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All Textbook Solutions for Single Variable Calculus: Early Transcendentals

18E 19E 20E 21E Find the exponential function f(x) = Cbx whose graph is given.23E Suppose you are offered a job that lasts one month. Which of the following method of payment do you prefer? I. One million dollars at the end of the month. ll. One cent on the first day of the month, two cents on the second day, four cents on the third day, and, in general 2n-1 cents on the nth day.25E Compare the functions f(x) = x5and g(x) = 5x by graphing both functions in several viewing rectangles. Find all points of intersection of the graphs correct to one decimal place. Which function grows more rapidly when x is large?Compare the functions f(x) = x10 and g(x) = ex by graphing both f and g in several viewing rectangles. When does the graph of g finally surpass the graph of f?28E 29E A bacteria culture starts with 500 bacteria and doubles in size every half hour. (a) How many bacteria are there after 3 hours? (b) How many bacteria are there after t hours? (c) How many bacteria are there after 40 minutes? (d) Graph the population function and estimate the time for the population to reach 100,000.The half-life of bismuth-210, 210Bi, is 5 days. (a) If a sample has a mass of 200 mg, find the amount remaining after 15 days. (b) Find the amount remaining after t days. (c) Estimate the amount remaining after 3 weeks. (d) Use a graph to estimate the time required for the mass to be reduced to 1 mg.32E 33E 34E Use a graphing calculator with exponential regression capability to model the population of the world with the data from 1950 to 2010 in Table 1 on page 49. Use the model to estimate the population in 1993 and to predict the population in the year 2020.36E 37E 38E (a) What is a one-to-one function? (b) How can you tell from the graph of a function whether it is one-to-one?2E 3E 4E A function is given by a table of values, a graph, a formula or a verbal description. Determine whether it is one-to-one.A function is given by a table of values, a graph, a formula or a verbal description. Determine whether it is one-to-one.7E A function is given by a table of values, a graph, a formula or a verbal description. Determine whether it is one-to-one. 8.9E 10E 11E 12E 13E 14E Assume that f is a one-to-one function. (a) If f(6) = 17, what is f1(17)? (b) If f1(3) = 2, what is f(2)?16E 17E 18E 19E 20E 21E 22E 23E 24E Find a formula for the inverse of the function. 25. y = ln(x + 3)Find a formula for the inverse of the function. 26. y=1ex1+ex Find an explicit formula for f1 and use it to graph f1, f, and the line y = x on the same screen. To check your work, see whether the graphs of f and f1 are reflections about the line. 27. f(x)=4x+3 28E 29E 30E 31E 32E 33E 34E Find the exact value of each expression. 35. (a) log2 32 (b) log8 2 Find the exact value of each expression. 35. (a) log51125 (b) ln(1/e2)37E Find the exact value of each expression. 38. (a) eln2 (b) eln(lne3)39E 40E Express the given quantity as a single logarithm. 41. 13ln(x+2)3+12[lnxln(x2+3x+2)2]Use Formula 10 to evaluate each logarithm correct to six decimal places. (a) log5 10 (b) log3 57 43E 44E 45E 46E 47E Make a rough sketch of the graph of each function. Do not use a calculator. Just use the graphs given in Figures 12 and 13 and, if necessary, the transformations of Section 1.3. 48. (a) y = ln(x) (b) y = ln |x|(a) What are the domain and range of f? (b) What is the x-intercept of the graph of f? (c) Sketch the graph of f. 49. f(x) = ln x + 2 (a) What are the domain and range of f? (b) What is the x-intercept of the graph of f? (c) Sketch the graph of f. 50. f(x) = ln(x 1) 1 Solve each equation for x. 51. (a) e74x=6 (b) ln(3x 10) = 2 52E 53E 54E 55E 56E 57E 58E 59E 60E 61E When a camera flash goes off, the batteries immediately begin to recharge the flash's capacitor, which stores electric charge given by Q(t) = Q0(1 e1/a) (The maximum charge capacity is Q0 and t is measured in seconds.) (a) Find the inverse of this function and explain its meaning. (b) How long does it take to recharge the capacitor to 90% of capacity if a = 2?63E 64E 65E 66E 67E 68E 69E Simplify the expression. 70. tan(sin1 x)71E Simplify the expression. 