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All Textbook Solutions for Calculus: Early Transcendentals

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 0 a b, then In a In b.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 x 0, then (ln x)6 = 6 In x.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. lf x 0 and a l, then lnxlna=lnxa 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. tan1(1)=3/4 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. tan1x=sin1xcos1x 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 x is any real number, thenx2=x.Let f be the function whose graph is given. (a) Estimate the value of f(2). (b) Estimate the values of x such that f(x) = 3. (c) State the domain of f (d) State the range of f. (e) On what interval is f increasing? (f) Is f one-to-one? Explain. (g) Is f even, odd, or neither even nor odd? Explain.The graph of g is given. (a) State the value of g(2). (b) Why is g one-to-one? (c) Estimate the value of g1(2). (d) Estimate the domain of g1 (e) Sketch the graph of g1 lf f(x) = x2 2x + 3, evaluate the difference quotient f(a+h)f(a)h Sketch a rough graph or the yield of a crop as a function of the amount of fertilizer used.Find the domain and range of the function. Write your answer in interval notation. 5. f(x) = 2/(3x 1)Find the domain and range of the function. Write your answer in interval notation. 6. g(x)=16x4 7RE Find the domain and range of the function. Write your answer in interval notation. 8. F(t) = 3 + cos 2t 9RE The graph of .f is given. Draw the graphs of the following functions. (a) y = f(x 8) (b) y = f(x) (c) y = 2 f(x) (d) y=12f(x)1 (e) y = f1(x) (f) y = f1(x + 3)11RE Use transformations to sketch the graph of the function. y=2x 13RE Use transformations to sketch the graph of the function. y = In(x + 1)Use transformations to sketch the graph of the function. f(x) = cos 2x Use transformations to sketch the graph of the function. f(x)={xifx0ex1ifx0 Determine whether f is even, odd, or neither even nor odd. (a) f(x)=2x53x2+2. (h) f(x) = x3 x7 (c) f(x)=ex2 d) f(x) = 1 + sin x Find an expression for the function whose graph consists of the line segment from point (2, 2) to the point (1, 0) together with the top half of the circle with center the origin and radius 1.If f(x) = In x and g(x) = x2 9. find the functions (a) fg (b) gf (c) ff (d) gg, and their domains.Express the function F(x)=1/x+x as a composition of three functions.A small-appliance manufacturer finds that it costs 9000 to produce 1000 toaster ovens a week and 12,000 to produce 1500 toaster ovens a week. (a) Express the cost as a function of the number of toaster ovens produced, assuming that it is linear. Then sketch the graph. (b) What is the slope or the graph and what does it represent? (c) What is the y-intercept of the graph and what does it represent?If f(x) = 2x + In x, find f1(2).Find the inverse function of f(x)=x+12x+1.Find the exact value of each expression. 64. (a) tan13 (b) arctan (1) 25. Find the exact value of each expression. (a) e2 ln 3 (b) log10 25 + log10 4 (c) tan{arcsin 12) (d) sin(cos1(45))26RE The half-life of palladium-100, 100Pd, is four days. (So half of any given quantity of 100Pd will disintegrate in four days.) The initial mass of a sample is one gram. (a) Find the mass that remains after 16 days. (b) Find the mass m(t) that remains after t days. (c) Find the inverse of this function and explain its meaning. (d) When will the mass be reduced to 0.01g?The 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).One of the legs of a right triangle has length 4 cm. Express the length of the altitude perpendicular to the hypotenuse as a function of the length of the hypotenuse.The altitude perpendicular to the hypotenuse of a right triangle is 12 cm. Express the length of the hypotenuse as a function of the perimeter.Solve the equation |2x 1| |x + 5| = 3.Solve the inequality |x 1| |x 3| 5.5P Sketch the graph of the function g(x) = |x2 1 | |x2 4|.7P 8P The 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{sin x, cos x} (c) f(x) = max{x2, 2 + x, 2 x}Sketch the region in the plane defined by each of the following equations or inequalities. (a) max{x, 2y} = 1 (b) 1 max{x, 2y} 1 (c) max{x, y2} = 1 Evaluate (log2 3)(log3 4)(log4 5)(log31 32).