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All Textbook Solutions for Calculus: Early Transcendentals (3rd Edition)
Basketball shot A basketball is shot with an initial velocity of v ft/s at an angle of 45 to the floor. The center of the basketball is 8 ft above the floor at a horizontal distance of 18 feet from the center of the basketball hoop when it is released. The height h (in feet) of the center of the basketball after it has traveled a horizontal distance of x feet is modeled by the function h(x)=32x2v2+x+8 (see figure). a. Find the initial velocity v if the center of the basketball passes the center of the hoop that is located 10ft above the floor. Assume the ball does not hit the front of the hoop (otherwise it might not pass though the center of the hoop). The validity of this assumption is explored in the remainder of this exercise. b. During the flight of the basketball, show that the distance s from the center of the basketball to the front of the hoop is s=(x17.25)2+(4x281+x2)2. (Hint: The diameter of the basketball hoop is 18 inches.) c. Determine whether the assumption that the basketball does not hit the front of the hoop in part (a) is valid. Use the fact that the diameter of a womens basketball is about 9.23 inches. (Hint: The ball will hit the front of the hoop if, during its flight, the distance from the center of the ball to the front of the hoop is less than the radius of the basketball.) d. A mens basketball has a diameter of about 9.5 inches. Would this larger ball lead to a different conclusion than in part (c)?Fermats Principle a. Two poles of heights m and n are separated by a horizontal distance d. A rope is stretched from the top of one pole to the ground and then to the top of the other pole. Show that the configuration that requires the least amount of rope occurs when 1 = 2 (see figure). b. Fermats Principle states that when light travels between two points in the same medium (at a constant speed), it travels on the path that minimizes the travel time. Show that when light from a source A reflects off a surface and is received at point B, the angle of incidence equals the angle of reflection, or 1 = 2 (see figure).56EMaking silos A grain silo consists of a cylindrical concrete tower surmounted by a metal hemispherical dome. The metal in the dome costs 1.5 times as much as the concrete (per unit of surface area). If the volume of the silo is 750 m3, what are the dimensions of the silo (radius and height of the cylindrical tower) that minimize the cost of the materials? Assume the silo has no floor and no flat ceiling under the dome.58EMinimizing related functions Complete each of the following parts. a. What value of x minimizes f(x) = (x a1)2 + (x a2)2, for constants a1 and a2? b. What value of x minimizes f(x) = (x a1)2 + (x a2)2 + (x a3)2, for constants a1, a2, and a3? c. Based on the answers to parts (a) and (b), make a conjecture about the values of x that minimize f(x)=(xa1)2+(xa2)2++(xan)2, for a positive integer n and constants a1, a2, , an. Use calculus to verify your conjecture.Searchlight problemnarrow beam A searchlight is 100 m from the nearest point on a straight highway (see figure). As it rotates, the searchlight casts a horizontal beam that intersects the highway in a point. If the light revolves at a rate of /6 rad/s, find the rate at which the beam sweeps along the highway as a function of . For what value of is this rate maximized?Metal rain gutters A rain gutter is made from sheets of metal 9 in wide. The gutters have a 3-in base and two 3-in sides, folded up at angle (see figure). What angle maximizes the cross-sectional area of the gutter?Gliding mammals Many species of small mammals (such as flying squirrels and marsupial gliders) have the ability to walk and glide. Recent research suggests that these animals choose the most energy-efficient means of travel. According to one empirical model, the energy required for a glider with body mass m to walk a horizontal distance D is 8.46 Dm2/3 (where m is measured in grams, D is measured in meters, and energy is measured in microliters of oxygen consumed in respiration). The energy cost of climbing to a height D tan and gliding a distance D at an angle below the horizontal is modeled by 1.36 m D tan (where = 0 represents horizontal flight and 45 represents controlled falling). Therefore, the function S(m,)=8.46m2/31.36mtan gives the energy difference per horizontal meter traveled between walking and gliding: If S 0 for given values of m and , then it is more costly to walk than glide. a. For what glide angles is it more efficient for a 200-gram animal to glide rather that walk? b. Find the threshold function = g(m) that gives the curve along which walking and gliding are equally efficient. Is it an increasing or decreasing function of body mass? c. To make gliding more efficient than walking, do larger gliders have a larger or smaller selection of glide angles than smaller gliders? d. Let = 25, (a typical glide angle). Graph S as a function of m, for 0 m 3000. For what values of m is gliding more efficient? e. For = 25, what value of m (call it m) maximizes S? f. Does m, as defined in part (e), increase or decrease with increasing ? That is, as a glider reduces its glide angle, does its optimal size become larger or smaller? g. Assuming Dumbo is a gliding elephant whose weight is 1 metric ton (106 g), what glide angle would Dumbo use to be more efficient at gliding than walking? (Source: Energetic savings and the body size distribution of gliding mammals, R. Dial, Evolutionary Ecology Research 5, 2003)Watching a Ferris wheel An observer stands 20 m from the bottom of a Ferris wheel on a line that is perpendicular to the face of the wheel, with her eyes at the level of the bottom of the wheel. The wheel revolves at a rate of rad/min, and the observers line of sight with a specific seat on the Ferris wheel makes an angle with the horizontal (see figure). At what time during a full revolution is changing most rapidly?Crease-length problem A rectangular sheet of paper of width a and length b, where 0 a b, is folded by taking one comer of the sheet and placing it at a point P on the opposite long side of the sheet (see figure). The fold is then flattened to form a crease across the sheet. Assuming that the fold is made so that there is no flap extending beyond the original sheet, find the point P that produces the crease of minimum length. What is the length of that crease?Crankshaft A crank of radius r rotates with an angular frequency . It is connected