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All Textbook Solutions for Calculus: Early Transcendentals (3rd Edition)
97EWhat is the radian measure of a 270 angle? What is the degree measure of a 5/4-rad angle?Evaluate cos (11/6) and sin (5/4).Use sin2+cos2=1 to prove that 1+cot2=csc2.Explain why sin1(sin0)=0, but sin1(sin2)2.Evaluate sec11 and tan11.Define the six trigonometric functions in terms of the sides of a right triangle.For the given angle corresponding to the point P(4, 3) in the figure, evaluate sin , cos , tan , cot , sec , and csc .A projectile is launched at an angle of above the horizontal with an initial speed of v ft/s and travels over level ground. The time of flight t (the time it takes, in seconds, for the projectile to return to the ground) is approximated by the equation t=vsin16. Determine the time of flight of a projectile if = /6 and v = 96.A boat approaches a 50-ft-high lighthouse whose base is at sea level. Let d be the distance between the boat and the base of the lighthouse. Let L be the distance between the boat and the top of the lighthouse. Let be the angle of elevation between the boat and the top of the lighthouse. a. Express d as a function of . b. Express L as a function of .How is the radian measure of an angle determined?Explain what is meant by the period of a trigonometric function. What are the periods of the six trigonometric functions?What are the three Pythagorean identities for the trigonometric functions?Given that sin=1/5 and =2/5, use trigonometric identities to find the values of tan , cot , sec , and csc .Solve the equation sin = 1, for 0 2.Solve the equation sin 2=1, for 02.Where is the tangent function undefined?What is the domain of the secant function?Explain why the domain of the sine function must be restricted in order to define its inverse function.Why do the values of cos1 x lie in the interval [0, ]?Evaluate cos1(cos(5/4)).Evaluate sin1(sin(11/6)).The function tan x is undefined at x = /2. How does this fact appear in the graph of y = tan1 x?State the domain and range of sec1 x.Evaluating trigonometric functions Without using a calculator, evaluate the following expressions or state that the quantity is undefined. 19.cos(2/3)Evaluating trigonometric functions Without using a calculator, evaluate the following expressions or state that the quantity is undefined. 20. sin(2/3)Evaluating trigonometric functions Without using a calculator, evaluate the following expressions or state that the quantity is undefined. 21. tan(3/4)Evaluating trigonometric functions Without using a calculator, evaluate the following expressions or state that the quantity is undefined. 22. tan(15/4)Evaluating trigonometric functions Without using a calculator, evaluate the following expressions or state that the quantity is undefined. 23. cot(13/3)Evaluating trigonometric functions Without using a calculator, evaluate the following expressions or state that the quantity is undefined. 24. sec(7/6)Evaluating trigonometric functions Without using a calculator, evaluate the following expressions or state that the quantity is undefined. 25. cot(17/3)Evaluating trigonometric functions Without using a calculator, evaluate the following expressions or state that the quantity is undefined. 26. sin(16/3)Evaluating trigonometric functions Without using a calculator, evaluate the following expressions or state that the quantity is undefined. 27. cos0Evaluating trigonometric functions Without using a calculator, evaluate the following expressions or state that the quantity is undefined. 28. sin(/2)Evaluating trigonometric functions Without using a calculator, evaluate the following expressions or state that the quantity is undefined. 29. cos()Evaluating trigonometric functions Without using a calculator, evaluate the following expressions or state that the quantity is undefined. 30. tan3Evaluating trigonometric functions Without using a calculator, evaluate the following expressions or state that the quantity is undefined. 31. sec(5/2)Evaluating trigonometric functions Without using a calculator, evaluate the following expressions or state that the quantity is undefined. 32. cotEvaluating trigonometric functions Without using a calculator, evaluate the following expressions or state that the quantity is undefined. 33. cos(/12) (Hint: use a half-angle formula.)Evaluating trigonometric functions Without using a calculator, evaluate the following expressions or state that the quantity is undefined. 34. sin(3/8)Solving trigonometric equations Solve the following equations. 37. tan x = 1Solving trigonometric equations Solve the following equations. 38. 2 cos + = 0Solving trigonometric equations Solve the following equations. 39. sin2=14,02Solving trigonometric equations Solve the following equations. 40. cos2=12,02Solving trigonometric equations Solve the following equations. 41. 2sinx1=0Solving trigonometric equations Solve the following equations. 44. sin2 1 = 0Solving trigonometric equations Solve the following equations. 44. sin cos = 0, 0 2Solving trigonometric equations Solve the following equations. 42. sin3x=22,0x2Solving trigonometric equations Solve the following equations. 43. cos 3x = sin 3x, 0 x 2Solving trigonometric equations Solve the following equations. 46. tan2 2 = 1, 0Solving trigonometric equations Solve the following equations. 45. sin2=15,02Solving trigonometric equations Solve the following equations. 46. cos3=37,0Projectile range A projectile is launched from the ground at an angle above the horizontal with an initial speed V in ft/s. The range (the horizontal distance traveled by the projectile over level ground) is approximated by the equation x=v232sin2. Find all launch angles that satisfy the following conditions; express your answers in degrees. 