Tangent Line Approximations
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1 60_009.qd //0 :8 PM Page SECTION.9 Dierentials Section.9 EXPLORATION Tangent Line Approimation Use a graphing utilit to graph. In the same viewing window, graph the tangent line to the graph o at the point,. Zoom in twice on the point o tangenc. Does our graphing utilit distinguish between the two graphs? Use the trace eature to compare the two graphs. As the -values get closer to, what can ou sa about the -values? Dierentials Understand the concept o a tangent line approimation. Compare the value o the dierential, d, with the actual change in,. Estimate a propagated error using a dierential. Find the dierential o a unction using dierentiation ormulas. Tangent Line Approimations Newton s Method (Section.8) is an eample o the use o a tangent line to a graph to approimate the graph. In this section, ou will stud other situations in which the graph o a unction can be approimated b a straight line. To begin, consider a unction that is dierentiable at c. The equation or the tangent line at the point c, c is given b c c c c c c and is called the tangent line approimation (or linear approimation) o at c. Because c is a constant, is a linear unction o. Moreover, b restricting the values o to be suicientl close to c, the values o can be used as approimations (to an desired accurac) o the values o the unction. In other words, as c, the limit o is c. EXAMPLE Using a Tangent Line Approimation Find the tangent line approimation o Tangent line sin at the point 0,. Then use a table to compare the -values o the linear unction with those o on an open interval containing 0. π () = + sin π π The tangent line approimation o at the point 0, Figure.6 Solution The derivative o is cos. First derivative So, the equation o the tangent line to the graph o at the point 0, is Tangent line approimation The table compares the values o given b this linear approimation with the values o near 0. Notice that the closer is to 0, the better the approimation is. This conclusion is reinorced b the graph shown in Figure.6. sin NOTE Be sure ou see that this linear approimation o sin depends on the point o tangenc. At a dierent point on the graph o, ou would obtain a dierent tangent line approimation.
2 60_009.qd //0 :8 PM Page 6 6 CHAPTER Applications o Dierentiation (c +, (c + )) (c,( (c)) c c + (c) (c) (c + ) When is small, c c is approimated b c. Figure.66 Dierentials When the tangent line to the graph o at the point c, c c c c Tangent line at c, c is used as an approimation o the graph o, the quantit c is called the change in, and is denoted b, as shown in Figure.66. When is small, the change in (denoted b ) can be approimated as shown. c c Actual change in c Approimate change in For such an approimation, the quantit is traditionall denoted b d, and is called the dierential o. The epression d is denoted b d, and is called the dierential o. Deinition o Dierentials Let represent a unction that is dierentiable on an open interval containing. The dierential o (denoted b d) is an nonzero real number. The dierential o (denoted b d) is d d. In man tpes o applications, the dierential o can be used as an approimation o the change in. That is, d or d. = = EXAMPLE Comparing and d Let. Find d when and d 0.0. Compare this value with or and 0.0. (, ) d The change in,, is approimated b the dierential o, d. Figure.67 Solution Because, ou have, and the dierential d is given b d d Now, using 0.0, the change in is.0 Dierential o Figure.67 shows the geometric comparison o d and. Tr comparing other values o d and. You will see that the values become closer to each other as d or approaches 0. In Eample, the tangent line to the graph o at is or g. Tangent line to the graph o at. For -values near, this line is close to the graph o, as shown in Figure.67. For instance, and g
3 60_009.qd //0 :8 PM Page 7 SECTION.9 Dierentials 7 Error Propagation Phsicists and engineers tend to make liberal use o the approimation o b d. One wa this occurs in practice is in the estimation o errors propagated b phsical measuring devices. For eample, i ou let represent the measured value o a variable and let represent the eact value, then is the error in measurement. Finall, i the measured value is used to compute another value, the dierence between and is the propagated error. Measurement Propagated error error Eact value Measured value EXAMPLE Estimation o Error The radius o a ball bearing is measured to be 0.7 inch, as shown in Figure.68. I the measurement is correct to within 0.0 inch, estimate the propagated error in the volume V o the ball bearing. 0.7 Ball bearing with measured radius that is correct to within 0.0 inch Figure.68 Solution The ormula or the volume o a sphere is V r, where r is the radius o the sphere. So, ou can write r 0.7 Measured radius and 0.0 r 0.0. Possible error To approimate the propagated error in the volume, dierentiate V to obtain dvdr r and write V dv Approimate V b dv. r dr 0.7 ±0.0 Substitute or r and dr. ±0.068 cubic inch. So, the volume has a propagated error o about 0.06 cubic inch. Would ou sa that the propagated error in Eample is large or small? The answer is best given in relative terms b comparing dv with V. The ratio dv V r dr r dr r 0.7 ±0.0 ±0.09 Ratio o dv to V Simpli. Substitute or dr and r. is called the relative error. The corresponding percent error is approimatel.9%.
