the regression line. c Use the linear model to estimate the oil consumption in 2002 and 2012. Years since 1985 Thousands of barrels of oil per day 0 5 10 15 20 25 60, 083 66, 533 70, 099 76, 784 84, 077 87, 302 Source: US Energy Information AdministrationThe table shows average US retail residential prices of electricity from 2000 to 2012, measured in cents per kilowatt hour. a Make a scatter plot. Is a linear model appropriate? b Find and graph the regression line. c Use your linear model from part b to estimate the average retail price of electricity in 2005 and 2013. Years since 2000 Cents/kWh 0 8.24 2 8.44 4 8.95 6 10.40 8 11.26 10 11.54 12 11.58 Source: US Energy Information AdministrationMany physical quantities are connected by inverse square laws, that is, by power functions of the form f(x)=kx2. In particular, the illumination of an object by a light source is inversely proportional to the square of the distance from the source. Suppose that after dark you are in a room with just one lamp and you are trying to read a book. The light is too dim and so you move halfway to the lamp. How much brighter is the light?30EThe table shows the number N of species of reptiles and amphibians inhabiting Caribbean islands and the area A of the island in square miles. a Use a power function to model N as a function of A. b The Caribbean island of Dominica has area 291 mi2. How many species of reptiles and amphibians would you expect to find on Dominica? Island A N Saba 4 5 Monserrat 40 9 Puerto Rico 3, 459 40 Jamaica 4, 411 39 Hispaniola 29, 418 84 Cuba 44, 218 76The table shows the mean average distances d of the planets from the sun taking the unit of measurement to be the distance from planet Earth to the sun and their periods T time of revolution in years. a Fit a power model to the data. b Keplers Third Law of Planetary Motion states that The square of the period of revolution of a planet is proportional to the cube of its mean distance from the sun. Does your model corroborate Keplers Third Law? Planet d T Mercury 0.387 0.241 Venus 0.723 0.615 Earth 1.000 1.000 Mars 1.523 1.881 Jupiter 5.203 11.861 Saturn 9.541 29.457 Uranus 19.190 84.008 Neptune 30.086 164.784Suppose the graph of f is given. Write equations for the graphs that are obtained from the graph of f as follows. a Shift 3 units upward. b Shift 3 units downwards. c Shift 3 units to the right. d Shift 3 units to the left. e Reflect about the x-axis. f Reflect about the y-axis. g Stretch vertically by a factor of 3. h Shrink vertically by a factor of 3.Explain how each graph is obtained from the graph of y=f(x). a y=f(x)+8 b y=f(x+8) c y=8f(x) d y=f(8x) e y=f(x)1 f y=8f(18x)The graph of y=f(x) is given. Match each equation with its graph and give reasons for your choices. a y=f(x4) b y=f(x)+3 c y=13f(x) d y=f(x+4) e y=2f(x+6)The graph of f is given. Draw the graphs of the following functions. a y=f(x)3 b y=f(x+1) c y=12f(x) d y=f(x)The graph of f is given. Use it to graph the following functions. a y=f(2x) b y=f(12x) c y=f(x) d y=f(x)The graph of y=3xx2 is given. Use transformations to create a function whose graph is as shown. 6.The graph of y=3xx2 is given. Use transformations to create a function whose graph is as shown. 7.a How is the graph of y=2sinx related to the graph of y=sinx? Use your answer and Figure 6 to sketch the graph of y=2sinx. b How is the graph of y=1+x related to the graph of y=x? Use your answer and Figure 4a to sketch the graph of y=1+xGraph the function by hand, not by plotting points, but by starting with the graph of one of the standard functions given in Section 1.2, and then applying the appropriate transformations. y=x2Graph the function by hand, not by plotting points, but by starting with the graph of one of the standard functions given in Section 1.2, and then applying the appropriate transformations. y=(x3)2Graph the function by hand, not by plotting points, but by starting with the graph of one of the standard functions given in Section 