47.Initial speed of 150 ft/s; range of 400 ftProjectile range A projectile is launched from the ground at an angle above the horizontal with an initial speed V in ft/s. The range (the horizontal distance traveled by the projectile over level ground) is approximated by the equation x=v232sin2. Find all launch angles that satisfy the following conditions; express your answers in degrees. 48.Initial speed of 160 fts/s; range of 350 ftInverse sines and cosines Without using a calculator, evaluate the following expressions or state that the quantity is undefined. 47. sin1 1Inverse sines and cosines Without using a calculator, evaluate the following expressions or state that the quantity is undefined. 48. cos1 (1)Inverse sines and cosines Without using a calculator, evaluate the following expressions or state that the quantity is undefined. 49. sin1(12)Inverse sines and cosines Without using a calculator, evaluate the following expressions or state that the quantity is undefined. 50. cos1(22)Inverse sines and cosines Without using a calculator, evaluate the following expressions or state that the quantity is undefined. 51. sin132Inverse sines and cosines Without using a calculator, evaluate the following expressions or state that the quantity is undefined. 52. cos1 2Inverse sines and cosines Without using a calculator, evaluate the following expressions or state that the quantity is undefined. 53. cos1(12)Inverse sines and cosines Without using a calculator, evaluate the following expressions or state that the quantity is undefined. 54. sin1 (1)Inverse sines and cosines Without using a calculator, evaluate the following expressions or state that the quantity is undefined. 55. cos (cos1 (1))Inverse sines and cosines Without using a calculator, evaluate the following expressions or state that the quantity is undefined. 56. cos1 (cos (7/6))Using right triangles Use a right-triangle sketch to complete the following exercises. 59.Suppose =cos1(5/13) Find sin and tanUsing right triangles Use a right-triangle sketch to complete the following exercises. 60.Suppose =tan1(4/3). Find sec and csc .Right-triangle relationships Draw a right triangle to simplify the given expressions. Assume x 0. 57. cos (sin1 x)Right-triangle relationships Draw a right triangle to simplify the given expressions. Assume x 0. 58. cos (sin1 (x/3))Right-triangle relationships Draw a right triangle to simplify the given expressions. Assume x 0. 59. sin (cos1 (x/2))Right-triangle relationships Draw a right triangle to simplify the given expressions. Assume x 0. 60. sin1 (cos ), for 0 /2Right-triangle relationships Draw a right triangle to simplify the given expressions. Assume x 0. 61. sin (2 cos1 x) (Hint: Use sin 2 = 2 sin cos .)Right-triangle relationships Draw a right triangle to simplify the given expressions. Assume x 0. 62. cos (2 sin1 x) (Hint: Use cos 2 = cos2 sin2 .)Trigonometric identities 29. Prove that sec=1cos.Trigonometric identities 30. Prove that tan=sincos.Trigonometric identities 31. Prove that tan2 + 1 = sec2 .Trigonometric identities 32. Prove that sincsc+cossec=1.Trigonometric identities 33. Prove that sec (/2 ) = csc .Trigonometric identities 34. Prove that sec (x + ) = sec x.Identities Prove the following identities. 73. cos1x+cos1(x)=74EEvaluating inverse trigonometric functions Without using a calculator, evaluate or simplify the following expressions. 67. tan1376EEvaluating inverse trigonometric functions Without using a calculator, evaluate or simplify the following expressions. 69. sec1 278E79EEvaluating inverse trigonometric functions Without using a calculator, evaluate or simplify the following expressions. 72. tan1 (tan (3/4))Evaluating inverse trigonometric functions Without using a calculator, evaluate or simplify the following expressions. 73. csc1 (sec 2)82ERight-triangle relationships Use a right triangle to simplify the given expressions. Assume x 0. 75. cos (tan1 x)Right-triangle relationships Use a right triangle to simplify the given expressions. Assume x 0. 76. tan (cos1 x)Right-triangle relationships Use a right triangle to simplify the given expressions. Assume x 0. 77. cos (sec1 x)Right-triangle relationships Use a right triangle to simplify the given expressions. Assume x 0. 78. cot (tan1 2x)Right-triangle relationships Use a right triangle to simplify the given expressions. Assume x 0. 79. sin(sec1(x2+164))88ERight-triangle pictures Express in terms of x using the inverse sine, inverse tangent, and inverse secant functions. 81.Right-triangle pictures Express in terms of x using the inverse sine, inverse tangent, and inverse secant functions. 82.Explain why or why not Determine whether the following statements are true and give an explanation or counterexample. a. sin (a + b) = sin a + sin b. b. The equation cos = 2 has multiple real solutions. c. The equation sin =12 has exactly one solution. d. The function sin (x/12) has a period of 12. e. Of the six basic trigonometric functions, only tangent and cotangent have a range of (, ). f. sin1xcos1x=tan1x. g. cos1 (cos (15/16)) = 15/16. h. sin1 x = 1/sin x.One function gives all six Given the following information about one trigonometric function, evaluate the other five functions. 84. sin=45 and 3/2One function gives all six Given the following information about one trigonometric function, evaluate the other five functions. 85. cos=513 and 0 /2One function gives all six Given the following information about one trigonometric function, evaluate the other five functions. 86. sec=53 and 3/2 2One function gives all six Given the following information about one trigonometric function, evaluate the other five functions. 87. csc=1312 and 0 /296EAmplitude and period Identify the amplitude and period of the following functions. 89. g() = 3 cos (/3)98EAmplitude and period Identify the amplitude and period of the following functions. 