3.9 Differentials. Tangent Line Approximations. Exploration. Using a Tangent Line Approximation

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1 3.9 Differentials Differentials Understand the concept of a tangent line approimation. Compare the value of the differential, d, with the actual change in,. Estimate a propagated error using a differential. Find the differential of a function using differentiation formulas. Eploration Tangent Line Approimation Use a graphing utilit to graph f. In the same viewing window, graph the tangent line to the graph of f at the point,. Zoom in twice on the point of tangenc. Does our graphing utilit distinguish between the two graphs? Use the trace feature to compare the two graphs. As the -values get closer to, what can ou sa about the -values? Tangent Line Approimations Newton s Method (Section 3.8) is an eample of the use of a tangent line to approimate the graph of a function. In this section, ou will stud other situations in which the graph of a function can be approimated b a straight line. To begin, consider a function f that is differentiable at c. The equation for the tangent line at the point c, f c is f c f c c f c f c c and is called the tangent line approimation (or linear approimation) of f at c. Because c is a constant, is a linear function of. Moreover, b restricting the values of to those sufficientl close to c, the values of can be used as approimations (to an desired degree of accurac) of the values of the function f. In other words, as approaches c, the limit of is f c. Using a Tangent Line Approimation See LarsonCalculus.com for an interactive version of this tpe of eample. π Tangent line f() = + sin π π The tangent line approimation of f at the point 0, Figure 3.65 Find the tangent line approimation of f sin at the point 0,. Then use a table to compare the -values of the linear function with those of f on an open interval containing 0. Solution f cos. The derivative of f is First derivative So, the equation of the tangent line to the graph of f at the point 0, is f 0 f Tangent line approimation The table compares the values of given b this linear approimation with the values of f near 0. Notice that the closer is to 0, the better the approimation. This conclusion is reinforced b the graph shown in Figure f sin REMARK Be sure ou see that this linear approimation of f sin depends on the point of tangenc. At a different point on the graph of f, ou would obtain a different tangent line approimation. Copright 0 Cengage Learning. All Rights Reserved. Ma not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third part content ma be suppressed from the ebook and/or echapter(s). Editorial review has deemed that an suppressed content does not materiall affect the overall learning eperience. Cengage Learning reserves the right to remove additional content at an time if subsequent rights restrictions require it.

2 3 Chapter 3 Applications of Differentiation (c + Δ, f(c + Δ)) (c, f(c)) c Δ Δ c + Δ f f (c)δ f(c) When is small, f c f c is approimated b f c. Figure 3.66 f(c + Δ) Differentials When the tangent line to the graph of f at the point c, f c f c f c c Tangent line at c, f c is used as an approimation of the graph of f, the quantit c is called the change in, and is denoted b, as shown in Figure When is small, the change in (denoted b ) can be approimated as shown. f c f c Actual change in f c Approimate change in For such an approimation, the quantit is traditionall denoted b d, and is called the differential of. The epression f d is denoted b d, and is called the differential of. Definition of Differentials Let f represent a function that is differentiable on an open interval containing. The differential of (denoted b d) is an nonzero real number. The differential of (denoted b d) is d f d. In man tpes of applications, the differential of can be used as an approimation of the change in. That is, d or f d. = Comparing and d = (, ) d Δ The change in,, is approimated b the differential of, d. Figure 3.67 Let. Find d when and d 0.0. Compare this value with for and 0.0. Solution Because f, ou have f, and the differential d is d f d f Differential of Now, using 0.0, the change in is f f f.0 f Figure 3.67 shows the geometric comparison of d and. Tr comparing other values of 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 of f at is. Tangent line to the graph of f at. For -values near, this line is close to the graph of f, as shown in Figure 3.67 and in the table f Copright 0 Cengage Learning. All Rights Reserved. Ma not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third part content ma be suppressed from the ebook and/or echapter(s). Editorial review has deemed that an suppressed content does not materiall affect the overall learning eperience. Cengage Learning reserves the right to remove additional content at an time if subsequent rights restrictions require it.

3 Error Propagation 3.9 Differentials 33 Phsicists and engineers tend to make liberal use of the approimation of b d. One wa this occurs in practice is in the estimation of errors propagated b phsical measuring devices. For eample, if ou let represent the measured value of a variable and let represent the eact value, then is the error in measurement. Finall, if the measured value is used to compute another value f, then the difference between f and f is the propagated error. Measurement error Propagated error f f Eact value Measured value Estimation of Error The measured radius of a ball bearing is 0.7 inch, as shown in the figure. The measurement is correct to within 0.0 inch. Estimate the propagated error in the volume V of the ball bearing. Solution sphere is V 3 r 3 The formula for the volume of a 0.7 Ball bearing with measured radius that is correct to within 0.0 inch. where r is the radius of 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, differentiate V dv dr r and write V dv Approimate V b dv. r dr 0.7 ±0.0 Substitute for r and dr. ± cubic inch. So, the volume has a propagated error of about 0.06 cubic inch. to obtain Would ou sa that the propagated error in Eample 3 is large or small? The answer is best given in relative terms b comparing dv with V. The ratio dv V r dr 3 r 3 3 dr r ±0.0 ±0.09 Ratio of dv to V Simplif. Substitute for dr and r. is called the relative error. The corresponding percent error is approimatel.9%. Dmitr Kalinovsk/Shutterstock.com Copright 0 Cengage Learning. All Rights Reserved. Ma not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third part content ma be suppressed from the ebook and/or echapter(s). Editorial review has deemed that an suppressed content does not materiall affect the overall learning eperience. Cengage Learning reserves the right to remove additional content at an time if subsequent rights restrictions require it.