to a piston by a connecting rod of length L (see figure). The acceleration of the piston varies with the position of the crank according to the function a()=2r(cos+rcos2L). For fixed , L, and r, find the values of , with 0 2, for which the acceleration of the piston is a maximum and minimum.Maximum angle Find the value of x that maximizes in the figure.Sum of isosceles distances a. An isosceles triangle has a base of length 4 and two sides of length 22. Let P be a point on the perpendicular bisector of the base. Find the location P that minimizes the sum of the distances between P and the three vertices. b. Assume in part (a) that the height of the isosceles triangle is h 0 and its base has length 4. Show that the location of P that gives a minimum solution is independent of h for h23.Cylinder and cones (Putnam Exam 1938) Right circular cones of height h and radius r are attached to each end of a right circular cylinder of height h and radius r, forming a double-pointed object. For a given surface area A, what are the dimensions r and h that maximize the volume of the object?Slowest shortcut Suppose you are standing in a field near a straight section of railroad tracks just as the locomotive of a train passes the point nearest to you, which is 14 mi away. The train, with length 13 mi, is traveling at 20 mi /hr. If you start running in a straight line across the field, how slowly can you run and still catch the train? In which direction should you run?Rectangles in triangles Find the dimensions and area of the rectangle of maximum area that can be inscribed in the following figures. a. A right triangle with a given hypotenuse length L b. An equilateral triangle with a given side length L c. A right triangle with a given area A d. An arbitrary triangle with a given area A (The result applies to any triangle, but first consider triangles for which all the angles are less than or equal to 90.)71EAnother pen problem A rancher is building a horse pen on the corner of her property using 1000 ft of fencing. Because of the unusual shape of her property, the pen must be built in the shape of a trapezoid (see figure). a. Determine the lengths of the sides that maximize the area of the pen. b. Suppose there is already a fence along the side of the property opposite the side of length y. Find the lengths of the sides that maximize the area of the pen, using 1000 ft of fencing.Minimum-length roads A house is located at each comer of a square with side lengths of 1 mi. What is the length of the shortest road system with straight roads that connects all of the houses by roads (that is, a road system that allows one to drive from any house to any other house)? (Hint: Place two points inside the square at which roads meet.) (Source: Problems for Mathematicians Young and Old, P. Halmos, MAA, 1991)The arbelos An arbelos is the region enclosed by three mutually tangent semicircles; it is the region inside the larger semicircle and outside the two smaller semicircles (see figure). a. Given an arbelos in which the diameter of the largest circle is 1, what positions of point B maximize the area of the arbelos? b. Show that the area of the arbelos is the area of a circle whose diameter is the distance BD in the figure.75ETurning a corner with a pole a. What is the length of the longest pole that can be carried horizontally around a comer at which a 3-ft corridor and a 4 ft corridor meet at right angles? b. What is the length of the longest pole that can be carried horizontally around a comer at which a corridor that is a feet wide and a corridor that is b feet wide meet at right angles? c. What is the length of the longest pole that can be carried horizontally around a corner at which a corridor that is a = 5 ft wide and a corridor that is b = 5 ft wide meet at an angle of 120? d. What is the length of the longest pole that can be carried around a corner at which a corridor that is a feet wide and a corridor that is b feet wide meet at right angles, assuming there is an 8-ft ceiling and that you may tilt the pole at any angle?Tree notch (Putnam Exam 1938, rephrased) A notch is cut in a cylindrical vertical tree trunk. The notch penetrates to the axis of the cylinder and is bounded by two half-planes that intersect on a diameter D of the tree. The angle between the two half-planes is . Prove that for a given tree and fixed angle , the volume of the notch is minimized by taking the bounding planes at equal angles to the horizontal plane that also passes through D.78EA challenging pen problem A farmer uses 200 meters of fencing to build two triangular pens against a barn (see figure); the pens are constructed with three sides and a diagonal dividing fence. What dimensions maximize the area of the pen?80ESketch the graph of a function f that is concave up on an interval containing the point a. Sketch the linear approximation to f at a. Is the graph of the linear approximation above or below the graph of f?
In Example 1, suppose you travel one mile in 75 seconds. What is the average speed given by the linear approximation formula? What is the exact average speed? Explain the discrepancy between the two values.
3QC4QC5QCSketch the graph of a smooth function f and label a point P(a, (f(a)) on the curve. Draw the line that represents the linear approximation to f at P.Suppose you find the linear approximation to a differentiable function at a local maximum of that function. Describe the graph of the linear approximation.How is linear approximation used to approximate the value of a function f near a point at which f and f are easily evaluated?How can linear approximation be used to approximate the change in y = f(x) given a change in x?Suppose f is differentiable on (,),f(1)=2, and f(1)=3. Find the linear approximation to f at x = 1 and use it to approximate f(1.1).Suppose f is differentiable on (,) and the equation of the line tangent to the graph of f at x = 2 is y = 5x 3. Use the linear approximation to f at x = 2 to approximate f(2.01).Linear approximation Estimate f(3.85) given that f(4) = 3 and f(4) = 2.Linear approximation Estimate f(5.1) given that f(5) = 10 and f(5) = 2.Given a function f that is differentiable on its domain, write and explain the relationship between the differentials dx and dy.Does the differential dy represent the change in f or the change in the linear approximation to f? Explain.Suppose f is differentiable on (,), f(5.01)f(5)=0.25. Use linear approximation to estimate the value of f(5).Suppose f is differentiable on (,), f(5.99)=7 and f(6)=7.002. Use linear approximation to estimate the value of f(6).Estimating speed Use the linear approximation given in Example 1 to answer the following questions. 