91. q(x) = 3.6 cos (x/24)Law of cosines Use the figure to prove the law of cosines (which is a generalization of the Pythagorean theorem): c2 = a2 + b2 2ab cos .Little-known fact The shortest day of the year occurs on the winter solstice (near December 21) and the longest day of the year occurs on the summer solstice (near June 21). However, the latest sunrise and the earliest sunset do not occur on the winter solstice, and the earliest sunrise and the latest sunset do not occur on the summer solstice. At latitude 40 north, the latest sunrise occurs on January 4 at 7:25 A.M. (14 days after the solstice), and the earliest sunset occurs on December 7 at 4:37 P.M. (14 days before the solstice). Similarly, the earliest sunrise occurs on July 2 at 4:30 A.M. (14 days after the solstice) and the latest sunset occurs on June 7 at 7:32 P.M. (14 days before the solstice). Using sine functions, devise a function s(t) that gives the time of sunrise t days after January 1 and a function S(t) that gives the time of sunset t days after January 1. Assume that s and S are measured in minutes and s = 0 and S = 0 correspond to 4:00 A.M. Graph the functions. Then graph the length of the day function D(t) = S(t) s(t) and show that the longest and shortest days occur on the solstices.Anchored sailboats A sailboat named Ditl is anchored 200 feet north and 300 feet east of an observer standing on shore, while a second sailboat named Windborne is anchored 250 feet north and 100 feet west of the observer. Find the angle between the two sailboats as determined by the observer on shore.Area of a circular sector Prove that the area of a sector of a circle of radius r associated with a central angle (measured in radians) is A=12r2.Graphing sine and cosine functions Beginning with the graphs of y = sin x or y = cos x, use shifting and scaling transformations to sketch the graph of the following functions. Use a graphing utility to check your work. 92. f(x) = 3 sin 2xGraphing sine and cosine functions Beginning with the graphs of y = sin x or y = cos x, use shifting and scaling transformations to sketch the graph of the following functions. Use a graphing utility to check your work. 93. g(x) = 2 cos (x/3)Graphing sine and cosine functions Beginning with the graphs of y = sin x or y = cos x, use shifting and scaling transformations to sketch the graph of the following functions. Use a graphing utility to check your work. 94. p(x) = 3 sin (2x /3) + 1Graphing sine and cosine functions Beginning with the graphs of y = sin x or y = cos x, use shifting and scaling transformations to sketch the graph of the following functions. Use a graphing utility to check your work. 95. q(x) = 3.6 cos (x/24) + 2108EDesigner functions Design a sine function with the given properties. 97. It has a period of 24 hr with a minimum value of 10 at t = 3 hr and a maximum value of 16 at t = 15 hr.Field goal attempt Near the end of the 1950 Rose Bowl football game between the University of California and Ohio State University. Ohio State was preparing to attempt a field goal from a distance of 23 yd from the endline at point A on the edge of the kicking region (see figure). But before the kick. Ohio State committed a penalty and the ball was backed up 5 yd to point B on the edge of the kicking region. After the game, the Ohio State coach claimed that his team deliberately committed a penalty to improve the kicking angle. Given that a successful kick must go between the uprights of the goal posts G1 and G2, is G1BG2 greater than G1AG2? (In 1950, the uprights were 23 ft 4 in apart, equidistant from the origin on the end line. The boundaries of the kicking region are 53 ft 4 in apart and are equidistant from the y-axis. (Source: The College Mathematics Journal 27, 4, Sep 1996)A surprising result The Earth is approximately circular in cross section, with a circumference at the equator of 24,882 miles. Suppose we use two ropes to create two concentric circles: one by wrapping a rope around the equator and another using a rope 38 ft longer (see figure). How much space is between the ropes?Daylight function for 40 N Verify that the function D(t)=2.8sin(2365(t81))+12 has the following properties, where t is measured in days and D is the number of hours between sunrise and sunset. a. It has a period of 365 days. b. Its maximum and minimum values are 14.8 and 9.2, respectively, which occur approximately at t = 172 and t = 355, respectively (corresponding to the solstices). c. D(81) = 12 and D(264) 12 (corresponding to the equinoxes).Block on a spring A light block hangs at rest from the end of a spring when it is pulled down 10 cm and released. Assume the block oscillates with an amplitude of 10 cm on either side of its rest position with a period of 1.5 s. Find a trigonometric function d(t) that gives the displacement of the block t seconds after it is released, where d(t) 0 represents downward displacement.Viewing angles An auditorium with a flat floor has a large flat-panel television on one wall. The lower edge of the television is 3 ft above the floor, and the upper edge is 10 ft above the floor (see figure). Express in terms of x.Ladders Two ladders of length a lean against opposite walls of an alley with their feet touching (see figure). One ladder extends h feet up the wall and makes a 75 angle with the ground. The other ladder extends k feet up the opposite wall and makes a 45 angle with the ground. Find the width of the alley in terms of h. Assume the ground is horizontal and perpendicular to both walls.Pole in a corner A pole of length L is carried horizontally around a corner where a 3-ft-wide hallway meets a 4-ft-wide hallway. For 0 /2, find the relationship between L and at the moment when the pole simultaneously touches both walls and the corner P. Estimate when L = 10 ft.Explain why or why not Determine whether the following statements are true and give an explanation or counterexample. a. A function could have the property that f(x) = f(x), for all x. b. cos (a + b) = cos a + cos b, for all a and b in [0, 2]. c. If f is a linear function of the form f(x) = mx + b, then f(u + v) = f(u) + f(v), for all u and v. d. The function f(x) = 1 x has the property that f(f(x)) = x. e. The set {x: |x + 3| 4} can be drawn on the number line without lifting your pencil. f. log10(xy) = (log10 x)(log10 y) g. sin1 (sin (2)) = 0Functions Decide whether graph A, graph B, or both represent functions.One-to-one functions Decide whether f, g, or both represent one-to-one functions.Domain and range Determine the domain and range of the following functions. 