1.2, and then applying the appropriate transformations. y=x3+112EGraph the function by hand, not by plotting points, but by starting with the graph of one of the standard functions given in Section 1.2, and then applying the appropriate transformations. y=2cos3x14EGraph the function by hand, not by plotting points, but by starting with the graph of one of the standard functions given in Section 1.2, and then applying the appropriate transformations. y=x24x+5Graph the function by hand, not by plotting points, but by starting with the graph of one of the standard functions given in Section 1.2, and then applying the appropriate transformations. y=1+sinxGraph the function by hand, not by plotting points, but by starting with the graph of one of the standard functions given in Section 1.2, and then applying the appropriate transformations. y=2x18E19EGraph the function by hand, not by plotting points, but by starting with the graph of one of the standard functions given in Section 1.2, and then applying the appropriate transformations. y=|x|221E22EGraph the function by hand, not by plotting points, but by starting with the graph of one of the standard functions given in Section 1.2, and then applying the appropriate transformations. y=|x1|Graph the function by hand, not by plotting points, but by starting with the graph of one of the standard functions given in Section 1.2, and then applying the appropriate transformations. y=|cosx|25E26ESome of the highest tides in the world occur in the Bay of Fundy on the Atlantic Coast of Canada. At Hopewell Cape the water depth at low tide is about 2.0 m and at high tide it is about 12.0 m. The natural period of oscillation is about 12 hours and on June 30, 2009, high tide occurred at 6:45 AM. Find a function involving the cosine function that models the water depth Dt in meters as a function of time t in hours after midnight on that day.In a normal respiratory cycle the volume of air that moves into and out of the lungs is about 500 mL. The reserve and residue volumes of air that remain in the lungs occupy about 2000 mL and a single respiratory cycle for an average human takes about 4 seconds. Find a model for the total volume of air Vt in the lungs as a function of time.29EUse the given graph of f to sketch the graph of y=1/f(x). Which features of f are the most important in sketching y=1/f(x)? Explain how they are used.Find (a)f+g,(b)fg,(c)fg, and f/g state their domains. 31. f(x)=x3+2x2,g(x)=3x2132EFind the functions (a)fg,(b)gf,(c)ff, and d gg and their domains. f(x)=3x+5, g(x)=x2+x34E35E36EFind the functions (a)fg,(b)gf,(c)ff, and d gg and their domains. f(x)=x+1x, g(x)=x+1x+238E39EFind fgh. f(x)=|x4|, g(x)=2x, h(x)=x41EFind fgh. f(x)=tanx, g(x)=xx1, h(x)=x343E44E45EExpress the function in the form fg. G(x)=x1+x3Express the function in the form fg. v(t)=sec(t2)tan(t2)Express the function in the form fg. u(t)=tant1+tant49EExpress the function in the form fgh. H(x)=2+|x|8Express the function in the form fgh. S(t)=sin2(cost)52EUse the given graphs of f and g to evaluate each expression, or explain why it is undefined. a f(g(2)) b g(f(0)) c (fg)(0) d (gf)(6) e (gg)(2) f (ff)(4)Use the given graphs of f and g to estimate the value of f(g(x)) for x=5,4,3,...,5. Use these estimates to sketch a rough graph of fg.A stone is dropped into a lake, creating a circular ripple that travels outward at a speed of 60 cm/s. a Express the radius r of this circle as a function of the time t in seconds. b If A is the area of this circle as a function of the radius, find Ar and interpret it.A spherical balloon is being inflated and the radius of the balloon is increasing at a rate of 2 cm/s. a Express the radius r of the balloon as a function of the time t in seconds. b If V is the volume of the balloon as a function of the radius, find Vr and interpret it.A ship is moving at a speed of 30 km/h parallel to a straight shoreline. The ship is 6 km from shore and it passes a lighthouse at noon. a Express the distance s