7. If you travel one mile in 59 seconds, what is your approximate average speed? What is your exact speed? EXAMPLE 1 Useful driving math Suppose you are driving along a highway at a nearly constant speed and you record the number of seconds it takes to travel between two consecutive mile markers. If it takes 60 seconds to travel one mile, then your average speed is 1 mi/60 s or 60 mi/hr. Now suppose that you travel one mile in 60 + x seconds; for example, if it lakes 62 seconds, then x = 2, and if it takes 57 seconds, then x = 3. In this case, your average speed over one mile is 1 mi/(60 + x) s. Because there are 3600 s in 1 hr, the function s(x)=360060+x=3600(60+x)1 gives your average speed in mi/hr if you travel one mile in x seconds more or less than 60 seconds. For example, if you travel one mile in 62 seconds, then x = 2 and your average speed is s(2) 58.06 mi/hr. If you travel one mile in 57 seconds, then x = 3 and your average speed is s(3) 63.16 mi/hr. Because you dont want to use a calculator while driving, you need an easy approximation to this function. Use linear approximation to derive such a formula.14EEstimating time Suppose you want to travel D miles at a constant speed of (60 + x) mi/hr, where x could be positive or negative. The time in minutes required to travel D miles is T(x)=60D(60+x)1. 15.Show that the linear approximation to T at the point x = 0 is T(x)L(x)=D(1x60).16EEstimating time Suppose you want to travel D miles at a constant speed of (60 + x) mi/hr, where x could be positive or negative. The time in minutes required to travel D miles is T(x)=60D(60+x)1. 17.Use the result of Exercise 15 to approximate the amount of time it takes to drive 80 miles at 57 mi/hr. What is the exact time required?Estimating time Suppose you want to travel D miles at a constant speed of (60 + x) mi/hr, where x could be positive or negative. The time in minutes required to travel D miles is T(x)=60D(60+x)1. 18.Use the result of Exercise 15 to approximate the amount of time it takes to drive 93 miles at 63 mi/hr. What is the exact time required?Linear approximation Find the linear approximation to the following functions at the given point a. 19.f(x)=4x2+x;a=1Linear approximation Find the linear approximation to the following functions at the given point a. 20.f(x)=x35x+3;a=2Linear approximation Find the linear approximation to the following functions at the given point a. 21.g(t)=2t+9;a=4Linear approximation Find the linear approximation to the following functions at the given point a. 22.h(w)=5w1;a=1Linear approximation Find the linear approximation to the following functions at the given point a. 23.f(x)=e3x6;a=2Linear approximation Find the linear approximation to the following functions at the given point a. 27.f(x)=9(4x+11)2/3;a=4Linear approximation a.Write the equation of the line that represents the linear approximation to the following functions at the given point a. b.Use the linear approximation to estimate the given quantity. c.Compute the percent error in your approximation, 100|approximation exact|/|exact|, where the exact value is given by a calculator. 25.f(x)=12x2;a=2;f(2.1)Linear approximation a.Write the equation of the line that represents the linear approximation to the following functions at the given point a. b.Use the linear approximation to estimate the given quantity. c.Compute the percent error in your approximation, 100|approximation exact|/|exact|, where the exact value is given by a calculator. 26.f(x)=sinx;a=/4;f(0.75)Linear approximation a.Write the equation of the line that represents the linear approximation to the following functions at the given point a. b.Use the linear approximation to estimate the given quantity. c.Compute the percent error in your approximation, 100|approximation exact|/|exact|, where the exact value is given by a calculator. 27.f(x)=ln(1+x);a=0;f(0.9)Linear approximation a.Write the equation of the line that represents the linear approximation to the following functions at the given point a. b.Use the linear approximation to estimate the given quantity. c.Compute the percent error in your approximation, 100|approximation exact|/|exact|, where the exact value is given by a calculator. 28.f(x)=x/(x+1);a=1;f(1.1)Linear approximation a.Write the equation of the line that represents the linear approximation to the following functions at the given point a. b.Use the linear approximation to estimate the given quantity. c.Compute the percent error in your approximation, 100|approximation exact|/|exact|, where the exact value is given by a calculator. 29.f(x)=cosx;a=0;f(0.01)Linear approximation a.Write the equation of the line that represents the linear approximation to the following functions at the given point a. b.Use the linear approximation to estimate the given quantity. c.Compute the percent error in your approximation, 100|approximation exact|/|exact |, where the exact value is given by a calculator. 30.f(x)=ex;a=0;f(0.05)Linear approximation a.Write the equation of the line that represents the linear approximation to the following functions at the given point a. b.Use the linear approximation to estimate the given quantity. c.Compute the percent error in your approximation, 100|approximation exact|/|exact |, where the exact value is given by a calculator. 31.f(x)=(8+x)1/3;a=0;f(0.1)Linear approximation a.Write the equation of the line that represents the linear approximation to the following functions at the given point a. b.Use the linear approximation to estimate the given quantity. c.Compute the percent error in your approximation, 100|approximation exact|/|exact |, where the exact value is given by a calculator. 32.f(x)=x4;a=81;f(85)Linear approximation a.Write the equation of the line that represents the linear approximation to the following functions at the given point a. b.Use the linear approximation to estimate the given quantity. c.Compute the percent error in your approximation, 100|approximation exact|/|exact |, where the exact value is given by a calculator. 33.f(x)=1/(x+1);a=0;1/1.1Linear approximation a.Write the equation of the line that represents the linear approximation to the following functions at the given point a. b.Use the linear approximation to estimate the given quantity. c.Compute the percent error in your approximation, 100|approximation exact|/|exact|, where the exact value is given by a calculator. 