4. f(x)=x5+xDomain and range Determine the domain and range of the following functions. 5. f(w)=2w23w2w2Domain and range Determine the domain and range of the following functions. 6. g(x)=ln(x+6)Domain and range Determine the domain and range of the following functions. 7. h(z)=z22z3Suppose f and g are even functions with f(2)=2 and g(2)=2. Evaluate f(g(2)) and g(f(2)).Is it true that tan (tan1x)=x for all x? Is it true that tan1(tanx)=x for all x?Evaluating functions from graphs Assume f is an odd function and that both f and g are one-to-one. Use the (incomplete) graph of f and the graph of g to find the following function values. 10. f(g(4))Evaluating functions from graphs Assume f is an odd function and that both f and g are one-to-one. Use the (incomplete) graph of f and the graph of g to find the following function values. 11.g(f(4))Evaluating functions from graphs Assume f is an odd function and that both f and g are one-to-one. Use the (incomplete) graph of f and the graph of g to find the following function values. 12. f1(10)Evaluating functions from graphs Assume f is an odd function and that both f and g are one-to-one. Use the (incomplete) graph of f and the graph of g to find the following function values. 13.g1(5)Evaluating functions from graphs Assume f is an odd function and that both f and g are one-to-one. Use the (incomplete) graph of f and the graph of g to find the following function values. 14. f1(g1(4))Evaluating functions from graphs Assume f is an odd function and that both f and g are one-to-one. Use the (incomplete) graph of f and the graph of g to find the following function values. 15. g1(f(3))Evaluating functions from graphs Assume f is an odd function and that both f and g are one-to-one. Use the (incomplete) graph of f and the graph of g to find the following function values. 16.f1(8)Evaluating functions from graphs Assume f is an odd function and that both f and g are one-to-one. Use the (incomplete) graph of f and the graph of g to find the following function values. 17. f1(1+f(3))Evaluating functions from graphs Assume f is an odd function and that both f and g are one-to-one. Use the (incomplete) graph of f and the graph of g to find the following function values. 18. g(1f(f1(7)))Composite functions Let f(x) = x3, g(x) = sin x, and h(x)=x. a. Evaluate h(g(/2)). b. Find h(f(x)). c. Find f(g(h(x))). d. Find the domain of g f e. Find the range of f g.Composite functions Find functions f and g such that h = f g. a. h(x) = sin (x2 +1) b. h(x) = (x2 4)3 c. h(x) = ecos 2xSimplifying difference quotients Evaluate and simplify the difference quotients f(x+h)f(x)h and f(x)f(a)xa for each function. 17. f(x) = x2 2xSimplifying difference quotients Evaluate and simplify the difference quotients f(x+h)f(x)h and f(x)f(a)xa for each function. 18. f(x) = 4 5xSimplifying difference quotients Evaluate and simplify the difference quotients f(x+h)f(x)h and f(x)f(a)xa for each function. 19. f(x) = x3 + 2Simplifying difference quotients Evaluate and simplify the difference quotients f(x+h)f(x)h and f(x)f(a)xa for each function. 20. f(x)=7x+3Equations of lines In each part below, find an equation of the line with the given properties. Graph the line. a. The line passing through the points (2, 3) and (4, 2) b. The line with slope 34and x-intercept (4, 0) c. The line with intercepts (4, 0) and (0, 2)Population function The population of a small town was 500 in 2018 and is growing at a rate of 24 people per year. Find and graph the linear population function p(t) that gives the population of the town t years after 2018. Then use this model to predict the population in 2033.Boiling-point function Water boils at 212 F at sea level and at 200 F at an elevation of 6000 ft. Assume that the boiling point B varies linearly with altitude a. Find the function B = f(a) that describes the dependence. Comment on whether a linear function is a realistic model.Publishing costs A small publisher plans to spend 1000 for advertising a paperback book and estimates the printing cost is 2.50 per book. The publisher will receive 7 for each book sold. a. Find the function C = f(x) that gives the cost of producing x books. b. Find the function R = g(x) that gives the revenue from selling x books. c. Graph the cost and revenue functions; then find the number of books that must be sold for the publisher to break even.Graphing equations Graph the following equations. Use a graphing utility to check your work. a. 2x 3y + 10 = 0 b. y = x2 + 2x 3 c. x2 + 2x + y2 + 4y + 1 = 0 d. x2 2x + y2 8y + 5 = 0Graphing functions Sketch a graph of each function. 30. f(x)={2xifx13xifx1Graphing functions Sketch a graph of each function. 31. g(x)={42xifx1(x1)2ifx1Graphing functions Sketch a graph of each function. 32. h(x)={3x27x+2x2ifx26ifx=233RE34REGraphing absolute value Consider the function f(x)=2(x|x|) Express the function in two pieces without using the absolute value. Then graph the function.Root functions Graph the functions f(x) = x1/3 and g(x) = x1/4. Find all points where the two graphs intersect. For x 1, is f(x) g(x) or is g(x) f(x)?37RE38RETransformation of graphs How is the graph of y=x2+6x3 obtained from the graph of y=x2?Shifting and scaling The graph of f is shown in the figure. Graph the following functions. a. f(x + 1) b. 2f(x 1) c. f(x/2) d. f(2(x 1))Symmetry Identify the symmetry (if any) in the graphs of the following equations. a. y = cos 3x b. y = 3x4 3x2 + 1 c. y2 4x2 = 4Solving equations Solve each equation. 