(a) Show that the function f(x)=ln(x+x2+1) is an odd function. (b) Find the inverse function of f.Solve the inequality ln(x2 2x 2) 0.Use 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?Is it true that f(g+h)=fg+fh?Prove that if n is a positive integer, then 7n 1 is divisible by 6.Prove that 1 + 3 + 5 + + (2n l ) = n2.If fo(x) = x2 and fn+1(x) = fo(fn(x)) for n = 0, 1, 2,, find a formula for fn(x).(a) If fo(x)=12x and fn+1=fofnforn=0,1,2,, find an expression for fn(x) and use mathematical induction to prove it. (b) Graph f0, f1, f2, f3 on the same screen and describe the effects of repeated composition.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.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+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 Use 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) f(x)=11+e1/x Use 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) f(x)=x2+xx3+x2 Sketch 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)=1 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 Sketch the graph of an example of a function f that satisfies all of the given conditions. limx0f(x)=2,limx0+f(x)=0limx4f(x)=3,limx4+f(x)=0,f(0)=2,f(4)=1 Guess the value of the limit (if it exists) by evaluating the function at the given numbers (correct to six decimal places). limx3x23xx20, x= 3.1, 3.05, 3.01, 3.001, 3.0001, 2.9, 2.95, 2.99, 2.999, 2.9999 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 Guess the value of the limit (if it exists) by evaluating the function at the given numbers (correct to six decimal places). limt0e5t1t,t=0.5,0.1,0.01,0.001,0.0001 Guess the value of the limit (if it exists) by evaluating the function at the given numbers (correct to six decimal places). limh0(2+h)532h, h = 0.5, 0.1, 0.01, 0.001, 0.0001 Use a table of Values to estimate i:he value of the limit. If you have a graphing device. use it to confirm your result graphically. limx4lnxln4x4 24E 25E 26E 27E Use a table of Values to estimate i:he value of the limit. If you have a graphing device. use it to confirm your result graphically. limx0+x2lnx (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.(a) Estimate the value of limx0sinxsinx by graphing the function f(x) = (sin x)/(sin x). State your answer correct to two decimal places. (b) Check your answer in part (a) by evaluating f(x) for values of x that approach 0.Determine the infinite limit. limx5+x+1x5 Determine the infinite limit. limx5x+1x5 Determine the infinite limit. limx12x(x1)2 Determine the infinite limit. limx3x(x3)5 Determine the infinite limit. limx3+ln(x29)Determine the infinite limit. limx0+ln(sinx)Determine the infinite limit. limx(/2)+1xsecx Determine the infinite limit. limxcotx Determine the infinite limit. limx2xcscx Determine the infinite limit. limx2x22xx24x+4 Determine the infinite limit. limx2+x22x8x25x+6 Determine the infinite limit. limx0+(1xlnx)Determine the infinite limit. limx0(lnx2x2)(a) Find the vertical asymptotes of the function y=x2+13x2x2 (b) Confirm your answer to part (a) by graphing the function.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 (a) By graphing the function f(x) = (tan 4x)/x 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 pan (a) by evaluating f(x) for values of x that approach 0.(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 (a) 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 f(x) for x = 0.04, 0.02, 0.01, 0.005, 0.003, and 0.001. Guess again.(a) Evaluate h(x) = (tan x x)/x3 for x = 1, 0.5, 0.1 , 0.05, 0.0 1, and 0.005. (b) Guess the value of limx0tanxxx3 (c) Evaluate h(x) for successively smaller values of x until you finally reach a value of 0 for h(x). Are you still confident that your guess in pan (b) is correct? Explain why you eventually obtained 0 values. (In Section 4.4 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 they-axis to estimate the limit of h(x) as x approaches 0. Continue to zoom in until you observe distortions in the graph of h. Compare with the results of pan (c).Graph 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.Consider the function f(x) = tan1x. (a) Show that .f(x) = 0 for x=1,12,13, (b) Show that f(x) = 1 for x=4,45,49, (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(2 sin x) x Then find the exact equations of these asymptotes.In the theory of relativity, the mass of a particle with velocity v is m0=m01v2/c2 where mo is the mass of the particle at rest and c is the speed of light. What happens as v c?