34.f(x)=cosx;a=/4;cos0.8Linear approximation a.Write the equation of the line that represents the linear approximation to the following functions at the given point a. b.Use the linear approximation to estimate the given quantity. c.Compute the percent error in your approximation, 100|approximation exact|/|exact |, where the exact value is given by a calculator. 35.f(x)=ex;a=0;e0.03Linear approximation a.Write the equation of the line that represents the linear approximation to the following functions at the given point a. b.Use the linear approximation to estimate the given quantity. c.Compute the percent error in your approximation, 100|approximation exact|/|exact |, where the exact value is given by a calculator. 36.f(x)=tanx;a=0;tan3Estimations with linear approximation Use linear approximations to estimate the following quantities. Choose a value of a that produces a small error. 21. 1/203Estimations with linear approximation Use linear approximations to estimate the following quantities. Choose a value of a to produce a small error. 38.tan(2)Estimations with linear approximation Use linear approximations to estimate the following quantities. Choose a value of a that produces a small error. 23. 146Estimations with linear approximation Use linear approximations to estimate the following quantities. Choose a value of a that produces a small error. 24. 653Estimations with linear approximation Use linear approximations to estimate the following quantities. Choose a value of a that produces a small error. 25. ln (1.05)Estimations with linear approximation Use linear approximations to estimate the following quantities. Choose a value of a that produces a small error. 26.5/29Estimations with linear approximation Use linear approximations to estimate the following quantities. Choose a value of a that produces a small error. 27. e0.06Estimations with linear approximation Use linear approximations to estimate the following quantities. Choose a value of a that produces a small error. 28. 1/119Estimations with linear approximation Use linear approximations to estimate the following quantities. Choose a value of a that produces a small error. 29. 1/510346ELinear approximation and concavity Carry out the following steps for the given functions f and points a. a. Find the linear approximation L to the function f at the point a. b. Graph f and L on the same set of axes. c. Based on the graphs in part (a), state whether linear approximations to f near a are underestimates or overestimates. d. Compute f(a) to confirm your conclusion in part (c). 31. f(x)=2x,a=1Linear approximation and concavity Carry out the following steps for the given functions f and points a. a. Find the linear approximation L to the function f at the point a. b. Graph f and L on the same set of axes. c. Based on the graphs in part (a), state whether linear approximations to f near a are underestimates or overestimates. d. Compute f(a) to confirm your conclusion in part (c). 32. f(x) = 5 x2, a = 249ELinear approximation and concavity Carry out the following steps for the given functions f and points a. a. Find the linear approximation L to the function f at the point a. b. Graph f and L on the same set of axes. c. Based on the graphs in part (a), state whether linear approximations to f near a are underestimates or overestimates. d. Compute f(a) to confirm your conclusion in part (c). 34. f(x)=2cosx,a=451EIdeal Gas Law The pressure P, temperature T, and volume V of an ideal gas are related by PV = nRT, where n is the number of moles of the gas and R is the universal gas constant. For the purposes of this exercise, let nR = 1; therefore P = T/V. a. Suppose that the volume is held constant and the temperature increases by T = 0.05. What is the approximate change in the pressure? Does the pressure increase or decrease? b. Suppose that the temperature is held constant and the volume increases by V = 0.1. What is the approximate change in the pressure? Does the pressure increase or decrease? c. Suppose that the pressure is held constant and the volume increases by V = 0.1. What is the approximate change in the temperature? Does the temperature increase or decrease?Explain why or why not Determine whether the following statements are true and give an explanation or counterexample. a. The linear approximation to f(x) = x2 at x = 0 is L(x) = 0. b. Linear approximation at x = 0 provides a good approximation to f(x) = |x|. c. If f(x) = mx + b, then the linear approximation to f at any point is L(x) = f(x). d. When linear approximation is used to estimate values of ln x near x = e, the approximations are overestimates of the true values.54EApproximating changes 35. Approximate the change in the volume of a sphere when its radius changes from r = 5 ft to r = 5.1 ft (V(r)=43r3).Approximating changes 36. Approximate the change in the atmospheric pressure when the altitude increases from z = 2 km to z = 2.01 km (P(z) = 1000 ez/10).Approximating changes 37. Approximate the change in the volume of a right circular cylinder of fixed radius r = 20 cm when its height decreases from h = 12 cm to h = 11.9 cm (V(h) = r2 h).Approximating changes 38. Approximate the change in the volume of a right circular cone of fixed height h = 4 m when its radius increases from r = 3 m to r = 3.05 m (V(r) = r2 h/3).Approximating changes 39. Approximate the change in the lateral surface area (excluding the area of the base) of a right circular cone with fixed height h = 6 m when its radius decreases from r = 10 m to r = 9.9 m (S=rr2+h2).Approximating changes 40. Approximate the change in the magnitude of the electrostatic force between two charges when the distance between them increases from r = 20 m to r = 21 m (F(r) = 0.01/r2).Differentials Consider the following functions and express the relationship between a small change in x and the corresponding change in y in the form dy = f(x) dx. 41. f(x) = 2x + 1Differentials Consider the following functions and express the relationship between a small change in x and the corresponding change in y in the form dy = f(x) dx. 42. f(x) = sin2 xDifferentials Consider the following functions and express the relationship between a small change in x and the corresponding change in y in the form dy = f(x) dx. 43. f(x) = 1/x3Differentials Consider the following functions and express the relationship between a small change in x and the corresponding change in y in the form dy = f(x) dx. 44. f(x) = e2xDifferentials Consider the following functions and express the relationship between a small change in x and the