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All Textbook Solutions for Calculus (MindTap Course List)
Is there a number a such that limx23x2+ax+a+3x2+x2 exists? If so, find the value of a and the value of the limit.66EUse the given graph of f to find a number such that if |x1| then |f(x)1|0.2Use the given graph of f to find a number such that if 0|x3| then |f(x)2|0.5Use the given graph of f(x)=x to find a number such that if |x4| then |x2|0.44EUse a graph to find a number such that if |x4| then |tanx1|0.26EFor the limit limx2(x33x+4)=6 illustrate Definition 2 by finding values of that correspond to =0.2 and =0.1.8Ea Use a graph to find a number such that if 4xx+ then x2+4x4100 b What limit does part a suggest is true?Given that limxcsc2x=, illustrate Definition 6 by finding values of that correspond to a M=500 and b M=1000.A machinist is required to manufacture a circular metal disk with area 1000 cm2. a What radius produces such a disk? b If the machinist is allowed an error tolerance of 5cm2 in the area of the disk, how close to the ideal radius in part a must the machinist control the radius? c In terms of the , definition of limxaf(x)=L, what is x? What is f(x)? What is a? What is L? What value of is given? What is the corresponding value of ?A crystal growth furnace is used in research to determine how best to manufacture crystals used in electronic components for the space shuttle. For proper growth of the crystal, the temperature must be controlled accurately by adjusting the input power. Suppose the relationship is given by T(w)=0.1w2+2.155w+20 where T is the temperature in degrees Celsius and w is the power input in watts. a How much power is needed to maintain the temperature at 200C? b If the temperature is allowed to vary from 200C by up to 1C, what range of wattage is allowed for the input power? c In terms of the , definition of limxaf(x)=L, what is x? What is f(x)? What is a? What is L? What value of is given? What is the corresponding value of ?a Find a number such that if |x2|, then |4x8| where =0.1. b Repeat part a with =0.01.14E15E16EProve the statement using the , definition of a limit and illustrate with a diagram like Figure 9. FIGURE 9 limx3(14x)=1318EProve the statement using the , definition of a limit. limx12+4x3=2Prove the statement using the , definition of a limit. limx10(345x)=5Prove the statement using the , definition of a limit. limx4x22x8x4=622EProve the statement using the , definition of a limit. limxax=aProve the statement using the , definition of a limit. limxac=cProve the statement using the , definition of a limit. limx0x2=0Prove the statement using the , definition of a limit. limx0x3=0Prove the statement using the , definition of a limit. limx0|x|=0Prove the statement using the , definition of a limit. limx6+6+x8=0Prove the statement using the , definition of a limit. limx2(x24x+5)=1Prove the statement using the , definition of a limit. limx2(x2+2x7)=1Prove the statement using the , definition of a limit. limx2(x21)=332E33E34E35E36EProve that limxax=a if a0. [Hint:Usexa=|xa|x+a.]38EIf the function f is defined by f(x)={0ifxisrational1ifxisirrational Prove that limx0f(x) does not exist.40E41E42E43ESuppose that limxaf(x)= and limxag(x)=c, where c is a real number. Prove each statement. a limxa[f(x)+g(x)]= b limxa[f(x)g(x)]=ifc0 c limxa[f(x)g(x)]=ifc0Write an equation that expresses the fact that a function f is continuous at the number 4.If f is continuous on (,), what can you say about its graph?a From the graph of f, state the numbers at which f is discontinuous and explain why. b For each of the numbers stated in part a, determine whether f is continuous from the right, or from the left, or neither.4E5ESketch the graph of a function f that is continuous except for the stated discontinuity. Discontinuities at 1 and 4, but continuous from the left at 1 and from the right at 47ESketch the graph of a function f that is continuous except for the stated discontinuity. Neither left nor right continuous at 2, continuous only from the left at 2The toll T charged for driving on a certain stretch of a toll road is 5 except during rush hours between 7 AM and 10 AM and between 4 PM and 7 PM when the toll is 7. a Sketch a graph of T as a function of the time t, measured in hours part midnight. b Discuss the discontinuities of this function and their significance to someone who uses the road.Explain why each function is continuous or discontinuous. a The temperature at