42. 48=6e4kSolving equations Solve each equation. 43. log10x2+3log10x=log1032Solving equations Solve each equation. 44. ln3x+ln(x+2)=0Solving equations Solve each equation. 45. 3ln(5t+4)=12Solving equations Solve each equation. 46. 7y3=50Solving equations Solve each equation. 47. 12sin2=0,02Solving equations Solve each equation. 48. sin22=1/2,/2/2Solving equations Solve each equation. 49. 4cos22=3,/2/250RE51RE52RE53REExistence of inverses Determine the largest intervals on which the following functions have an inverse. 26. g(t) = 2 sin (t/3)Finding inverses Find the inverse function. 55. f(x)=64xFinding inverses Find the inverse function. 56. f(x)=3x4Finding inverses Find the inverse function. 57.f(x)=x24x+5, for x2Finding inverses Find the inverse function. 58. f(x)=4x2x2+10, for x0Finding inverses Find the inverse function. 59.f(x)=3x2+1, for x0Finding inverses Find the inverse function. 60. f(x)=1/x2 for x0Finding inverses Find the inverse function. 61. f(x)=ex2+1, for x0Finding inverses Find the inverse function. 62. f(x)=ln(x2+1) for x0Domain and range of an inverse Find the inverse of f(x)=6xx+2 Graph both f and f1 on the same set of axes. (Hint: The range of f(x) is [0, 6).)Graphing sine and cosine functions Use shifts and scalings to graph the following functions, and identify the amplitude and period. a. f(x) = 4 cos (x/2) b. g() = 2 sin (2/3) c. h() = cos (2( /4))Designing functions Find a trigonometric function f that satisfies each set of properties. Answers are not unique. a. It has a period of 6 with a minimum value of 2 at t = 0 and a maximum value of 2 at t = 3 b. It has a period of 24 with a maximum value of 20 at t = 6 and a minimum value of 10 at t = 18.66REMatching Match each function af with the corresponding graphs AF. a. f(x) = sin x b. f(x) = cos 2x c. f(x) = tan(x/2) d. f(x) = sec x e. f(x) = cot 2x f. f(x) = sin2 x68RE69REEvaluating sine Find the exact value of sin 5871REInverse sines and cosines Evaluate or simplify the following expressions without using a calculator. 36. sin132Inverse sines and cosines Evaluate or simplify the following expressions without using a calculator. 37. cos132Inverse sines and cosines Evaluate or simplify the following expressions without using a calculator. 38. cos1(12)Inverse sines and cosines Evaluate or simplify the following expressions without using a calculator. 39. sin1(1)Inverse sines and cosines Evaluate or simplify the following expressions without using a calculator. 40. cos (cos1 (1))77RE78RERight triangles Given that =sin11213, evaluate cos , tan , cot , sec , and csc .Right-triangle relationships Draw a right triangle to simplify the given expression. Assume x0 and 0/2 80.csc (cot1x)Right-triangle relationships Draw a right triangle to simplify the given expression. Assume x0 and 0/2. 81. sin(cos1(x/4))Right-triangle relationships Draw a right triangle to simplify the given expression. Assume x 0 and 0 /2. 46. tan (sec1 (x/2))83RERight-triangle relationships Draw a right triangle to simplify the given expression. Assume x 0 and 0 /2. 48. csc1 (sec )85REIdentities Prove the following identities. 86. sin1+cos=1cossin87RE88RESum of squared integers Let T(n)=12+22++n2, where n is a positive integer. It can be shown that T(n)=n(n+1)(2n+1)6 a. Make a table of T(n), for n=1,2,,10. b. How would you describe the domain of this function? c. What is the least value of n for which T(n)1000?Sum of integers Let S(n)=1+2++n, where n is a positive integer. It can be shown that S(n)=n(n+1)/2. a. Make a table ofS(n), for n=1,2,,10. b. How would you describe the domain of this function? c. What is the least value of n for which S(n)1000?Little-known fact The shortest day of the year occurs on the winter solstice (near December 21) and the longest day of the year occurs on the summer solstice (near June 21). However, the latest sunrise and the earliest sunset do not occur on the winter solstice, and the earliest sunrise and the latest sunset do not occur on the summer solstice. At latitude 40 north, the latest sunrise occurs on January 4 at 7:25 a.m. (14 days after the solstice), and the earliest sunset occurs on December 7 at 4:37 p.m. (14 days before the solstice). Similarly, the earliest sunrise occurs on July 2 at 4:30 a.m. (14 days after the solstice) and the latest sunset occurs on June 7 at 7:32 p.m. (14 days before the solstice). Using sine functions, devise a function S(t) that gives the time of sunrise t days after January 1 and a function S(t) that gives the time of sunset t days after January 1. Assume s and S are measured in minutes and that s = 0 and S = 0 correspond to 4:00 a.m. Graph the functions. Then graph the length of the day function D(t)=S(t)s(t) and show that the longest and shortest days occur on the solstices.In Example 1, what is the average velocity between t=2 and t=3? Example 1 Average Velocity A rock is launched vertically upward from the ground with a speed of 96 ft/s. Neglecting air resistance, a wall-known formula from physics states that the position of the rock after t seconds s given by the function s(t)=16t2+96t. The position s is measured in feet with s=0 corresponding to the ground. Find the average velocity of the rock between each pair of times. a. t=1s and t=3s b. t=1s and t=2sExplain the difference between average velocity and instantaneous velocity.In Figure 2.5, is mtan at t=2 greater than or less than mtan att=1?Suppose s(t) is the position of an object moving along a line at time t 0. What is the average velocity between the times t = a and t = b?Suppose s(t) is the position of an object moving along a line at time t 0. Describe a process for finding the