corresponding change in y in the form dy = f(x) dx. 45. f(x) = 2 a cos x, a constantDifferentials Consider the following functions and express the relationship between a small change in x and the corresponding change in y in the form dy = f(x) dx. 46. f(x) = (4 + x)/(4 x)Differentials Consider the following functions and express the relationship between a small change in x and the corresponding change in y in the form dy = f(x) dx. 47. f(x) = 3x3 4xDifferentials Consider the following functions and express the relationship between a small change in x and the corresponding change in y in the form dy = f(x) dx. 48. f(x) = sin1 xDifferentials Consider the following functions and express the relationship between a small change in x and the corresponding change in y in the form dy = f(x) dx. 49. f(x) = tan xDifferentials Consider the following functions and express the relationship between a small change in x and the corresponding change in y in the form dy = f(x) dx. 50. f(x) = ln (1 x)71EErrors in approximations Suppose f(x) = 1/(1 + x) is to be approximated near x = 0. Find the linear approximation to f at 0. Then complete the following table showing the errors in various approximations. Use a calculator to obtain the exact values. The percent error is 100|approximation exact |/| exact |. Comment on the behavior of the errors as x approaches 0.73EWhich of the following functions lead to an indeterminate form as x0:f(x)=x2/(x+2),g(x)=(tan3x)/x,orh(x)=(1cosx)/x2?2QCWhat is the form of the limit limx/2(x/2)(tanx)? Write it in the form 0/0.Explain why a limit of the form 0 is not an indeterminate form.Before proceeding, use your intuition and rank these classes of functions in order of their growth rates.Compare the growth rates of f(x)=x2 and g(x)=x3 as x. Compare the growth rates of f(x)=x2 and g(x)=10x2 as x.Explain with examples what is meant by the indeterminate form 0/0.Why are special methods, such as lHpitals Rule, needed to evaluate indeterminate forms (as opposed to substitution)?Explain the steps used to apply lHpitals Rule to a limit of the form 0/0.Give examples of each of the following. a.A limit with the indeterminate form 0/0 that equals 3. b.A limit with the indeterminate form 0/0 that equals 4.Give examples of each of the following. a. A limit with the indeterminate form 0 that equals 1. b. A limit with the indeterminate form 0 that equals 2.Which of the following limits can be evaluated without l'Hpitals Rule? Evaluate each limit. a.limx0sinxx3+2x+1 b.limx0sinxx3+2xExplain how to convert a limit of the form 0 to a limit of the form 0/0 or /.Give an example of a limit of the form / as x 0.9EEvaluate limx2x33x2+2xx2 using lHpitals Rule and then check your work by evaluating the limit using an appropriate Chapter 2 method.Explain why the form 1 is indeterminate and cannot be evaluated by substitution. Explain how the competing functions behave.Give the two-step method for attacking an indeterminate limit of the form limxaf(x)g(x).In terms of limits, what does it mean for f to grow faster than g as x ?In terms of limits, what does it mean for the rates of growth of f and g to be comparable as x ?Rank the functions x3, ln x, xx, and 2x in order of increasing growth rates as x .Rank the functions x100, ln x10, xx, and 10x in order of increasing growth rates as x .0/0 form Evaluate the following limits using lHpitals Rule. 13. limx2x22x86x+x20/0 form Evaluate the following limits using lHpitals Rule. 14. limx1x4+x3+2x+2x+1Limits Evaluate the following limits. Use lHpitals Rule when it is convenient and applicable. 19.limx1x2+2xx+3Limits Evaluate the following limits. Use lHpitals Rule when it is convenient and applicable. 20.limx0ex120x+50/0 form Evaluate the following limits using lHpitals Rule. 15. limx1lnx4xx230/0 form Evaluate the following limits using l'Hpitals Rule. 16. limx0ex1x2+3x23E/ form Evaluate the following limits. 38. limx4x32x2+6x3+40/0 form Evaluate the following limits using lHpitals Rule. 17. limxelnx1xe0/0 form Evaluate the following limits using lHpitals Rule. 18. limx14tan1xx1Limits Evaluate the following limits. Use lHpitals Rule when it is convenient and applicable. 27.limx0+1lnx1+lnxLimits Evaluate the following limits. Use lHpitals Rule when it is convenient and applicable. 28.limx0+x3xxx29E0/0 form Evaluate the following limits using l'Hpitals Rule. 20. limx2xsinx+x242x20/0 form Evaluate the following limits using lHpitals Rule. 21. limutanucotuu/40/0 form Evaluate the following limits using lHpitals Rule. 22. limz0tan4ztan7z0/0 form Evaluate the following limits. 23. limx01cos3x8x20/0 form Evaluate the following limits. 24. limx0sin23xx20/0 form Evaluate the following limits. 25. limxcosx+1(x)20/0 form Evaluate the following limits. 26. limx0=exx15x2/ form Evaluate the following limits. 39. limx/2tanx3/(2x)38E0/0 form Evaluate the following limits. 27. limx0exsinx1x4+8x3+12x20/0 form Evaluate the following limits. 28. limx0sinxx7x341E0/0 form Evaluate the following limits. 30. limxtan1x/21/x0/0 form Evaluate the following limits. 31. limx1x3x25x3x4+2x3x24x20/0 form Evaluate the following limits. 32. limx1xn1x1, n is a positive integer/ form Evaluate the following limits. 41. limxln(3x+5)ln(7x+3)+1/ form Evaluate the following limits. 42. limxln(3x+5ex)ln(7x+3e2x)0/0 form Evaluate the following limits. 33. limx3v1v25v30/0 form Evaluate the following limits. 34. limy2y2+y68y2y0/0 form Evaluate the following limits. 35. limx2x24x+4sin2(x)0/0 form Evaluate the following limits. 36. limx23x+232x251EMiscellaneous limits by any means Use analytical methods to evaluate the following limits. 91. limx1(1x11x1)0 form Evaluate the following limits. 45. limx0xcscx0 form Evaluate the following limits. 46. limx1(1x)tan(x2)0 form Evaluate the following limits. 47. limx0csc6xsin7x0 form Evaluate the following limits. 48. limx(csc(1/x)(e1/x1))0 form Evaluate the following limits. 49. limx/2(2x)secx0 form Evaluate the following limits. 50. limx0+(sinx)1xxLimits Evaluate the following limits. Use lHpitals Rule when it is convenient and applicable. 59.limx1+(1x11lnx)Limits Evaluate the following limits. Use lHpitals Rule when it is convenient and applicable. 60.limx1(xx1xlnx)form Evaluate the following limits. 51. limx0(cotx1x)form Evaluate the following limits. 53. lim/2(tansec)Limits Evaluate the following limits. Use lHpitals Rule when it is convenient and applicable. 63.limx(x2x2+16x2)64EMiscellaneous limits by any means Use analytical methods to evaluate the following limits. 87. limx(x2x4)Miscellaneous limits by any means Use analytical methods to evaluate the following limits. 