a specific location as a function of time b The temperature at a specific time as a function of the distance due west from New York City c The altitude above sea level as a function of the distance due west from New York City d The cost of a taxi ride as a function of the distance traveled e The current in the circuit for the lights in a room as a function of timeUse the definition of continuity and the properties of limits to show that the function is continuous at the given number a. f(x)=(x+2x3)4,a=112EUse the definition of continuity and the properties of limits to show that the function is continuous at the given number a. p(v)=23v2+1, a=114EUse the definition of continuity and the properties of limits to show that the function is continuous on the given interval. f(x)=x+x4, [4,)Use the definition of continuity and the properties of limits to show that the function is continuous on the given interval. g(x)=x13x+6, (,2)17E18E19EExplain why the function is discontinuous at the given number a. Sketch the graph of the function. f(x)={x2xx21ifx11ifx=1a=121E22EHow would you remove the discontinuity of f? In other words, how would you define f(2) in order to make f continuous at 2? f(x)=x2x2x224EExplain, using Theorems 4, 5, 7, and 9, why the function is continuous at every number in its domain. State the domain. F(x)=2x2x1x2+126EExplain, using Theorems 4, 5, 7, and 9, why the function is continuous at every number in its domain. State the domain. Q(x)=x23x3228E29E30E31E32ELocate the discontinuities of the function and illustrate by graphing. y=11+sinx34EUse continuity to evaluate the limit. limx2x20x2Use continuity to evaluate the limit. limxsin(x+sinx)37E38E39E40EFind the numbers at which f is discontinuous. At which of these numbers is f continuous from the right, from the left, or neither? Sketch the graph of f. f(x)={x2ifx1xif1x11/xifx1Find the numbers at which f is discontinuous. At which of these numbers is f continuous from the right, from the left, or neither? Sketch the graph of f. f(x)={x2+1ifx13xif1x4xifx4Find the numbers at which f is discontinuous. At which of these numbers is f continuous from the right, from the left, or neither? Sketch the graph of f. f(x)={x+2ifx02x2if0x12xifx1The gravitational force exerted by the planet Earth on a unit mass at a distance r from the center of the planet is F(r)={GMrR3ifrRGMr2ifrR Where M is the mass of Earth, R is its radius, and G is the gravitational constant. Is F a continuous function of r?For what value of the constant c is the function f continuous on (,)? f(x)={cx2+2xifx2x3cxifx2Find the values of a and b that makes f continuous everywhere. f(x)={x24x2ifx2ax2bx+3if2x32xa+bifx347E48E49E50EIf f(x)=x2+10 sin x, show that there is a number c such that f(c)=1000.52EUse the Intermediate Value Theorem to show that there is a root of the given equation in the specified interval. x4+x3=0,(1,2)Use the Intermediate Value Theorem to show that there is a root of the given equation in the specified interval. 2/x=xx, 2, 3Use the Intermediate Value Theorem to show that there is a root of the given equation in the specified interval. cosx=x, 0, 156E57E58Ea Prove that the equation has at least one real root. b Use your graphing device to find the root correct to three decimal places. x5x24=060EProve, without graphing, that the graph of the function has at least two x-intercepts in the specified interval. y=sinx3, 1, 262EProve that f is continuous at a if and only if limh0f(a+h)=f(a)64EProve that cosine is a continuous function.66EFor what values of x is f continuous? f(x)={0ifxisrational1ifxisirrational68EIs there a number that is exactly 1 more than its cube?70EShow that the function f(x)={x4sin(1/x)ifx00ifx=0 is continuous on (,).72EA Tibetan monk leaves the monastery at 7.00 AMand takes his usual path to the top of the mountain, arriving at 7.00 PM. The following morning he starts at 7.00 AMat the top and takes the same path back, arriving at the monastery at 7.00 PM. Use the Intermediate Value Theorem to show that there is a point on the path that the monk will cross at exactly the same time of day on both days.a What is a function? What are its domain and range? b What is the graph of a function? c How can you tell whether a given curve is the graph of a function?2CC3CC4CC5CC6CC7CC8CC9CC10CC11CC12CC13CC14CC15CC16CC17CC18CC19CC1TFQ2TFQ3TFQ4TFQ5TFQ6TFQ7TFQ8TFQ9TFQ10TFQ11TFQ12TFQDetermine whether statement is true or false. If it is true, explain why. If it is