4 60_009.qd //0 :8 PM Page 8 8 CHAPTER Applications o Dierentiation Calculating Dierentials Each o the dierentiation rules that ou studied in Chapter can be written in dierential orm. For eample, suppose u and v are dierentiable unctions o. B the deinition o dierentials, ou have du u d and dv v d. So, ou can write the dierential orm o the Product Rule as shown below. duv d uv d d uv vu d uv d vu d u dv v du Dierential o uv Product Rule Dierential Formulas Let u and v be dierentiable unctions o. Constant multiple: dcu c du Sum or dierence: du ± v du ± dv Product: Quotient: d uv u dv v du d u v du u dv v v EXAMPLE Finding Dierentials Mar Evans Picture Librar GOTTFRIED WILHELM LEIBNIZ (66 76) Both Leibniz and Newton are credited with creating calculus. It was Leibniz, however, who tried to broaden calculus b developing rules and ormal notation. He oten spent das choosing an appropriate notation or a new concept. a. b. c. d. Function sin cos The notation in Eample is called the Leibniz notation or derivatives and dierentials, named ater the German mathematician Gottried Wilhelm Leibniz. The beaut o this notation is that it provides an eas wa to remember several important calculus ormulas b making it seem as though the ormulas were derived rom algebraic manipulations o dierentials. For instance, in Leibniz notation, the Chain Rule d d du d du d Derivative d d d cos d d sin cos d d d Dierential d d d cos d d sin cos d d d would appear to be true because the du s divide out. Even though this reasoning is incorrect, the notation does help one remember the Chain Rule.
5 60_009.qd //0 :8 PM Page 9 SECTION.9 Dierentials 9 EXAMPLE Finding the Dierential o a Composite Function sin cos d d cos d Original unction Appl Chain Rule. Dierential orm EXAMPLE 6 Finding the Dierential o a Composite Function d d d Original unction Appl Chain Rule. Dierential orm Dierentials can be used to approimate unction values. To do this or the unction given b, ou use the ormula d d which is derived rom the approimation d. The ke to using this ormula is to choose a value or that makes the calculations easier, as shown in Eample 7. EXAMPLE 7 Approimating Function Values Use dierentials to approimate 6.. Solution Using, ou can write d d. Now, choosing 6 and d 0., ou obtain the ollowing approimation g() = + 8 () = (6, ) The tangent line approimation to at 6 is the line g 8. For -values near 6, the graphs o and g are close together, as shown in Figure.69. For instance, and g Figure.69 In act, i ou use a graphing utilit to zoom in near the point o tangenc 6,, ou will see that the two graphs appear to coincide. Notice also that as ou move arther awa rom the point o tangenc, the linear approimation is less accurate.