instantaneous velocity at t = a.Basic Skills 7. Average velocity The function s(t) represents the position of an object at time t moving along a line. Suppose s(2) = 136 and s(3) = 156. Find the average velocity of the object over the interval of time [2, 3].Average velocity The function s(t) represents the position of an object at time t moving along a line. Suppose s(1) = 84 and s(4) = 144. Find the average velocity of the object over the interval of time [1, 4].Average velocity The table gives the position s(t) of an object moving along a line at time t, over a two-second interval. Find the average velocity of the object over the following intervals. a. [0, 2] b. [0, 1.5] c. [0, 1] d. [0, 0.5]Average velocity The graph gives the position s(t) of an object moving along a line at time t, over a 2.5-second interval. Find the average velocity of the object over the following intervals. a. [0.5, 2.5] b. [0.5, 2] c. [0.5, 1.5] d. [0.5, 1]Instantaneous velocity The following table gives the position s(t) of an object moving along a line at time t. Determine the average velocities over the time intervals [1, 1.01], [1, 1.001 ], and [1, 1.0001]. Then make a conjecture about the value of the instantaneous velocity at t = 1.Instantaneous velocity The following table gives the position s(t) of an object moving along a line at time t. Determine the average velocities over the time intervals [2, 2.01], (2, 2.001], and [2, 2.0001]. Then make a conjecture about the value of the instantaneous velocity at t = 2.What is the slope of the secant Line that passes through the points (a, f(a)) and (b, f(b)) on the graph of f?Describe a process for finding the slope of the line tangent to the graph of f at (a, f(a)).Describe the parallels between finding the instantaneous velocity of an object at a point in time and finding the slope of the line tangent to the graph of a function at a point on the graph.Given the functionf(x)=16x2+64x, complete the following. a. Find the slopes of the secant lines that pass though the points (x,f(x)) and (2,f(2)), x=1.5,1.9,1.99,1.999, and 1.9999 (see figure), b. b Make a conjecture about the value of the limit of the slopes of the secant lines that pass through (x,f(x)) and (2,f(2)) as x approaches 2. c. What is the relationship between your answer to part (b) and the slope of the line tangent to the curve at x=2(see figure)?Average velocity The position of an object moving vertically along a line is given by the function s(t) = 16t2 + 128t. Find the average velocity of the object over the following intervals. a. [1, 4] b. [1, 3] c. [1, 2] d. [1, 1 + h], where h 0 is a real numberAverage velocity The position of an object moving vertically along a line is given by the function s(t) = 4.9t2 + 30t + 20. Find the average velocity of the object over the following intervals. a. [0, 3] b. [0, 2] c. [0, 1] d. [0, h], where h 0 is a real numberAverage velocity Consider the position function s(t) = 16t2 + 100t representing the position of an object moving vertically along a line. Sketch a graph of s with the secant line passing through (0.5, s(0.5)) and (2, s(2)). Determine the slope of the secant line and explain its relationship to the moving object.Average velocity Consider the position function s(t) = sin t representing the position of an object moving along a line on the end of a spring. Sketch a graph of s together with a secant line passing through (0, s(0)) and (0.5, s(0.5)). Determine the slope of the secant line and explain its relationship to the moving object.Instantaneous velocity Consider the position function s(t)=16t2+128t (Exercise 13). Complete the following table with the appropriate average velocities. Then make a conjecture about the value of the instantaneous velocity at t=1.Instantaneous velocity Consider the position function s(t)=4.9t2+30t+20 (Exercise 14). Complete the following table with the appropriate average velocities. Then make a conjecture about the value of the instantaneous velocity at t=2Instantaneous velocity Consider the position function s(t) = 16t2 + 100t. Complete the following table with the appropriate average velocities. Then make a conjecture about the value of the instantaneous velocity at t = 3. Time interval Average velocity [2, 3] [2.9, 3] [2.99, 3] [2.999, 3] [2.9999, 3]Instantaneous velocity Consider the position function s(t) = 3 sin t that describes a block bouncing vertically on a spring. Complete the following table with the appropriate average velocities. Then make a conjecture about the value of the instantaneous velocity at t = /2. Time interval Average velocity [/2, ] [/2, /2 + 0.1] [/2, /2 + 0.01] [/2, /2 + 0.001] [/2, /2 + 0.0001]Instantaneous velocity For the following position functions, make a table of average velocities similar to those in Exercises 1920 and make a conjecture about the instantaneous velocity at the indicated time. 21. s(t) = 16t2 + 80t + 60 at t = 3Instantaneous velocity For the following position functions, make a table of average velocities similar to those in Exercises 1920 and make a conjecture about the instantaneous velocity at the indicated time. 22. s(t) = 20 cos t at t = /2Instantaneous velocity For the following position functions, make a table of average velocities similar to those in Exercises 1920 and make a conjecture about the instantaneous velocity at the indicated time. 23. s(t) = 40 sin 2t at t = 0Instantaneous velocity For the following position functions, make a table of average velocities similar to those in Exercises 1920 and make a conjecture about the instantaneous velocity at the indicated time. 24. s(t) = 20/(t +1) at t = 0Slopes of tangent lines For the following functions, make a table of slopes of secant lines and make a conjecture about the slope of the tangent line at the indicated point. 