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All Textbook Solutions for Single Variable Calculus: Early Transcendentals, Volume I
68E69E70E71E72E73E74E75E76E77E(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?Discuss four ways of representing a function. Illustrate your discussion with examples.3RCC4RCC5RCC6RCC7RCC8RCC9RCC10RCC11RCC12RCC13RCC1RQDetermine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. If f(s) = f(t), then s = t.3RQ4RQ5RQ6RQ7RQDetermine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. You can always divide by ex.9RQ10RQ11RQ12RQ13RQ14RQ1RE2RElf f(x) = x2 2x + 3, evaluate the difference quotient f(a+h)f(a)h4RE5RE6RE7RE8RESuppose that the graph of .f is given. Describe how the graphs of the following functions can be obtained from the graph of .f. (a) y =.f(x) + 8 (b) y = f(x + 8) (c) y = 1 + 2.f(x) (d) y = f(x 2) 2 (c) y = f(x) (f) y = f1(x)10RE11RE12RE13RE14RE15RE16RE17RE18RE19RE20RE22RE23RE24RE25RE26RE27REThe population of a certain species in a limited environment with initial population 100 and carrying capacity 1000 is P(t)=100,000100+900et where t is measured in years. (a) Graph this function and estimated how long it takes for the population to reach 900. (b) Find the inverse of this function and explain its meaning. (c) Use the inverse function to find the time required for the population to reach 900. Compare with the result of part (a).1P2P3P4P5P6P7P8P9P10P11P12P13PUse indirect reasoning to prove that log2 5 is an irrational number.A driver sets out on a journey. For the first half of the distance she drives at the leisurely pace of 30 mi/h; she drives the second half at 60 mi/h. What is her average speed on this trip?16P17P18P19P20PA Lank holds 1000 gallons o f 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. t(min) 5 10 15 20 25 30 V(gal) 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 compiles 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. t(min) 36 38 40 42 44 Heartbeats 2560 2661 2806 2948 3080 The monitor estimates this value by calculating the slope of a secant line. Use the data to estimate the patient's heart rate after 42 minutes using the secant line between the points with the given values of t. (a) t = 36 and t = 42 (b) t = 38 and t = 42 (c) t = 40 and t = 42 (d) t = 42 and t = 44 What are your conclusions?The point P(2, 1) lies on the curve y = 1/(1 x). (a) If Q is the point (x, 1/(1 x)), 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 P(0.5, 0) lies on the curve y = cos x. (a) If Q is the point (x, cos x), 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 result of part (a), guess the value of the slope of the tangent line to the curve at P(0.5, 0). (c) Using the slope from part (b), find an equation of the tangent line to the curve at P(0.5, 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 40 ft/s, its height in feet t seconds later is given by y = 40t 16t2. (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) 0.0 I seconds (b) Estimate the instantaneous 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 t seconds later is given by y = 10t 1.86t2. (a) Find the average velocity over the given time intervals: (i) [1, 2] (ii) [1, 1.5] (iii) [ 1, 1.1] (iv) [ 1, 1.0 1] (v) [ 1, 1.001] (b) Estimate the instantaneous velocity when t = 1.The table shows the position of a motorcyclist after accelerating from rest. (a) Find the average velocity for each tune 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 = 2 sin t + 3 cos t, 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 P(1, 0) lies on the curve y = sin(l0/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) Use a graph of the curve to explain why the slopes of the secant lines in part (a) arc 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)=3andlimx1f(x)=7 In this situation is it possible that 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)5EFor 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 exist, 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)8E9EA 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.11E12EUse the graph of the function f to state the value of each limit, if it exists. If it does not exist, explain why. (a) limx0f(x) (b) limx0+f(x) (c) limx0f(x)14E15E16E17E18E19EGuess the value of the limit (if it exists) by evaluating the function at the given numbers (correct to six decimal