between the lighthouse and the ship as a function of d, the distance the ship has traveled since noon; that is, find f so that s=f(d) b Express d as a function of t, the time elapsed since noon; that is, find g so that d=g(t). c Find fg. What does this function represent?58EThe Heaviside function H is defined by H(t)={0ift01ift0 It is used in the study of electric circuits to represent the sudden surge of electric current, or voltage, when a switch is instantaneously turned on. a Sketch the graph of the Heaviside function. b Sketch the graph of the voltage Vt in a circuit if the switch is turned on at time t=0 and 120 volts are applied instantaneously to the circuit. Write a formula for Vt in terms of Ht. c Sketch the graph of the voltage Vt in a circuit if the switch is turned on at time t=5 seconds and 240 volts are applied instantaneously to the circuit. Write a formula for Vt in terms of Ht. Note that starting at t = 5 corresponds to a translation.60ELet f and g be linear functions with equations f(x)=m1x+b1 and g(x)=m2x+b2. Is fg also a linear function? If so, what is the slope of its graph?If you invest x dollars at 4 interest compounded annually, then the amount Ax of the investment after one year is A(x)=1.04x. Find AA,AAA, and AAAA. What do these compositions represent? Find a formula for the composition of n copies of A.a If g(x)=2x+1 and h(x)=4x2+4x+7, find a function f such that fg=h. Think about what operations you would have to perform on the formula for g to end up with the formula for h. b If f(x)=3x+5 and h(x)=3x2+3x+2, find a function g such that fg=h.If f(x)=x+4 and h(x)=4x1, find a function g such that gf=h.Suppose g is an even function and let h=fg. Is h always an even function?Suppose g is an odd function and let h=fg. Is h always an odd function? What if f is odd? What if f is even?A tank holds 1000 gallons of water, which drains from the bottom of the tank in half an hour. The values in the table show the volume V of water remaining in the tank in gallons after t minutes. tmin 5 10 15 20 25 30 Vgal 694 444 250 111 28 0 a If P is the point 15, 250 on the graph of V, find the slopes of the secant lines PQ when Q is the point on the graph with t=5, 10, 20, 25, and 30. b Estimate the slope of the tangent line at P by averaging the slopes of two secant lines. c Use a graph of the function to estimate the slope of the tangent line at P. This slope represents the rate at which the water is flowing from the tank after 15 minutes.A cardiac monitor is used to measure the heart rate of a patient after surgery. It complies the number of heartbeats after t minutes. When the data in the table are graphed, the slope of the tangent line represents the heart rate in beats per minute. tmin 36 38 40 42 44 Heartbeats 2530 2661 2806 2948 3080 The monitor estimates this value by calculating the slope of a secant line. Use the data to estimate the patients heart rate after 42 minutes using the secant line between the points with the given values of t. a t=36andt=42 b t=38andt=42 c t=40andt=42 d t=42andt=44The point P (2,1) lies on the curve y=1/(1x). a If Q is the point (x,1/(14), use your calculator to find the slope of the secant line PQ correct to six decimal places for the following values of x: i 1.5 ii 1.9 iii 1.99 iv 1.999 v 2.5 vi 2.1 vii 2.01 viii 2.001 b Using the results of part a, guess the value of the slope of the tangent line to the curve at P(2,1). c Using the slope from part b, find an equation of the tangent line to the curve at P(2,1).The point P0.5, 0 lies on the curve y=cosx. a If Q is the point (x,cosx), use your calculator to find the slope of the secant line PQ correct to six decimal places for the following values of x: i 0 ii 0.4 iii 0.49 iv 0.499 v 1 vi 0.6 vii 0.51 viii 0.501 b Using the results of part a, guess the value of the slope of the tangent line to the curve at P0.5, 0. c Using the slope from partb, find an equation of the tangent line to the curve at P0.5, 0. d Sketch the curve, two of the secant