(a) Use numerical and graphical evidence to guess the value of the limit limx1x31x1 (b) How close to I does x have to be to ensure that the function in pan (a) is within a distance 0.5 of its limit?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)Evaluate the limit and justify each step by indicating the appropriate Limit Law(s). limx3(5x33x2+x6)Evaluate the limit and justify each step by indicating the appropriate Limit Law(s). limx1(x43x)(x2+5x+3)Evaluate the limit and justify each step by indicating the appropriate Limit Law(s). limt2t422t23t+2 Evaluate the limit and justify each step by indicating the appropriate Limit Law(s). limu2u4+3u+6 Evaluate the limit and justify each step by indicating the appropriate Limit Law(s). limx8(1+x3)(26x2+x3)Evaluate the limit and justify each step by indicating the appropriate Limit Law(s). limt2(t22t33t+5)2 Evaluate the limit and justify each step by indicating the appropriate Limit Law(s). limx22x2+13x2 (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.Evaluate the limit, if it exists. limx2x2+x5x5 Evaluate the limit, if it exists. limx3x2+3xx2x12 Evaluate the limit, if it exists. limx5x25x+6x5 Evaluate the limit, if it exists. limx4x2+3xx2x12 Evaluate the limit, if it exists. limt3t292t2+7t+3 Evaluate the limit, if it exists. limx12x2+3x+1x22x3 Evaluate the limit, if it exists. limh0(5+h)225h Evaluate the limit, if it exists. limh0(2+h)38h Evaluate the limit, if it exists. limx2x+2x3+8 Evaluate the limit, if it exists. limt1t41t31 Evaluate the limit, if it exists. limh09+h3h Evaluate the limit, if it exists. limu24u+13u2 Evaluate the limit, if it exists. limx31x13x3 Evaluate the limit, if it exists. limh0(3+h)131h Evaluate the limit, if it exists. limt01+t1tt Evaluate the limit, if it exists. limt0(1t1t2+t)Evaluate the limit, if it exists. limx164x16xx2 Evaluate the limit, if it exists. limx2x24x+4x43x24 Evaluate the limit, if it exists. limt0(1t1+t1t)Evaluate the limit, if it exists. limx4x2+95x+4 Evaluate the limit, if it exists. limh0(x+h)x3h Evaluate the limit, if it exists. limh01(xh)21x2h (a) Estimate the value of limx0x1+3x1 by graphing the function f(x)=x/(1+3x1). (b) Make a table of values of f(x) for x close to 0 and guess the value of the limit. (c) Use the Limit Laws to prove that your guess in correct.(a) Use a graph of f(x)=3+x3x to estimate the value of limx0f(x) to two decimal places. (b) Use a table of values of f(x) to estimate the limit to four decimal places. (c) Use the Limit Laws to find the exact value of the limit.Use 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.Use the Squeeze Theorem to show that limx0x3+x2sinx=0 Illustrate by graphing the functions f, q, and h (in the notation of the Squeeze Theorem) on the same screen.If 4x 9 f(x) x2 4x + 7 for x 0, find limx4f(x)If 2x g(x) x4 x2 + 2 for all x, evaluate limx1g(x)Prove that limx0x4cos2x=0.Prove that limxxesin(/x)=0.Find the limit, if it exists. If the limit does not exist, explain why. limx3(2x+x3)Find the limit, if it exists. If the limit does not exist, explain why. limx62x+12x+6 Find the limit, if it exists. If the limit does not exist, explain why. limx0.52x12x3x2 Find the limit, if it exists. If the limit does not exist, explain why. limx22x2+x Find the limit, if it exists. If the limit does not exist, explain why. limx(1x1x)Find the limit, if it exists. If the limit does not exist, explain why. limx0+(1x1x)The signum (or sign)function, denoted by sgn, is defined by sgnx={1ifx00ifx=01ifx0 (a) Sketch the graph of this function. (b) Find each of the following limits or explain why it does not exist. (i) limx0+sgnx (ii) limx0sgnx (iii) limx0sgnx (iv) limx0sgnx Let 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.Let g(x)=x2+x6x2 (a) Find (i) limx2+g(x) (ii) limx2g(x) (b) Does limx2 g(x) exist'! (c) Sketch the graph of g.Let f(x)={x2+1ifx1(x2)2ifx1 (a) Find limx0 f(x).and lim x0+ f(x) (b) Does limx0 f(x) exist? (c) Sketch the graph of f.Let B(t)={412tift2t+cift2 Find the value of c so that limt2B(t) exists.l.