86. limxx2ln(cos1x)Miscellaneous limits by any means Use analytical methods to evaluate the following limits. 95. limn1+2++nn2Miscellaneous limits by any means Use analytical methods to evaluate the following limits. 92. limx0+x1/lnxMiscellaneous limits by any means Use analytical methods to evaluate the following limits. 93. limxlog2xlog3xMiscellaneous limits by any means Use analytical methods to evaluate the following limits. 94. limx(log2xlog3x)Miscellaneous limits by any means Use analytical methods to evaluate the following limits. 85. limx65x+2521/x1/6Miscellaneous limits by any means Use analytical methods to evaluate the following limits. 88. limx/2(2x)tanxMiscellaneous limits by any means Use analytical methods to evaluate the following limits. 89. limxx3(1xsin1x)Miscellaneous limits by any means Use analytical methods to evaluate the following limits. 90. limx(x2e1/xx2x)Limits Evaluate the following limits. Use lHpitals Rule when it is convenient and applicable. 75.limx0+x2xLimits Evaluate the following limits. Use lHpitals Rule when it is convenient and applicable. 76.limx0(1+4x)3/xLimits Evaluate the following limits. Use lHpitals Rule when it is convenient and applicable. 77.lim/2(tan)cosLimits Evaluate the following limits. Use lHpitals Rule when it is convenient and applicable. 78. lim0+(sin)tanLimits Evaluate the following limits. Use lHpitals Rule when it is convenient and applicable. 79. lim0+(1+x)cotxLimits Evaluate the following limits. Use lHpitals Rule when it is convenient and applicable. 80. lim0(1+ax)x, for a constant aLimits Evaluate the following limits. Use lHpitals Rule when it is convenient and applicable. 81. lim0(eax+x)1/x, for a constant aLimits Evaluate the following limits. Use lHpitals Rule when it is convenient and applicable. 82. limz(1+10z2)z2Limits Evaluate the following limits. Use lHpitals Rule when it is convenient and applicable. 83. limx0(x+cosx)1/xAn optics limit The theory of interference of coherent oscillators requires the limit lim2msin2(N/2)sin2(/2), where N is a positive integer and m is any integer. Show that the value of this limit is N2.Compound interest Suppose you make a deposit of P into a savings account that earns interest at a rate of 100r % per year. a. Show that if interest is compounded once per year, then the balance after t years is B(t) = P(1 + r)t. b. If interest is compounded m times per year, then the balance after t years is B(t) = P(1 + r/m)mt. For example, m = 12 corresponds to monthly compounding, and the interest rate for each month is r/12. In the limit m , the compounding is said to be continuous. Show that with continuous compounding, the balance after t years is B(t) = Pert.Two methods Evaluate the following limits in two different ways: Use the methods of Chapter 2 and use lHpitals Rule. 83. limx2x3x2+15x3+2xTwo methods Evaluate the following limits in two different ways. Use the methods of Chapter 2 and use lHpitals Rule. 87. limx0e2x+4ex5e2x1More limits Evaluate the following limits. 88. limx0(sinxx)1/x2More limits Evaluate the following limits. 89. limx1xlnxx+1xln2xMore limits Evaluate the following limits. 90. limx1xlnx+lnx2x+2x2ln3x88-94. More limits Evaluate the following limits.
91.
More limits Evaluate the following limits. 92. limnn2ln(nsin1n)More limits Evaluate the following limits. 93. limx0axbxx, for positive constants a and bMore limits Evaluate the following limits. 94. limx0(1+ax)b/x, for positive constants a and bComparing growth rates Use limit methods to determine which of the two given functions grows faster or state that they have comparable growth rates. 69. x10; e0.01xComparing growth rates Use limit methods to determine which of the two given functions grows faster or state that they have comparable growth rates. 70. x2 ln x; ln2 xComparing growth rates Use limit methods to determine which of the two given functions grows faster or state that they have comparable growth rates. 71. ln x20; ln xComparing growth rates Use limit methods to determine which of the two given functions grows faster or state that they have comparable growth rates. 72. ln x; ln (ln x)Comparing growth rates Use limit methods to determine which of the two given functions grows faster or state that they have comparable growth rates. 73. 100x; xxComparing growth rates Use limit methods to determine which of the two given functions grows faster or state that they have comparable growth rates. 74. x2 ln x; x3Comparing growth rates Use limit methods to determine which of the two given functions grows faster or state that they have comparable growth rates. 75. x20; 1.00001x102EComparing growth rates Use limit methods to determine which of the two given functions grows faster or state that they have comparable growth rates. 79. ex2; e10x104EExplain why or why not Determine whether the following statements are true and give an explanation or counterexample. a. By lHpitals Rule, limx2x2x21=limx212x=14. b. limx0(xsinx)=limx0f(x)g(x)=limx0f(x)limx0g(x)=(limx01)(limx0cosx)=1. c. limx0x1/x is an indeterminate form. d. The number 1 raised to any fixed power is 1. Therefore, because (1 + x) 1 as x 0, (1 + x)1/x 1 as x 0. e. The functions ln x100 and ln x have comparable growth rates as x . f. The function ex grows faster than 2x as x .Graphing functions Make a complete graph of the following functions using the graphing guidelines outlined in Section 4.4. 106. g(x)=x2lnxGraphing functions Make a complete graph of the following functions using the graphing guidelines outlined in Section 4.4. 107. f(x)=xlnxGraphing functions Make a complete graph of the following functions using the graphing guidelines outlined in Section 4.4. 108. f(x)=lnxx2Graphing functions Make a complete graph of the following functions using the graphing guidelines outlined in Section 4.4. 109. p(x)=xex2/2Algorithm complexity The complexity of a computer algorithm is the number of operations or steps the algorithm needs to complete its task assuming there are n pieces of input (for example, the number of steps needed to put n numbers in ascending order). Four algorithms for doing the same task have complexities of A: n3/2, B: n log2 n, C: n(log2 n)2, and D: nlog2n. Rank the algorithms in order of increasing efficiency for large values of n. Graph the complexities as they vary with n and comment on your observations.LHpital loops Consider the limit limx0ax+bcx+d, where a, b, c, and d are positive real numbers. Show that lHpitals Rule fails for this limit. Find the limit using another method.General result Let a and b be positive real numbers. Evaluate limx(axa2x2bx) in terms of a and b.Exponential functions and powers Show that any exponential function bx, for b 1, grows faster than xp, for p 0.Exponentials with different bases Show that f(x) = ax grows faster than g(x) = bx as x if 1 b a.Logs with different bases Show that f(x) = loga x and g(x) = logb x, where a l and b 1. grow at a comparable rate as x .Factorial growth rate The factorial function is defined for positive integers as n!