false, explain why or given an example that disproves the statement. If limx5f(x)=0 and limx5g(x)=0, then limx5[f(x)/g(x)] does not exist.14TFQ15TFQ16TFQ17TFQ18TFQDetermine whether statement is true or false. If it is true, explain why. If it is false, explain why or given an example that disproves the statement. If the line x=1 is a vertical asymptote of y=f(x), then f is not defined at 1.Determine whether statement is true or false. If it is true, explain why. If it is false, explain why or given an example that disproves the statement. If f(1)0 and f(3)0, then there exists a number c between 1 and 3 such that f(c)=0.21TFQ22TFQ23TFQ24TFQ25TFQ26TFQ27TFQLet f be the function whose graph is given, a Estimate the value of f2. b Estimate the value of x such that fx=3. c State the domain of f. d State the range of f. e On what interval is f increasing? f Is f even, odd, or neither even nor odd? Explain.Determine whether each curve is the graph of a function of x. If it is, state the domain and range of the function.If f(x)=x22x+3, evaluate the different quotient f(a+h)f(a)h4EFind the domain and range of the function. Write your answer in interval notation. f(x)=2/(3x1)6EFind the domain and range of the function. Write your answer in interval notation. y=1+sinx8E9EThe graph of f is given. Draw the graphs of the following functions. a y=f(x8) b y=f(x) c y=2f(x) d y=12f(x)1Use transformations to sketch the graph of the function. y=(x2)312EUse transformations to sketch the graph of the function. y=x22x+214EUse transformations to sketch the graph of the function. f(x)=cos2x16EDetermine whether f is even, odd, or neither even nor odd. a f(x)=2x53x2+2 b f(x)=x3x7 c f(x)=cos(x2) d f(x)=1+sinx18EIf f(x)=x and g(x)=sinx, find the functions a fg, b gf, c ff, d gg, and their domains.20ELife expectancy improved dramatically in the 20th century. The table gives the life expectancy at birth in years of males born in the United States. Use a scatter plot to choose an appropriate type of model. Use your model to predict the life span of a male born in the year 2010. Birth Type Life expectancy 1900 48.3 1910 51.1 1920 55.2 1930 57.4 1940 62.5 1950 65.6 1960 66.6 1970 67.1 1980 70.0 1990 71.8 2000 73.022EThe graph of f is given, a Find each limit, or explain why it does not exist. i limx2+f(x) ii limx3+f(x) iii limx3f(x) iv limx4f(x) v limx0f(x) vi limx2f(x) b State the equations of the vertical asymptotes. c At what numbers if f discontinuous? Explain.24E25E26E27EFind the limit. limx1+x29x2+2x329E30EFind the limit. limr9r(r9)432EFind the limit. limu1u41u3+5u26u34EFind the limit. lims164ss1636EFind the limit. limx011x2x38EIf 2x1f(x)x2 for 0x3, find limx1f(x).40EProve the statement using the precise definition of a limit. limx2(145x)=442EProve the statement using the precise definition of a limit. limx2(x23x)=244ELet f(x)={xifx03xif0x3(x3)2ifx3 a Evaluate each limit, if it exists. i limx0+f(x) ii limx0f(x) iii limx0f(x) iv limx3f(x) v limx3+f(x) vi limx3f(x) b Where is f discontinuous? c Sketch the graph of f.46EShow that the function is continuous on its domain. State the domain. h(x)=x4+x3cosx48EUse the Intermediate Value Theorem to show that there is a root of the equation in the given interval. x5x3+3x5=0,(1, 2)50E51E52ESolve the equation |2x1||x+5|=3.2PSketch the graph of the function f(x)=|x24|x|+3|.4P5P6PThe notation max {a,b,...} means the largest of the numbers a, b, Sketch the graph of each function. a f(x)=max{x,1/x} b f(x)=max{sinx,cosx} c f(x)=max{x2,2+x,2x}8P9P10P11P12P13P14P15P16P17PThe figure shows a point P on the parabola y=x2 and the point Q where the perpendicular bisector of OP intersects the yaxis. As P approaches the origin along the parabola, what happens to Q? Does it have a limiting position? If so, find it.19P20P21PA fixed point of a function f is a number c in its domain such that f(c)=c. The function doesnt move c; it stays fixed. a Sketch the graph of a continuous function with domain [0,1] whose range also lies in [0,1]. Locate a fixed point of f. b Try to draw the graph of a continuous function with domain [0,1] and range in [0,1] that does not have a fixed point. What is the obstacle? c Use the Intermediate Value Theorem to prove that any continuous function with domain [0,1] and range in [0,1] must have a fixed point.23Pa The figure shows an isosceles triangle ABC with B=C. The bisector of angle B intersects the side AC at the point P. Suppose that the BC remains fixed but the altitude |AM| of the