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.(a) If the symbol denotes the greatest integer function defined in Example I 0, evaluate (i) limx2+x (ii) limx2x (iii) limx2.4x (b) If n is an integer, evaluate (i) limxnx (ii) limxn+x (c) For what values of a does limx0x exist?Let f(x)=cosx,x. (a) Sketch the graph of f. (b) Evaluate each limit, if it exists. (i) limx0f(x) (ii) limx(/2)f(x) (iii) limx(/2)+f(x) (iv) limx(/2)f(x) (c) For what values of a does lim x0 f(x) exist ?If f(x)=x+x, show that lim x0 f(x) exists hut is not equal to f(2).In the theory of relativity, the Lorentz contraction formula L=L01v2/c2 expresses the length L of an object as a function of its velocity v with respect to an observer, where L0 is the length of the object at rest and c is the speed of light. Find limvc L and interpret the result. Why is a left-hand limit necessary?If p is a polynomial, Show that limxa p(x) = p(a)If r is a rational function. use Exercise 57 to show that limxa r(x) = r(a) for every number a in the domain of r.If limx1f(x)8x1=10, find limx1f(x).If limx0f(x)x2=5, find the following limits. (a) limx0f(x) (b) limx0f(x)x If f(x)={x2ifxisrational0ifxisirrational prove that limx0f(x)=0.Show by means of an example that limxa[f(x)+g(x)] may exist even though neither limxaf(x)norlimxag(x)exists.Show by means of an example that limxa[f(x)g(x)] may exist even though neither limxa f(x) nor limxa g(x) exists.Evaluate limx26x23x1.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 66E Use the given graph of f to find a number such that if |x 1| then |f(x) 1| 0.2 Use the given graph of f to find a number such that if 0|x 3| then |f(x) 2 | 0.5 Use the given graph of f(x)=x to find a number such that if |x 4 | then x20.4 Use the given graph of f(x) =x2 to find a number such that if |x 1| then |x2 1 | 12 Use a graph to find a number such that if |x4|then|tanx1|0.2 Use a graph to find a number such that if |x4|then|2xx2+40.4|0.1 For the limit limx2(x33x+4)=6 illustrate Definition 2 by finding values of that correspond =0.2 and 0.1. Definition 2 For the limit limx0e2x1x=2 illustrate Definition 2 by finding values of that correspond to = 0.5 and = 0. 1. Definition 2 (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?Given that limxcsc2=, 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 5 cm2 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.lw2 + 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 al 200C? (b) If the temperature is allowed to vary from 200C by up to 1C, what range of wan age is allowed for the input power? (c) In terms of thee, , 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 va1ue of ?(a) Find a number such that if |x 2| , then |4x 8| , where = 0.1. (b) Repeat part (a) with = 0.01 Given that limx2(5x7)=3, illustrate Definition 2 by finding values of that correspond to . = 0.1, = 0.05, and = 0.01. Definition 2 Prove the statement using the , definition of a limit and illustrate with a diagram like Figure 9. FIGURE9 limx3(1+13x)=2 Prove the statement using the , definition of a limit and illustrate with a diagram like Figure 9. FIGURE9 limx4(2x5)=3 Prove the statement using the , definition of a limit and illustrate with a diagram like Figure 9. FIGURE9 limx3(14x)=13 Prove the statement using the , definition of a limit and illustrate with a diagram like Figure 9. FIGURE9 limx2(3x+5)=1 Prove the statement using the , definition of a limit. limx12+4x3=2 Prove the statement using the , definition of a limit. limx10(345x)=5 Prove the statement using the , definition of a limit. limx4x22x8x4=6 22E Prove the statement using the , definition of a limit. limxax=a Prove the statement using the , definition of a limit. limxac=c Prove the statement using the , definition of a limit. limx0x2=0 26E Prove the statement using the , definition of a limit. limx0x=0 