72. sin(2 arccos x)73E 74E 75E 76E 77E Explain what each of the following means and illustrate with sketch. (a) limxaf(x)=L (b) limxa+f(x)=L (c) limxaf(x)=L (d) limxaf(x)= (e) limxf(x)=L 2RCC 3RCC 4RCC 5RCC 6RCC 7RCC 8RCC 9RCC 10RCC 11RCC 12RCC 13RCC 14RCC 15RCC 16RCC 1RQ 2RQ 3RQ 4RQ 5RQ 6RQ Determine 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 limx5f(x)=0andlimx5g(x)=0,thenlimx5[f(x)/g(x)] does not exist.8RQ 9RQ 10RQ 11RQ 12RQ 13RQ 14RQ 15RQ 16RQ 17RQ Determine 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 is continuous on [ 1 , 1] and f(1) = 4 and f(1) = 3, then there exists. a number r such that | r | 1 and f(r) = .19RQ 20RQ 21RQ Determine 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'(r) exists, then limxrf(x)=f(r).23RQ Determine 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. The equation x10 10x2 + 5 = 0 has a root in the interval (0, 2).25RQ 26RQ 1RE 2RE 3RE 4RE 5RE 6RE 7RE 8RE 9RE 10RE 11RE 12RE 13RE 14RE 15RE 16RE 17RE 18RE 19RE 20RE 21RE 22RE If 2x 1 f(x) x2 for 0 x 3, find limx1f(x).24RE 25RE 26RE 27RE 28RE 29RE 30RE 31RE 32RE 33RE Use the Intermediate Value Theorem to show that there is a root of the equation in the given interval. cosx=ex2,(0,1)35RE 36RE 37RE According to Boyle's Law, if the temperature of a confined gas is held fixed, !hen the product of the pressure P and the volume V is a constant. Suppose that, for a certain gas, PV = 800, where P is measured in pounds per square inch and Vis measured in cubic inches. (a) Find the average rate of change of P as V increases from 200 in3 to 250 in3. (b) Express V as a function of P and show that the instantaneous rate of change of V with respect to P is inversely proportional to the square of P.39RE 40RE 41RE 42RE 43RE 44RE 45RE 46RE 47RE The figure shows the graphs of f, f', and f". Identify each curve, and explain your choices.49RE 50RE 51RE 52RE 53RE 54RE 1P Find numbers a and b such that limx0ax+b2x=1.3P The figure shows a point P on the parabola y = x2 and the point Q where the perpendicular bisector of OP intersects the y-axis. As P approaches the origin along the parabola, what happens to Q? Does it have a limiting position? If so, find it.5P 6P 7P 8P 9P 10P 11P 12P 13P Suppose f is a function with the property that | f(x) | x2 for all x. Show that f(0) = 0. Then show that f'(0) = 0.A 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.1E 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)For the function f whose graph is given, state the value of each quantity, if it exists. If it does not exit, 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 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)For the function A whose graph is shown, state the following. (a) limx3A(x) (b) limx3A(x) (c) limx3+A(x) (d) limx1A(x) (e) The equations of the vertical asymptotes For 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.10E Sketch the graph of the function and use it to determine the values of a for which limxaf(x) exists. f(x)={1+xifx1x2if1x12xifx1 Sketch the graph of the function and use it to determine the values of a for which limxaf(x) exists. f(x)={1+sinxifx0cosxif0xsinxifx0 13E 14E 15E Sketch the graph of an example of a function f that satisfies all of the given conditions. limx0f(x)=1,limx3f(x)=2,limx3+f(x)=2,f(0)=1,f(3)=1 Sketch 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)=1 18E 19E Guess 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, 3.0001 21E 22E 23E 24E 25E 26E 27E 28E 29E 30E 31E 32E Determine the infinite limit. limx12x(x1)2 34E 35E 36E 37E 38E 39E 40E 41E 42E 43E 44E Determine limx11x31 and limx1+1x31 (a) by evaluating f(x) = l/(x3 1) for values of x that approach 1 from the left and from the right, (b) by reasoning as in Example 9, and (c) from a graph of f EXAMPLE 9 FIGURE 15 46E (a) Estimate the value of the limit limx0 (1 + x)1/xto five decimal places. Does this number look familiar? (b) Illustrate part (a) by graphing the function y = (I + x)1/x 48E 49E 50E 51E 52E 53E 54E 55E Given 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)3E 4E 5E 6E 7E 8E 9E (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.11E Evaluate the limit, if it exists. limx3x2+3xx2x12 13E 14E 15E Evaluate the limit, if it exists. limx12x2+3x+1x22x3 17E 18E 19E 20E 21E 22E 23E Evaluate the limit, if it exists. limh0(3+h)131h 25E 26E 27E 28E

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