4 3 Chapter 3 Applications of Differentiation Calculating Differentials Each of the differentiation rules that ou studied in Chapter can be written in differential form. For eample, let u and v be differentiable functions of. B the definition of differentials, ou have du u d and dv v d. So, ou can write the differential form of the Product Rule as shown below. d uv d uv d d uv vu d uv d vu d u dv v du Differential of uv. Product Rule Differential Formulas Let u and v be differentiable functions of. Constant multiple: Sum or difference: Product: Quotient: d cu c du d u ± v du ± dv d uv u dv v du d u v du u dv v v Finding Differentials 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 formal notation. He often spent das choosing an appropriate notation for a new concept. See LarsonCalculus.com to read more of this biograph. a. b. c. d. e. Function Derivative Differential sin cos The notation in Eample is called the Leibniz notation for derivatives and differentials, named after the German mathematician Gottfried Wilhelm Leibniz. The beaut of this notation is that it provides an eas wa to remember several important calculus formulas b making it seem as though the formulas were derived from algebraic manipulations of differentials. For instance, in Leibniz notation, the Chain Rule d d du d du d d d d d d cos d d sin cos d d d d d 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. Mar Evans Picture Librar/The Image Works Copright 0 Cengage Learning. All Rights Reserved. Ma not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third part content ma be suppressed from the ebook and/or echapter(s). Editorial review has deemed that an suppressed content does not materiall affect the overall learning eperience. Cengage Learning reserves the right to remove additional content at an time if subsequent rights restrictions require it.

5 3.9 Differentials 35 Finding the Differential of a Composite Function f sin 3 f 3 cos 3 d f d 3 cos 3 d Original function Appl Chain Rule. Differential form Finding the Differential of a Composite Function f d f d f d Original function Appl Chain Rule. Differential form Differentials can be used to approimate function values. To do this for the function given b f, use the formula REMARK This formula is equivalent to the tangent line approimation given earlier in this section. f f d f f d which is derived from the approimation f f d. The ke to using this formula is to choose a value for that makes the calculations easier, as shown in Eample 7. Use differentials to approimate 6.5. Approimating Function Values Solution Using f, ou can write f f f d d. Now, choosing 6 and d 0.5, ou obtain the following approimation. f g() = + 8 f() = (6, ) The tangent line approimation to f at 6 is the line g 8. For -values near 6, the graphs of f and g are close together, as shown in Figure For instance, and f g Figure 3.68 In fact, if ou use a graphing utilit to zoom in near the point of tangenc 6,, ou will see that the two graphs appear to coincide. Notice also that as ou move farther awa from the point of tangenc, the linear approimation becomes less accurate. Copright 0 Cengage Learning. All Rights Reserved. Ma not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third part content ma be suppressed from the ebook and/or echapter(s). Editorial review has deemed that an suppressed content does not materiall affect the overall learning eperience. Cengage Learning reserves the right to remove additional content at an time if subsequent rights restrictions require it.

6 36 Chapter 3 Applications of Differentiation 3.9 Eercises See CalcChat.com for tutorial help and worked-out solutions to odd-numbered eercises. Using a Tangent Line Approimation In Eercises 6, find the tangent line approimation T to the graph of f at the given point. Use this linear approimation to complete the table. Comparing and d In Eercises 7 0, use the information to evaluate and compare and d Function -Value Differential of Finding a Differential In Eercises 0, find the differential d of the given function sin 0. Using Differentials In Eercises and, use differentials and the graph of f to approimate (a) f.9 and (b) f.0. To print an enlarged cop of the graph, go to MathGraphs.com (, ) f T. f,. f 6,, 3. f 5,, 3. f, 5. f sin,, sin 6. f csc, tan. csc f 3 5 sec 5 3 (, ), 3, 3 d 0. 6 d 0. f, csc d 0.0 d Using Differentials In Eercises 3 and, use differentials and the graph of to approimate (a) g.93 and (b) g 3. given that g g g 5 ( 3, ) 5. Area The measurement of the side of a square floor tile is 0 inches, with a possible error of inch. (a) Use differentials to approimate the possible propagated error in computing the area of the square. (b) Approimate the percent error in computing the area of the square. 6. Area The measurement of the radius of a circle is 6 inches, with a possible error of inch. (a) Use differentials to approimate the possible propagated error in computing the area of the circle. (b) Approimate the percent error in computing the area of the circle. 7. Area The measurements of the base and altitude of a triangle are found to be 36 and 50 centimeters, respectivel. The possible error in each measurement is 0.5 centimeter. (a) Use differentials to approimate the possible propagated error in computing the area of the triangle. (b) Approimate the percent error in computing the area of the triangle. 8. Circumference The measurement of the circumference of a circle is found to be 6 centimeters, with a possible error of 0.9 centimeter. (a) Approimate the percent error in computing the area of the circle. (b) Estimate the maimum allowable percent error in measuring the circumference if the error in computing the area cannot eceed 3%. 9. Volume and Surface Area The measurement of the edge of a cube is found to be 5 inches, with a possible error of 0.03 inch. (a) Use differentials to approimate the possible propagated error in computing the volume of the cube. (b) Use differentials to approimate the possible propagated error in computing the surface area of the cube. (c) Approimate the percent errors in parts (a) and (b). 3 3 (3, 3) g 3 5 Copright 0 Cengage Learning. All Rights Reserved. Ma not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third part content ma be suppressed from the ebook and/or echapter(s). Editorial review has deemed that an suppressed content does not materiall affect the overall learning eperience. Cengage Learning reserves the right to remove additional content at an time if subsequent rights restrictions require it.