=n(n1)(n2)321. For example, 5! = 5 4 3 2 1 = 120. A valuable result that gives good approximations to n! for large values of n is Stirlings formula, n!2nnnen. Use this formula and a calculator to determine where the factorial function appears in the ranking of growth rates given in Theorem 4.15. (See the Guided Project Stirlings Formula.)A geometric limit Let f() be the area of the triangle ABP (see figure) and let g() be the area of the region between the chord PB and the arc PB. Evaluate lim0g()/f().118EExponentials vs. super exponentials Show that xx grows faster than bx as x , for b 1.Exponential growth rates a. For what values of b 0 does bx grow faster than as ex as x ? b. Compare the growth rates of ex and eax as x , for a 0.Verity that setting y = 0 in the equation yf(xn)=f(xn)(xxn) and solving for x gives the formula for Newtons method.What happens if you apply Newtons method to the function f(x)=x?Give a geometric explanation of Newtons method.2EA graph of f and the lines tangent to f at x = 1, 2, and 3 are given. If x0 = 3, find the values of x1, x2, and x3 that are obtained by applying Newtons method.A graph of f and the lines tangent to f at x = 3, 2, and 3 are given. If x0 = 3, find the values of x1, x2, and x3 that are obtained by applying Newtons method.Let f(x)=2x36x2+4x. Use Newtons method to find x1 given that x0 = 1.4. Use the graph of f (see figure) and an appropriate tangent line to illustrate how x1 is obtained from x0.The function f(x)=4xx2+4 is differentiable and has a local maximum at x = 2, where f(2)=0 (see figure). a.Use a graphical explanation to show that Newtons method applied to f(x)=0 fails to produce a value x1 for the initial approximation x0 = 2. b.Use the formula for Newtons method to show that Newtons method fails to produce a value x1 with the initial approximation x0 2.How do you decide when to terminate Newtons method?Give the formula for Newtons method for the function f(x) = x2 5Formulating Newtons method Write the formula for Newtons method and use the given initial approximation to compute the approximations x1 and x2. 5. f(x) = x2 6; x0 = 3Formulating Newtons method Write the formula for Newtons method and use the given initial approximation to compute the approximations x1 and x2. 6. f(x) = x2 2x 3; x0 = 2Formulating Newtons method Write the formula for Newtons method and use the given initial approximation to compute the approximations x1 and x2. 7. f(x) = ex x; x0 = ln 2Formulating Newtons method Write the formula for Newtons method and use the given initial approximation to compute the approximations x1 and x2. 8. f(x) = x3 2; x0 = 2Finding roots with Newtons method For the given function f and initial approximation x0 use Newtons method to approximate a root of f. Stop calculating approximations when two successive approximations agree to five digits to the right of the decimal point after rounding. Show your work by making a table similar to that in Example 1. 13. f(x)=x210;x0=3Finding roots with Newtons method For the given function f and initial approximation x0 use Newtons method to approximate a root of f. Stop calculating approximations when two successive approximations agree to five digits to the right of the decimal point after rounding. Show your work by making a table similar to that in Example 1. 14. f(x)=x3+x2+1;x0=1.5Finding roots with Newtons method For the given function f and initial approximation x0 use Newtons method to approximate a root of f. Stop calculating approximations when two successive approximations agree to five digits to the right of the decimal point after rounding. Show your work by making a table similar to that in Example 1. 15. f(x)=sinx+x1;x0=0.5Finding roots with Newtons method For the given function f and initial approximation x0 use Newtons method to approximate a root of f. Stop calculating approximations when two successive approximations agree to five digits to the right of the decimal point after rounding. Show your work by making a table similar to that in Example 1. 16. f(x)=ex+x5;x0=1.6Finding roots with Newtons method For the given function f and initial approximation x0 use Newtons method to approximate a root of f. Stop calculating approximations when two successive approximations agree to five digits to the right of the decimal point after rounding. Show your work by making a table similar to that in Example 1. 17. f(x)=tanx2x;x0=1.2Finding roots with Newtons method For the given function f and initial approximation x0 use Newtons method to approximate a root of f. Stop calculating approximations when two successive approximations agree to five digits to the right of the decimal point after rounding. Show your work by making a table similar to that in Example 1. 18. f(x)=xln(x+1)1;x0=1.7Finding roots with Newtons method For the given function f and initial approximation x0 use Newtons method to approximate a root of f. Stop calculating approximations when two successive approximations agree to five digits to the right of the decimal point after rounding. Show your work by making a table similar to that in Example 1. 19. f(x)=cos1xx;x0=0.75More root finding Find all the roots of the following functions. Use preliminary analysis and graphing to determine good initial approximations. 32. f(x)=cosxx7More root finding Find all the roots of the following functions. Use preliminary analysis and graphing to determine good initial approximations. 33. f(x) = cos 2x x2 + 2xMore root finding Find all the roots of the following functions. Use preliminary analysis and graphing to determine good initial approximations. 34. f(x)=x6secx on [0, 8]More root finding Find all the roots of the following functions. Use preliminary analysis and graphing to determine good initial approximations. 35. f(x)=exx+45More root finding Find all the roots of the following functions. Use preliminary analysis and graphing to determine good initial approximations. 36. f(x)=x55x34120More root finding Find all the roots of the following functions. Use preliminary analysis and graphing to determine good initial approximations. 