Prove the statement using the , definition of a limit. limx6+6+x8=0 Prove the statement using the , definition of a limit. limx2(x24x+5)=1 Prove the statement using the , definition of a limit. limx2(x2+2x7)=1 Prove the statement using the , definition of a limit. limx2(x21)=3 Prove the statement using the , definition of a limit. limx2x3=8 Verify that another possible choice of for showing that limx3x2=9 in Example 4 is = min{2, /8}. EXAMPLE 4 Verify, by a geometric argument, that the largest possible choice of for showing that limx3x2=9 is =9+3 Prove that limx21x=12.Prove that limxax=aifa0. [Hint:Usexa=xax+a]If 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 2 If the function f is defined by f(x)={0ifxisrational1ifxisirrational prove that limx0f(x) does not exist.By comparing Definitions 2, 3, and 4, prove Theorem 2.3.1. Definition 2 Definition 3 Definition 4 How close to 3 do we have to take x so that 1(x+3)410,000 Prove, using Definition 6, that limx31(x+3)4= Definition 6 Prove that limx0+lnx=Suppose 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)]=ifc0 Write 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 2 Sketch 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 4 Sketch the graph of a function f that is continuous except for the stated discontinuity. Removable discontinuity at 3, jump discontinuity at 5 Sketch 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 2 The 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 past 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 time Use 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=1 Use 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=2 Use 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=1 Use the definition of continuity and the properties of limits to show that the function is continuous at the given number a. f(x)=3x45x+x2+43,a=2 Use 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,]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)Explain why the function is discontinuous at the given number a. Sketch the graph of the function. f(x)=1x+2a=2 Explain why the function is discontinuous at the given number a. Sketch the graph of the function. f(x)={1x+2ifx21ifx=2a=2 Explain why the function is discontinuous at the given number a. Sketch the graph of the function. f(x)={x+3ifx12xifx1a=1 Explain why the function is discontinuous at the given number a. Sketch the graph of the function. f(x)={x2xx21ifx11ifx=1a=1 Explain why the function is discontinuous at the given number a. Sketch the graph of the function. f(x)={cosxifx00ifx=01x2ifx0a=0 Explain why the function is discontinuous at the given number a. Sketch the graph of the function. f(x)={2x25x3x3ifx36ifx=2a=3 How 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)=x2x2x2 How 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)=x31x24 Explain, 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 F(x)=2x2x1x2+1 Explain, 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 G(x)=x2+12x2x1 Explain, 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 Q(x)=x23x32 Explain, 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 R(t)=esint2+cost Explain, 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)Explain, 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 B(x)=tanx4+x2 Explain, 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+1x Explain, 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 N(r)=tan1(1+er2)Locate the discontinuities of the function and illustrate by graphing. y=11+e1/x Locate the discontinuities of the function and illustrate by graphing. y=ln(tan2x)Use continuity to evaluate the limit. limx2x20x2 Use continuity to evaluate the limit. limxsin(x+sinx)Use continuity to evaluate the limit. limx1ln(5x21+x)Use continuity to evaluate the limit. limx43x22x4 Show that f is continuous on ( , ). f(x)={1x2ifx1lnxifx1 Show that f is continuous on ( , ). f(x)={sinxifx/4cosxifx/4 Find 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/xifx1 Find 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)={2xifx13xif1x4xifx4 Find 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+2ifx0exif0x12xifx1 