triangle approaches 0, so A approaches the midpoint M of BC. What happens to P during this process? Does it have a limiting position? If so, find it. b Try to sketch the path traced out by P during the process. Then find an equation of this curve and use this equation to sketch the curve.25PA curve has equation y=f(x). a Write an expression for the slope of the secant line through the points P(3,f(3)) and Q(x,f(x)). b Write an expression for the slope of the tangent line at P.Graph the curve y=sinx in the viewing rectangles [2,2] by [2,2],[1,1] by [1,1], and [0.5,0.5] by [0.5,0.5]. What do you notice about the curve as you zoom in toward the origin?a Find the slope of the tangent line to the parabola y=4xx2 at the point 1, 3 i using Definition 1 ii using Equation 2 b Find an equation of the tangent line in part a. c Graph the parabola and the tangent line. As a check on your work, zoom in toward the point 1, 3 until the parabola and the tangent line are indistinguishable.a Find the slope of the tangent line to the curve y=xx3 at the point 1, 0 i using Definition 1 ii using Equation 2 b Find an equation of the tangent line in part a. c Graph the curve and the tangent line in successively smaller viewing rectangles centered at 1, 0 until the curve and the line appear to coincide.5E6EFind an equation of the tangent line to the curve at the given point. y=x, 1, 1Find an equation of the tangent line to the curve at the given point. y=2x+1x+2, 1, 1a Find the slope of the tangent to the curve y=3+4x22x3 at the point where x=a. b Find equations of the tangent lines at the points 1, 5 and 2, 3. c Graph the curve and both tangents on a common screen.a Find the slope of the tangent to the curve y=1/x at the point where x=a. b Find equations of the tangent lines at the points 1, 1 and (4,12) c Graph the curve and both tangents on a common screen.a A particle starts by moving to the right along a horizontal line; the graph of its position function is shown in the figure. When is the particle moving to the right? Moving to the left? Standing still? b Draw a graph of the velocity function.Shown are graphs of the position functions of two runners. A and B, who run a 100-meter race and finish in a tie. a Describe and compare how the runners run the race. b At what time is the distance between the runners the greatest? c At what time do they have the same velocity?If a ball is thrown into the air with a velocity of 40 ft/s, its height in feet after t seconds is given by y=40t16t2. Find the velocity when t=2.If a rock is thrown upward on the planet Mars with a velocity of 10 m/s, its height in meters after t seconds is given by H=10t1.86t2. a Find the velocity of the rock after one second. b Find the velocity of the rock when t=a. c When will the rock hit the surface? d With what velocity will the rock hit the surface?15EThe displacement in feet of a particle moving in a straight line is given by s=12t26t+23, where t is measured in seconds. a Find the average velocity over each time interval: i 4, 8 ii 6, 8 iii 8, 10 iv 8, 12 b Find the instantaneous velocity when. t=8. c Draw the graph of s as a function of t and draw the secant lines whose slopes are the average velocities in part a. Then draw the tangent line whose slope is the instantaneous velocity in part b.17EThe graph of a function f is shown. a Find the average rate of change of f on the interval 20, 60. b Identify an interval on which the average rate of change of f is 0. c Which interval gives a larger average rate of change, 40, 60 or 40, 70? d Compute f(40)f(10)4010 What does this value represent geometrically?19EFind an equation of the tangent line to the graph of y=g(x) at x=5 if g(5)=3 and g(5)=4.If an equation of the tangent line to the curve y=f(x) at the point where a=2 is y=4x5, find f(2) and f(2).If the tangent line to y=f(x) at 4, 3 passes through the point 0, 2, find f(4) and f(4).Sketch the graph of a function f for which f(0)=0, f(0)=3, f(1)=0, and f(2)=1.Sketch the graph of a function g for which g(0)=g(2)=g(4)=0, g(1)=g(3)=0, g(0)=g(4)=1, g(2)=1, limx5g(x)=, and limx1+g(x)=.Sketch the graph of a function g that is continuous on its domain (5,5) and where g(0)=1, g(0)=1, g(2)=0, limx5+g(x)=, and limx5g(x)=3.26E27E28E29E30E31E32EFind f(a). f(x)=2t+1t+334EFind f(a). f(x)=12x36EEach limit represents the derivative of some function f at some number a. State such an f and a in each case. limh09+h3h38EEach limit represents the derivative of some function f at