7 3.9 Differentials Volume and Surface Area The radius of a spherical balloon is measured as 8 inches, with a possible error of 0.0 inch. (a) Use differentials to approimate the possible propagated error in computing the volume of the sphere. (b) Use differentials to approimate the possible propagated error in computing the surface area of the sphere. (c) Approimate the percent errors in parts (a) and (b). 3. Stopping Distance The total stopping distance T of a vehicle is T where T is in feet and is the speed in miles per hour. Approimate the change and percent change in total stopping distance as speed changes from 5 to 6 miles per hour. 3. HOW DO YOU SEE IT? The graph shows the profit P (in dollars) from selling units of an item. Use the graph to determine which is greater, the change in profit when the production level changes from 00 to 0 units or the change in profit when the production level changes from 900 to 90 units. Eplain our reasoning Profit (in dollars) 33. Pendulum The period of a pendulum is given b where L is the length of the pendulum in feet, 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), find the approimate error in this pendulum clock in da. 3. Ohm s Law A current of I amperes passes through a resistor of R ohms. Ohm s Law states that the voltage E applied to the resistor is E IR. 0,000 9,000 8,000 7,000 6,000 5,000,000 3,000,000,000 T L g P Number of units The voltage is constant. Show that the magnitude of the relative error in R caused b a change in I is equal in magnitude to the relative error in I. 35. Projectile Motion The range R of a projectile is where v 0 is the initial velocit in feet per second and is the angle of elevation. Use differentials to approimate the change in the range when v feet per second and is changed from 0 to. 36. Surveing A surveor standing 50 feet from the base of a large tree measures the angle of elevation to the top of the tree as 7.5. How accuratel must the angle be measured if the percent error in estimating the height of the tree is to be less than 6%? Approimating Function Values In Eercises 37 0, use differentials to approimate the value of the epression. Compare our answer with that of a calculator Verifing a Tangent Line Approimation In Eercises and, verif the tangent line approimation of the function at the given point. Then use a graphing utilit to graph the function and its approimation in the same viewing window. Function Approimation Point. f. R v 0 sin 3 f tan True or False? In Eercises 7 50, determine whether the statement is true or false. If it is false, eplain wh or give an eample that shows it is false. 7. If c, then d d. d 8. If a b, then d. WRITING ABOUT CONCEPTS 9. If is differentiable, then lim d , 0, 0 3. Comparing and d Describe the change in accurac of d as an approimation for when is decreased.. Describing Terms When using differentials, what is meant b the terms propagated error, relative error, and percent error? Using Differentials In Eercises 5 and 6, give a short eplanation of wh the approimation is valid tan If f, f is increasing and differentiable, and > 0, then d. Copright 0 Cengage Learning. All Rights Reserved. Ma not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third part content ma be suppressed from the ebook and/or echapter(s). Editorial review has deemed that an suppressed content does not materiall affect the overall learning eperience. Cengage Learning reserves the right to remove additional content at an time if subsequent rights restrictions require it.

8 Answers to Odd-Numbered Eercises A35 Section 3.9 (page 36). T f T T f T T cos sin f T ; d ; d d 3. sec tan d sin d d 9 d. (a) 0.9 (b).0 3. (a) (b) (a) ± 5 8 in. (b) 0.65% 7. (a) ±0.75 cm (b) about.9% 9. (a) ±0.5 in. 3 (b) ±5. in. (c) 0.6%; 0.% mi; About 7.3% 33. (a) % (b) 6 sec 3.6 min ft 37. f, d d f Calculator: f, d d 3 f Calculator:.998. f 0 f The value of d becomes closer to the value of as decreases. 5. f ; d d f True 9. True 6 (0, ) f 6 Copright 0 Cengage Learning. All Rights Reserved. Ma not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third part content ma be suppressed from the ebook and/or echapter(s). Editorial review has deemed that an suppressed content does not materiall affect the overall learning eperience. Cengage Learning reserves the right to remove additional content at an time if subsequent rights restrictions require it.