25. f(x) = 2x2 at x = 2Slopes of tangent lines For the following functions, make a table of slopes of secant lines and make a conjecture about the slope of the tangent line at the indicated point. 26. f(x) = 3 cos x at x = /2Slopes of tangent lines For the following functions, make a table of slopes of secant lines and make a conjecture about the slope of the tangent line at the indicated point. 27. f(x) = ex at x = 0Slopes of tangent lines For the following functions, make a table of slopes of secant lines and make a conjecture about the slope of the tangent line at the indicated point. 28. f(x) = x3 x at x = 1Tangent lines with zero slope a. Graph the function f(x) = x2 4x + 3. b. Identify the point (a, f(a)) at which the function has a tangent line with zero slope. c. Confirm your answer to part (b) by making a table of slopes of secant lines to approximate the slope of the tangent line at this point.Tangent lines with zero slope a. Graph the function f(x) = 4 x2. b. Identify the point (a, f(a)) at which the function has a tangent line with zero slope. c. Consider the point (a, f(a)) found in part (b). Is it true that the secant line between (a h, f(a h)) and (a + h, f(a + h)) has slope zero for any value of h 0?Zero velocity A projectile is fired vertically upward and has a position given by s(t) = 16t2 + 128t + 192, for 0 t 9. a. Graph the position function, for 0 t 9. b. From the graph of the position function, identify the time at which the projectile has an instantaneous velocity of zero; call this time t = a. c. Confirm your answer to part (b) by making a table of average velocities to approximate the instantaneous velocity at t = a. d. For what values of t on the interval [0, 9] is the instantaneous velocity positive (the projectile moves upward)? e. For what values of t on the interval [0, 9] is the instantaneous velocity negative (the projectile moves downward)?Impact speed A rock is dropped off the edge of a cliff, and its distance s (in feet) from the top of the cliff after t seconds is s(t) = 16t2. Assume the distance from the top of the cliff to the ground is 96 ft. a. When will the rock strike the ground? b. Make a table of average velocities and approximate the velocity at which the rock strikes the ground.Slope of tangent line Given the function f(x) = 1 cos x and the points A(/2, f(/2)), B(/2 + 0.05, (/2 + 0.05)), C(/2 + 0.5, f(/2 + 0.5)), and D(, f()) (see figure), find the slopes of the secant lines through A and D, A and C. and A and B. Then use your calculations to make a conjecture about the slope of the line tangent to the graph of f at x = /2.In Example 1, suppose we redefine the function at one point so that f(1)=1 Does this change the value of limx1f(x)? Example 1 Finding Limits from a Graph Use the graph of f (Figure 2.7) to determine the following values, if possible. a. f(1) and limx1f(x) b. f(2) and limx2f(x) c. f(3) and limx3f(x)Why is the graph of y=cos(1/x) difficult to plot nearx=0, as suggested by Figure 2.14?Explain the meaning of limxaf(x)=L.True or false: When limxaf(x) exists, it always equals f(a). Explain.Finding limits from a graph Use the graph of h in the figure to find the following values or state that they do not exist. a. h(2) b. limx2h(x) c. h(4) d. limx4h(x) e. limx5h(x)Finding limits from a graph Use the graph of g in the figure to find the following values or state that they do not exist. a. g(0) b. limx0g(x) c. g(1) d. limx1g(x)Finding limits from a graph Use the graph of f in the figure to find the following values or state that they do not exist. a. f(1) b. limx1f(x) c. f(0) d. limx0f(x)Finding limits from a graph Use the graph of f in the figure to find the following values or state that they do not exist. a. f(2) b. limx2f(x) c. limx4f(x) d. limx5f(x)Estimating a limit from tables Let f(x)=x24x2. a. Calculate f(x) for each value of x in the following table. b. Make a conjecture about the value of limx2x24x2.Estimating a limit from tables Let f(x)=x31x1. a. Calculate f(x) for each value of x in the following table. b. Make a conjecture about the value of limx1x31x1.Estimating a limit numerically Let g(t)=t9t3. a. Make two tables, one showing values of g for t = 8.9, 8.99, and 8.999 and one showing values of g for t = 9.1, 9.01, and 9.001. b. Make a conjecture about the value of limt9t9t3.Estimating a limit numerically Let f(x) = (1 + x)1/x. a. Make two tables, one showing values of f for x = 0.01, 0.001, 0.0001, and 0.00001 and one showing values of f for x = 0.01, 0.001, 0.0001, and 0.00001. Round your answers to five digits. b. Estimate the value of limx0(1+x)1/x. c. What mathematical constant does limx0(1+x)1/x appear to equal?Explain the meaning of limxa+f(x)=L.Explain the meaning of limxaf(x)=L.If limxaf(x)=L and limxa+f(x)=M, where L and M are finite real numbers, then how are L and M related if limxaf(x) exists?Let g(x)=x34x8|x2| a. Calculate g(x) for each value of x in the following table b. Make a conjecture about the values of limx2g(x),limx2+g(x),and limx2g(x) state that they do not exist.Use the graph of f in the figure to find the following values or state that they do not exist. If a limit does not exist, explain why. a. f(1) b. limx1f(x) c. limx1+f(x) d. limx1f(x)What are the potential problems of using a graphing utility to estimate limxaf(x)?Finding limits from a graph Use the graph of f in the figure to find the following values or state that they do not exist. If a limit does not exist, explain why. a. f(1) b. limx1f(x) c. limx1+f(x) d. limx1f(x) e. f(3) f. limx3f(x) g. limx3+f(x) h. limx3f(x) i. f(2) j. limx2f(x) k. limx2+f(x) l. limx2f(x)One-sided and two-sided limits Use the graph of g in the figure to find the following values or state that they do not exist. If a limit does not exist, explain why. a. g(2) b. limx2g(x) c. limx2+g(x) d. limx2g(x) e. g(3) f. limx3g(x) g. limx3+g(x) h. g(4) i. limx4g(x)Evaluating limits graphically Sketch a graph of f and use it to make a conjecture about the values of f(a),limxaf(x),limxa+f(x) and limxaf(x) or state that they do not exist. 