some number a. State such an f and a in each case. limx2x664x240E41E42E43EA particle moves along a straight line with equation of motion s=f(t), where s is measured in meters and t in seconds. Find the velocity and the speed when t=4. f(t)=10+45t+145EA roast turkey is taken from an oven when its temperature has reached 185F and is placed on a table in a room where the temperature is 75F. The graph shows how the temperature of the turkey decreases and eventually approaches room temperature. By measuring the slope of the tangent, estimate the rate of change of the temperature after an hour.Researchers measured the average blood alcohol concentration Ct of eight men starting one hour after consumption of 30 mL of ethanol corresponding to two alcoholic drinks. t hours 1.0 1.5 2.0 2.5 3.0 Ct g/dL 0.033 0.024 0.018 0.012 0.007 a Find the average rate of change of C with respect to t over each time interval: i 1.0, 2.0ii 1.5, 2.0 iii 2.0, 2.5iv 2.0, 3.0 In each case, include the units. b Estimate the instantaneous rate of change at t=2 and interpret your result. What are the units? Source: Adapted from P. Wilkinson el al., Pharmacokinetics of Ethanol after Oral Administration in the Fasting State, Journal of Pharmacokinetics and Biopharmaceutics 5 1977: 20724.48E49EThe table shows values of the viral load V(t) in HIV patient 303, measured in RNA copies/mL, t days after ABT-538 treatment was begun. t 4 8 11 15 22 V(t) 53 18 9.4 5.2 3.6 a Find the average rate of change of V with respect to t over each time interval: i 4, 11ii 8, 11 iii 11, 15iv 11, 22 What are the units? b Estimate and interpret the value of the derivative V(11). Source: Adapted from D. Ho et al., Rapid Turnover of Plasma Virions and CD4 Lymphocytes in HIV-1 Infection, Nature 373 1995; 12326.The cost in dollars of producing x units of a certain commodity is C(x)=5000+10x+0.05x2. a Find the average rate of change of C with respect to x when the production level is changed i from x=100 to x=105 ii from x = 100 to x= 101 b Find the instantaneous rate of change of C with respect to x when x = 100. This is called the marginal cost. Its significance will be explained in Section 3.7.If a cylindrical tank holds 100, 000 gallons of water, which can be drained from the bottom of the tank in an hour, then Torricellis Law gives the volume V of water remaining in the tank after t minutes as V(t)=100.000(1160t)20t60 Find the rate at which the water is flowing out of the tank the instantaneous rate of change of V with respect to t as a function of t. What are its units? For times t = 0, 10, 20, 30, 40, 50, and 60 min, find the flow rate and the amount of wate remaining in the tank. Summarize your findings in a sentence or two. At what time is the flow rate the greatest? The least?The cost of producing x ounces of gold from a new gold mine is C=f(x) dollars. a What is the meaning of the derivative f(x)? What are its units? b What does the statement f(800)=17 mean? c Do you think the values of f(x) will increase or decrease in the short term? What about the long term? Explain.The number of bacteria after t hours in a controlled laboratory experiment is n=f(t). a What is the meaning of the derivative f(5)? What are its units? b Suppose there is an unlimited amount of space and nutrients for the bacteria. Which do you think is larger, f(5) or f(10)? If the supply of nutrients is limited, would that affect your conclusion? Explain.Let H(t) be the daily cost in dollars to heat an office building when the outside temperature is t degrees Fahrenheit. a What is the meaning of H(58)? What are its units? b Would you expect H(58) to be positive or negative? Explain.The quantity in pounds of a gourmet ground coffee that is sold by a coffee company at a price of p dollars per pound is Q=f(p). a What is the meaning of the derivative f(8)? What are its units? b Is f(8) positive or negative? Explain.The quantity of oxygen that can dissolve in water depends on the temperature of the water. So thermal pollution influences the oxygen content of water. The graph shows how oxygen solubility S varies as a function of the water temperature T. a What is the meaning of the derivative S(T)? What are its units? b Estimate the value of S(16) and interpret it.The graph shows the influence of the temperature T on the maximum sustainable swimming speed S of Coho salmon. a What is the meaning of the derivative S(T)? What are its units? b Estimate the values of S(15) and S(25) and interpret them.