lines, and the tangent line.If a ball is thrown into the air with a velocity of 40ft/s, its height in feet t seconds later is given by y=40t16t2. a Find the average velocity for the time period beginning when t=2 and lasting i 0.5 seconds ii 0.1 seconds iii 0.05 seconds iv 0.01 seconds b estimate the instantaneous velocity when t=2.6EThe table shows the position of a motorcyclist after accelerating from rest. tseconds 0 1 2 3 4 5 6 sfeet 0 4.9 20.6 46.5 79.2 124.8 176.7 a Find the average velocity for each time period: i 2, 4 ii 3, 4 iii 4, 5 iv 4, 6 b Use the graph of s as a function of t to estimate the instantaneous velocity when t=3.The displacement in centimeters of a particle moving back and forth along a straight line is given by the equation of motion s=2sint+3cost, where t is measured in seconds. a Find the average velocity during each time period: i 1, 2 ii 1, 1.1 iii 1, 1.01 iv 1, 1.001 b Estimate the instantaneous velocity of the particle when t=1.The point P1, 0 lies on the curve y=sin(10/x). a If Q is the point (x,sin(10/x)), find the slope of the secant line PQ correct to four decimal places for x=2, 1.5, 1.4, 1.3, 1.2, 1.1, 0.5, 0.6, 0.7, 0.8 and 0.9. Do the slopes appear to be approaching a limit. b Uses a graph of the curve to explain why the slopes of the secant lines in part a are not close to the slope of the tangent line at P. c By choosing appropriate secant lines, estimate the slope of the tangent line at P.Explain in your own words what is meant by the equation limx2f(x)=5 Is it possible for this statement to be true and yet f(2)=3? Explain.Explain what it means to say that limx1f(x)=3 and limx1+f(x)=7 Is this situation is it possible than limx1f(x) exists? Explain.Explain the meaning of each of the following. a limx3f(x)= b limx4+f(x)=Use the given graph of f to state the value of each quantity, if it exists. If it does not exist, explain why. a limx2f(x) b limx2+f(x) c limx2f(x) d f(2) e limx4f(x) f f(4)For the function f whose graph is given, state the value of each quantity, if it exists. If it does not exist, explain why. a limx1f(x) b limx3f(x) c limx3+f(x) d limx3f(x) e f(3)For the function h whose graph is given, state the value of each quantity, if it exists. If it does not exist, explain why. a limx3h(x) b limx3+h(x) c limx3h(x) d h(3) e limx0h(x) f limx0+h(x) g limx0h(x) h h(0) i limx2h(x) j h(2) k limx5+h(x) l limx5h(x)For the function g whose graph is given, state the value of each quantity, if it exists. If it does not exists, explain why. a limt0g(t) b limt0+g(t) c limt0g(t) d limt2g(t) e limt2+g(t) f limt2g(t) g g(2) h limt4g(t)For the fiunction A whose graph is shown, state the following. a limx3A(x) b limx2+A(x) c limx2+A(x) d limx1A(x) e The equations of the vertical asymptotesFor the function f whose graph is shown, state the following. a limx7f(x) b limx3f(x) c limx0f(x) d limx6f(x) e limx6+f(x) f The equations of the vertical asymptotes.A patient receives a 150-mg injection of a drug every 4 hours. The graph shows the amount f(t) of the drug in the bloodstream after t hours. Find limt12f(t) and limt12+f(t) and explain the significance of these one-sided limits.Sketch the graph of the function and use it to determine the values of a for which limxaf(x) exists. f(x)={1+xifx1x2if1x12xifx112E13E14ESketch the graph of an example of a function f that satisfies all of the given conditions. limx0f(x)=1,limx0+f(x)=2,f(0)=116ESketch the graph of an example of a function f that satisfies all of the given conditions. limx3+f(x)=4,limx3f(x)=2,limx2f(x)=2,f(3)=3,f(2)=118E19E20E21E22EUse a table of values to estimate the value of the limit. If you have a graphing device, use it to confirm your result graphically. lim0sin3tan224E25E26E27E28EDetermine the infinite limit. limx5+x+1x530EDetermine the infinite limit. limx12x(x1)232EDetermine the infinite limit. limx2+x1x2(x+2)34EDetermine the infinite limit. limx(/2)+1xsecx36EDetermine the infinite limit. limx2xcscx38E39E40E41E42Ea Evaluate the function