The 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+2xifx2x3cxifx2 Find the values of a and h that make f continuous everywhere. f(x)={x24x2ifx2ax2bx+3ifxx32xa+bifx2 Suppose f and g are continuous functions such that g(2) = 6 and limx2[3f(x)+f(x)g(x)]=36. Find f(2).Let f(x)=1/xandg(x)=1/x2. (a) Find (f g)(x). (b) Is f g continuous everywhere? Explain.Which of the following functions .f has a removable discontinuity at a? If the discontinuity is removable, find a function g that agrees with f for x a and is continuous at a. (a) f(x)=x41x1,a=1 (b) f(x)=x3x22xx2,a=2 (c) f(x)=sinx,a=Suppose that a function f is continuous on [0, 1] except at 0.25 and that f(0) = 1 and f(l) = 3. Let N = 2. Sketch two possible graphs of f, one showing that f might not satisfy the conclusion of the Intermediate Value Theorem and one showing that f might still satisfy the conclusion of the Intermediate Value Theorem (even though it doesnt satisfy the hypothesis).If f(x) = x2 + 10 sin x, show that there is a number c such that f(c) = 1000.Suppose f is continuous on [1, 5] and the only solutions of the equation f(x) = 6 arc x = 1 and x = 4.lf f(2) = 8, explain why f(3) 6.Use the Intermediate Value Theorem to show that there is a root of the given equation in the specified interval. x4 + x 3 = 0, (1,2)Use the Intermediate Value Theorem to show that there is a root of the given equation in the specified interval. lnx=xx,(2,3)Use the Intermediate Value Theorem to show that there is a root of the given equation in the specified interval. ex = 3 2x, (0,1)Use the Intermediate Value Theorem to show that there is a root of the given equation in the specified interval. sin x = x2 x, (1,2)(a) Prove that the equation has at least one real root. (b) Use your calculator to find an interval of length 0.01 that contains a root. cos x = x3 (a) Prove that the equation has at least one real root. (b) Use your calculator to find an interval of length 0.01 that contains a root. ln x = 3 2x (a) Prove that the equation has at least one real root. (b) Use your graphing device to find the root. correct to three decimal places. 100ex/100 = 0.0 1x2 (a) Prove that the equation has at least one real root. (b) Use your graphing device to find the root. correct to three decimal places. arctan x = 1 x Prove, without graphing, that the graph of the function has at least two x-intercepts in the specified interval. y = sin x3, (1, 2)Prove, without graphing, that the graph of the function has at least two x-intercepts in the specified interval. y = x2 3 + 1/x, (0, 2)Prove that f is continuous at a if and only if limh0f(a+h)=f(a)To prove that sine is continuous, we need to show that limxasinx=sina for every real number a. By Exercise 63 an equivalent statement is that limh0sin(a+h)=sina Use (6) to show that. this is true.Prove that cosine is a continuous function.66E For what values of x is f continuous? f(x)={0ifxisrational1ifxisirrational For what values of x is g continuous? g(x)={0ifxisrational1ifxisirrational Is there a number that is exactly 1 more than its cube?If a and b are positive numbers, prove that the equation ax3+2x21+bx3+x2=0 has at least one solution in the interval (1, 1).Show that the function f(x)={x4sin(1/x)ifx00ifx=0 is continuous on ( , ).(a) Show that the absolute value function F(x) = | x | is continuous everywhere. (b) Prove that if f is a continuous function on an interval, then so is | f |. (c) Is the converse or the statement in part (b) also true? In other word, if | f | is continuous, does it follow that f is continuous? If so, prove it. If not, find a counterexample.Explain in your own words tile meaning of each of the following. (a) limxf(x)=5 (b) limxf(x)=3 (a) Can the graph of y = f(x) intersect a vertical asymptote? Can it intersect a horizontal asymptote? Illustrate bysketching graphs. (b) How many horizontal asymptotes can the graph of y = .f(x) have? Sketch graphs to illustrate the possibilities.For the function f whose graph is given, state the following. (a) limxf(x) (b) limxf(x) (c) limx1f(x) (d) limx3f(x) (e) The equations of the asymptotes

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