37. f(x) = ln x x2 + 3x 1More root finding Find all the roots of the following functions. Use preliminary analysis and graphing to determine good initial approximations. 38. f(x) = x2(x 100) + 1Finding intersection points Use Newtons method to approximate all the intersection points of the following pairs of curves. Some preliminary graphing or analysis may help in choosing good initial approximations. 15. y = sin x and y=x228EFinding intersection points Use Newtons method to approximate all the intersection points of the following pairs of curves. Some preliminary graphing or analysis may help in choosing good initial approximations. 17. y=1x and y = 4 x230EFinding intersection points Use Newtons method to approximate all the intersection points of the following pairs of curves. Some preliminary graphing or analysis may help in choosing good initial approximations. 19. y=4x and y = x2 + 132E33E34E35EInvestment problem A one-time investment of 2500 is deposited in a 5-year savings account paying a fixed annual interest rate r, with monthly compounding. The amount of money in the account after 5 years is A(r)=2500(a+r12)60. a.Use Newtons method to find the value of r f the goal is to have 3200 in the account after 5 years. b.Verify your answer to part (a) algebraically.Applications 45. A damped oscillator The displacement of a particular object as it bounces vertically up and down on a spring is given by y(t) = 2.5et cos 2t, where the initial displacement is y(0) = 2.5 and y = 0 corresponds to the rest position (see figure). a. Find the time at which the object first passes the rest position, y = 0. b. Find the time and the displacement when the object reaches its lowest point. c. Find the time at which the object passes the rest position for the second time. d. Find the time and the displacement when the object reaches its high point for the second time.The sinc function The sinc function, sinc(x)=sinxx for x 0, sinc (0) = 1. appears frequently in signal-processing applications. a. Graph the sinc function on [2, 2]. b. Locate the first local minimum and the first local maximum of sinc (x), for r 0.Estimating roots The values of various roots can be approximated using Newtons method. For example, to approximate the value of 103. we let x=103 and cube both sides of the equation to obtain x3 = 10. or x3 10 = 0. Therefore, 103 is a root of p(x) = x3 10, which we can approximate by applying Newtons method. Approximate each value of r by first finding a polynomial with integer coefficients that has a root r. Use an appropriate value of xo and stop calculating approximations when two successive approximations agree to five digits to the right of the decimal point after rounding. 39. r=71/440E41E42E43ENewtons method and curve sketching Use Newtons method to find approximate answers to the following questions. 22. Where are all the local extrema of f(x) = 3x4 + 8x3 + 12x2 + 48x located?Newtons method and curve sketching Use Newtons method to find approximate answers to the following questions. 23. Where are the inflection points of f(x)=95x5152x4+73x3+30x2+1 located?46E47EFixed points An important question about many functions concerns the existence and location of fixed points. A fixed point of f is a value of x that satisfies the equation f(x)=x;it corresponds to a point at which the graph of f intersects the line y = x. Find all the fixed points of the following functions. Use preliminary analysis and graphing to determine good initial approximations. 48. f(x)=x23x2+x+1Fixed points An important question about many functions concerns the existence and location of fixed points. A fixed point of f is a value of x that satisfies the equation f(x)=x;it corresponds to a point at which the graph of f intersects the line y = x. Find all the fixed points of the following functions. Use preliminary analysis and graphing to determine good initial approximations. 49. f(x)=cosxFixed points An important question about many functions concerns the existence and location of fixed points. A fixed point of f is a value of x that satisfies the equation f(x) = x; it corresponds to a point at which the graph off intersects the line y = x. Find all the fixed points of the following functions. Use preliminary analysis and graphing to determine good initial approximations. 30. f(x)=tanx2 on (, )Fixed points An important question about many functions concerns the existence and location of fixed points. A fixed point of f is a value of x that satisfies the equation f(x) = x; it corresponds to a point at which the graph off intersects the line y = x. Find all the fixed points of the following functions. Use preliminary analysis and graphing to determine good initial approximations. 31. f(x) = 2x cos x on [0, 2]Pitfalls of Newtons method Let f(x)=x1+x2, which has just one root, r = 0. 52. Use the initial approximation x0=1/3 to complete the following steps. a.Use Newtons method to find the exact values of x1 and x2. b.State the values of x3,x4,x5, without performing any additional calculations. c.Use a graph of f to illustrate why Newtons method produces the values found in part (b). d.Why does Newtons method fail to approximate the root r = 0 if x0=1/3?53EApproximating square roots Let a 0 be given and suppose we want to approximate a using Newtons method. a. Explain why the square root problem is equivalent to finding the positive root of f(x) = x2 a. b. Show that Newtons method applied to this function takes the form (sometimes called the Babylonian method) xn+1=12(xn+axn),forn=0,1,2,. c. How would you choose initial approximations to approximate 13 and 73? d. Approximate 13 and 73 with at least 10 significant digits.55E56EAn eigenvalue problem A certain kind of differential equation (see Chapter 9) leads to the root-finding problem tan =, where the roots are called eigenvalues. Find the first three positive eigenvalues of this problem.58E59E60EVerify by differentiation that x4 is an antiderivative of 4x3.Find the family of antiderivatives for each of f(x)=ex,g(x)=4x3, and h(x) = sec2 x.Use differentiation to verify result 6 in Table 4.9: cscxcotxdx=cscx+C.4QCPosition is an antiderivative of velocity. But there are infinitely many antiderivatives that differ by a constant. Explain how two objects can have the same velocity function but two afferent position functions.Fill in the blanks with either of the words the derivative or an antiderivative: If F(x) = f(x), then f is _____ of F and F is _____ of f.Describe the set of antiderivatives of f(x) = 0.