places). limx3x23xx29, x= 2.5, 2.9, 2.95, 2.99, 2.999, 2.9999, 3.5, 3.1, 3.05, 3.01, 3.001, 3.000121E22E23E24E25E26E27E28E(a) By graphing the function f(x) = (cos 2x cos x)/x2 and zooming in toward the point where the graph crosses the y-axis, estimate the value of limx0 f(x) (b) Check your answer in part (a) by evaluating f(x) for values of x that approach 0.30E31E32E33E34E35E36EDetermine the infinite limit. limx(/2)+1xsecx38E39E40E41E42E43E44E45E46E47E49E50EGraph the function f(x) = sin(/x) of Example 4 in the viewing rectangle [ l, 1] by [1 , 1]. Then zoom in toward the origin several times. Comment on the behavior of this function.52E53E54E55EGiven that limx2f(x)=4limx2g(x)=2limx2h(x)=0 find the limits that ex.ist. 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)f(x) (f) limx2g(x)f(x)Tire graphs of f and g are given. Use them to evaluate each limit, if it exists. If the 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) limx3f(x)g(x) (e) limx2[x2f(x)] (f) f(1)+limx1g(x)3E4E5E6E7E8E9E(a) What is wrong with the following equation? x2+x6x2=x+3 (b) In view of part (a). explain why the equation limx2x2+x6x2=limx2x+3 is correct.11E12EEvaluate the limit, if it exists. limx5x25x+6x514E15E16E17E18E19E20E21E22E23E24E25E26EEvaluate the limit, if it exists. limx164x16xx228E29E30E31E32E33E34EUse the Squeeze Theorem to show that limx0(x2cos20x)=0. Illustrate by graphing the functions f(x) = x2, g(x) = x2cos 20x, and h(x) = x2 on the same screen.36E37E38E39E40E41E42E43E44E45E46E47E48E49E50E51El.et g(x)={xifx13ifx=12xif1x2x3ifx2 (a) Evaluate each of the following, if it exists. (i) limx1g(x) (ii) limx1g(x) (iii) g(1) (iv) limx2g(x) (v) limx2+g(x) (vi) limx2g(x) (b) Sketch the graph of g.53E54E55E56EIf p is a polynomial, Show that limxa p(x) = p(a)58E59E60E61E62E63E64E65E66EUse the given graph of f to find a number such that if |x 1| then |f(x) 1| 0.22E3E4E5E6EFor the limit limx2(x33x+4)=6 illustrate Definition 2 by finding values of that correspond =0.2 and 0.1. Definition 28E(a) Use a graph to find a number such that if 2 x 2 + then 1ln(x1)100 (b) What limit docs part (a) suggest is true?10E11E12E13E14E15EProve the statement using the , definition of a limit and illustrate with a diagram like Figure 9. FIGURE9 limx4(2x5)=317E18EProve the statement using the , definition of a limit. limx12+4x3=220E21E22E23EProve the statement using the , definition of a limit. limxac=c25E26E27E28E29E30E31E32E33E34E36E37EIf H is the Heaviside function defined in Example 2.2.6, prove, using Definition 2, that limt0H(t) does not exist. [Hint: Use an indirect proof as follows. Suppose that the limit is L Take = 12 in the definition of a limit and try to arrive at a contradiction.] Definition 239E40E41E42E43ESuppose that limxaf(x)=andlimxag(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.From the graph of g, state the intervals on which g is continuous.Sketch the graph of a function f that is continuous except for the stated discontinuity. Discontinuous but continuous from the right, at 2Sketch the graph of a function f that is continuous except for the stated discontinuity. Discontinuities at 1and 4, but continuous from the left at 1 and from the right at 4Sketch the graph of a function f that is continuous except for the stated discontinuity. Removable discontinuity at 3, jump discontinuity at 5Sketch 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 29EExplain 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 time11EUse the definition of continuity and the properties of limits to show that the function is continuous at the given number a. g(t)=t2+5t2t+1,a=213E14EUse the definition of continuity and the properties of limits to show that the function is continuous on the given interval. f(x)=x+x1,[4,]16E17E18E19E20E21E22EHow 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)=x2x2x2How 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)=x31x2425E26E27E28EExplain, using Theorems 4, 5, 7, and 9, why the function is continuous at every number in its domain. State the domain. Theorem 4 Theorem 5 Theorem 7 Theorem 9 A(t)=arcsin(1+2t)30EExplain, using Theorems 4, 5, 7, and 9, why the function is continuous at every number in its domain. State the domain. Theorem 4 Theorem 5 Theorem 7 Theorem 9 M(x)=1+1x32E33E34E35E36E37E38EShow that f is continuous on ( , ). f(x)={1x2ifx1lnxifx140E41E42E43E44E