19. f(x)={x2+1ifx13ifx1;a=1Evaluating limits graphically Sketch a graph of f and use it to make a conjecture about the values of f(a),limxaf(x),limxa+f(x), and limxaf(x) or state that they do not exist. 20. f(x)={3xifx2x1ifx2;a=2Evaluating limits graphically Sketch a graph of f and use it to make a conjecture about the values of f(a),limxaf(x),limxa+f(x), and limxaf(x) or state that they do not exist. 21.f(x)={xifx43ifx=4x+1ifx4;a=4Evaluating limits graphically Sketch a graph of f and use it to make a conjecture about the values of f(a),limxaf(x),limxa+f(x) and limxaf(x) or state that they do not exist. 22.f(x)=|x+2|+2;a=2Evaluating limits graphically Sketch a graph of f and use it to make a conjecture about the values of f(a),limxaf(x),limxa+f(x), and limxaf(x) or state that they do not exist. 23.f(x)=x225x5;a=5Evaluating limits graphically Sketch a graph of f and use it to make a conjecture about the values of f(a),limxaf(x),limxa+f(x), and limxaf(x) or state that they do not exist. 24. f(x)=x100x10;a=100Evaluating limits graphically Sketch a graph of f and use it to make a conjecture about the values of f(a),limxaf(x),limxa+f(x) and limxaf(x) or state that they do not exist. 25. f(x)=x2+x2x1;a=1Evaluating limits graphically Sketch a graph of f and use it to make a conjecture about the values of f(a),limxaf(x),limxa+f(x), and limxaf(x) or state that they do not exist. 26.f(x)=1x4x21;a=1Estimating limits graphically and numerically Use a graph of f to estimate limxaf(x) or to show that the limit does not exist. Evaluate f(x) near x=a to support your conjecture. 27.f(x)=x2ln|x2|;a=2Estimating limits graphically and numerically Use a graph of f to estimate limxaf(x) or to show that the limit does not exist. Evaluate f(x) near x=a to support your conjecture. 28.f(x)=e2x2x1x2;a=0Estimating limits graphically and numerically Use a graph of f estimate limxaf(x)or to show that the limit does not exist. Evaluate f(x) near x = a to support your conjecture. 29.f(x)=1cos(2x2)(x1)2;a=1Estimating limits graphically and numerically Use a graph of f estimate limxaf(x)or to show that the limit does not exist. Evaluate f(x) near x = a to support your conjecture. 30.f(x)=3sinx2cosx+2x;a=0Estimating limits graphically and numerically Use a graph of f estimate limxaf(x)or to show that the limit does not exist. Evaluate f(x) near x = a to support your conjecture. 31.f(x)=sin(x+1)|x+1|;a=1Estimating limits graphically and numerically Use a graph of f estimate limxaf(x)or to show that the limit does not exist. Evaluate f(x) near x = a to support your conjecture. 32.f(x)=x34x2+3x|x3|;a=3Further Explorations 27. Explain why or why not Determine whether the following statements are true and give an explanation or counterexample. a. The value of limx3x29x3 does not exist. b. The value of limxaf(x) is always found by computing f(a). c. The value of limxaf(x) does not exist if f(a) is undefined. d. limx0x=0. e. limx/2cotx=0.The Heaviside function The Heaviside function is used in engineering applications to model flipping a switch. It is defined as H(x)={0ifx01ifx0. a.Sketch a graph of H on the interval [1, 1]. b.Does limx0H(x) exist?Postage rates Assume postage for sending a first-class letter in the United States is 0.47 for the first ounce (up to and including 1 oz) plus 0.21 for each additional ounce (up to and including each additional ounce). aGraph the function p = f(w) that gives the postage p for sending a letter that weighs w ounces, for 0 w 3.5. b.Evaluate limw2.3f(w) c.Does limw3f(w) exist? ExplainCalculator limits Estimate the following limits using graphs or tables. 36. limh0(1+2h)1/h2e2+hCalculator limits Estimate the following limits using graphs or tables. 37.limx/2cot3xcosxCalculator limits Estimate the following limits using graphs or tables. 38.limx118(x31)x31Calculator limits Estimate the following limits using graphs or tables. 39.limx19(2xx4x3)1x3/4Calculator limits Estimate the following limits using graphs or tables. 40.limx06x3xxln16Calculator limits Estimate the following limits using graphs or tables. 41.limh0ln(1+h)hCalculator limits Estimate the following limits using graphs or tables. 42.limh04h1hln(h+2)Strange behavior near x = 0 a. Create a table of values of sin (1/x), for x=2,23,25,27,29, and 211. Describe the pattern of values you observe. b. Why does a graphing utility have difficulty plotting the graph of y = sin (1/x) near x = 0 (see figure)? c. What do you conclude about limx0sin(1/x)?Strange behavior near x = 0 a. Create a table of values of tan (3/x) for x = 12/, 12/(3),12/(5), , 12/(11). Describe the general pattern in the values you observe. b. Use a graphing utility to graph y = tan (3/x). Why do graphing utilities have difficulty plotting the graph near x = 0? c. What do you conclude about limx0tan(3/x)?Sketching graphs of functions Sketch the graph of a function with the given properties. You do not need to find a formula for the function. 45.f(2)=1,limx2f(x)=3Sketching graphs of functions Sketch the graph of a function with the given properties. You do not need to find a formula for the function. 28. f(1) = 0, f(2) = 4, f(3) = 6, limx2f(x)=3, limx2+f(x)=5Sketching graphs of functions Sketch the graph of a function with the given properties. You do not need to find a formula for the function. 29. g(1) = 0, g(2) = 1, g(3) = 2, limx2g(x)=0, limx3g(x)=1, limx3+g(x)=2Sketching graphs of functions Sketch the graph of a function with the given properties. You do not need to find a formula for the function. 30. h(1) = 2, limx1h(x)=0, limx1+h(x)=3, h(1)=limx1h(x)=1, limx1+h(x)=4Sketching graphs of functions Sketch the graph of a function with the given properties. You do not need to find a formula for the function. 31. p(0) = 2, limx0p(x)=0, limx2p(x) does not exist, p(2)=limx2+p(x)=1