f(x)=x2(2x/1000) for x=1, 0.8,0.6,0.4,0.2,0.1, and 0.05, and guess the value of limx0(x22x1000) b Evaluate fx for x=0.04,0.02,0.01,0.005,0.003, and 0.001. Guess again.a Evaluate h(x)=(tanxx)/x3 for x=1,0.5,0.1,0.05,0.01, and 0.005. b Guess the value of limx0tanxxx3. c Evaluate hx for successively smaller values of x until you finally reach a value of 0 for hx. Are you still confident that your guess in part b is correct? Explain why you eventually obtained values of 0 for hx. In Section 6.8 a method for evaluating this limit will be explained. d Graph the function h in the viewing rectangle [1,1] by [0,1]. Then zoom in toward the point where the graph crosses the y-axis to estimate the limit of hx as x approaches 0. Continue to zoom in until you observe distortions in the graph of h. Compare with the results of part c.45EConsider the function f(x)=tan1x. a Show that f(x)=0 for x=1,12,13,... b Show that f(x)=1 for x=1,12,13,... c What can you conclude about limx0+tan1x?Use a graph to estimate the equations of all the vertical asymptotes of the curve y=tan(2sinx)x Then find the exact equations of these asymptotes.In the theory of relativity, the mass of a particle with velocity v is m=m01v2/c2 Where mo the mass of the particle at rest and c is is the speed of light. What happens as vc?a Use numerical and graphical evidence to guess the value of the limit limx1x31x1 b How close to 1 does x have to be to ensure that the function in part a is within a distance 0.5 of its limit?Given that limx2f(x)=4limx2g(x)=2limx2h(x)=0 Find the limits that exist. If the limit does not exist, explain why. a limx2[f(x)+5g(x)] b limx2[g(x)]3 c limx2f(x) d limx23f(x)g(x) e limx2g(x)h(x) f limx2g(x)h(x)f(x)The graphs of f and g are given. Use them to evaluate each limit, if it limit does not exist, explain why. a limx2[f(x)+g(x)] b limx0[f(x)g(x)] c limx1[f(x)g(x)] d limx3[f(x)g(x)] e limx2[x2f(x)] f f(1)+limx1g(x)3E4E5E6E7E8EEvaluate the limit and justify each step by indicating the appropriate Limit Lawss. limx22x2+13x2a What is wrong with the following equation? x2+x6x2=x+3 b In view of part a, explain why the equation limx2x2+x6x2=limx2(x+3) is correctEvaluate the limit, if it exists. limx5x26x+5x512EEvaluate the limit, if it exists. limx5x25x+6x514EEvaluate the limit, if it exists. limt3t292t2+7t+316EEvaluate the limit, if it exists. limh0(5+h)225h18EEvaluate the limit, if it exists. limx2x+2x3+820EEvaluate the limit, if it exists. limh09+h3h22EEvaluate the limit, if it exists. limx31x13x324EEvaluate the limit, if it exists. limt01+t1tt26EEvaluate the limit, if it exists. limx164x16xx228EEvaluate the limit, if it exists. limt0(1t1+t1t)30EEvaluate the limit, if it exists. limh0(x+h)3x3hEvaluate the limit, if it exists. limh01(x+h)21x2h33E34EUse the Squeeze Theorem to show that limx0(x2cos20x)=0. Illustrate by graphing the functions f(x)=x2,g(x)=x2cos20x, and h(x)=x2 on the same screen.36EIf 4x9f(x)x24x+7 for x0, find limx4f(x).38E39EProve that limx0+x[1+sin2(2/x)]=0.Find the limit, if it exists. If the limit does not exist, explain why. limx3(2x+|x3|)42EFind the limit, if it exists. If the limit does not exist, explain why. limx0.52x1|2x3x2|44EFind the limit, if it exists. If the limit does not exist, explain why. limx0(1x1|x|)46E47ELet. g(x)=sgn(sinx). a Find each of the following limits or explain why it does not exist. i limx0+g(x) ii limx0g(x) iii limx0g(x) iv limx+g(x) v limxg(x) vi limxg(x) b For which values of a does limxag(x) not exist? c Sketch a graph of g.49E50ELet B(t)={412tift2t+cift2 Find the value of c so that limt2B(t) exists.52Ea If the symbol denotes the greatest integer function defined in Example 10, evaluate i limx2+x ii limx2x iii limx2.4x b If n is a integer, evaluate i limxnx ii limxn+x c For what values of a does limxax exists?54E55E56E57E58E59E60EIf f(x)={x2ifxisrational0ifxisirrational prove that limx0f(x)=0.62EShow by means of an example that limxa[f(x)g(x)] may exist even though neither limxaf(x) nor limxag(x) exists.64E