6 60_009.qd //0 :8 PM Page 0 0 CHAPTER Applications o Dierentiation Eercises or Section.9 In Eercises 6, ind the equation o the tangent line T to the graph o at the given point. Use this linear approimation to complete the table... T In Eercises 7 0, use the inormation to evaluate and compare and d. In Eercises 0, ind the dierential d o the given unction In Eercises, use dierentials and the graph o to approimate (a).9 and (b).0. To print an enlarged cop o the graph, go to the website 6, (, ),,..,,,,. 6. sin, csc,, sin, csc d 0. d 0. d 0.0 d cot 8. sin sec cos 6 (, ).. In Eercises and 6, use dierentials and the graph o approimate (a) g.9 and (b) g. given that g See or worked-out solutions to odd-numbered eercises. (, ) g (, ) 7. Area The measurement o the side o a square is ound to be inches, with a possible error o 6 inch. Use dierentials to approimate the possible propagated error in computing the area o the square. 8. Area The measurements o the base and altitude o a triangle are ound to be 6 and 0 centimeters, respectivel. The possible error in each measurement is 0. centimeter. Use dierentials to approimate the possible propagated error in computing the area o the triangle. 9. Area The measurement o the radius o the end o a log is ound to be inches, with a possible error o inch. Use dierentials to approimate the possible propagated error in computing the area o the end o the log. 0. Volume and Surace Area The measurement o the edge o a cube is ound to be inches, with a possible error o 0.0 inch. Use dierentials to approimate the maimum possible propagated error in computing (a) the volume o the cube and (b) the surace area o the cube.. Area The measurement o a side o a square is ound to be centimeters, with a possible error o 0.0 centimeter. (a) Approimate the percent error in computing the area o the square. (b) Estimate the maimum allowable percent error in measuring the side i the error in computing the area cannot eceed.%.. Circumerence The measurement o the circumerence o a circle is ound to be 6 inches, with a possible error o. inches. (a) Approimate the percent error in computing the area o the circle. (, ) (, ) g g to
7 60_009.qd //0 :8 PM Page SECTION.9 Dierentials (b) Estimate the maimum allowable percent error in measuring the circumerence i the error in computing the area cannot eceed %.. Volume and Surace Area The radius o a sphere is measured to be 6 inches, with a possible error o 0.0 inch. Use dierentials to approimate the maimum possible error in calculating (a) the volume o the sphere, (b) the surace area o the sphere, and (c) the relative errors in parts (a) and (b).. Proit The proit P or a compan is given b P Approimate the change and percent change in proit as production changes rom to 0 units. Volume In Eercises and 6, the thickness o each shell is 0. centimeter. Use dierentials to approimate the volume o each shell.. 0. cm 6. cm 0 cm 7. Pendulum The period o a pendulum is given b T L g where L is the length o the pendulum in eet, g is the acceleration due to gravit, and T is the time in seconds. The pendulum has been subjected to an increase in temperature such that the length has increased b %. (a) Find the approimate percent change in the period. (b) Using the result in part (a), ind the approimate error in this pendulum clock in da. 8. Ohm s Law A current o I amperes passes through a resistor o R ohms. Ohm s Law states that the voltage E applied to the resistor is E IR. I the voltage is constant, show that the magnitude o the relative error in R caused b a change in I is equal in magnitude to the relative error in I. 9. Triangle Measurements The measurement o one side o a right triangle is ound to be 9. inches, and the angle opposite that side is with a possible error o. (a) Approimate the percent error in computing the length o the hpotenuse. (b) Estimate the maimum allowable percent error in measuring the angle i the error in computing the length o the hpotenuse cannot eceed %. 0. Area Approimate the percent error in computing the area o the triangle in Eercise cm 00 cm. Projectile Motion The range R o a projectile is where v 0 is the initial velocit in eet per second and is the angle o elevation. I v 0 00 eet per second and is changed rom 0 to, use dierentials to approimate the change in the range.. Surveing A surveor standing 0 eet rom the base o a large tree measures the angle o elevation to the top o the tree as 7.. How accuratel must the angle be measured i the percent error in estimating the height o the tree is to be less than 6%? In Eercises 6, use dierentials to approimate the value o the epression. Compare our answer with that o a calculator Writing In Eercises 7 and 8, give a short eplanation o wh the approimation is valid tan In Eercises 9, veri the tangent line approimation o the unction at the given point. Then use a graphing utilit to graph the unction and its approimation in the same viewing window R v 0 sin Function tan Approimation Writing About Concepts Point 0,, 0, 0 0,. Describe the change in accurac o d as an approimation or when is decreased.. When using dierentials, what is meant b the terms propagated error, relative error, and percent error? True or False? In Eercises 8, determine whether the statement is true or alse. I it is alse, eplain wh or give an eample that shows it is alse.. I c, then d d. 6. I a b, then dd. 7. I is dierentiable, then lim d I, is increasing and dierentiable, and > 0, then d.
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