Tangent Line Approximations

Tangent Line Approximations Newton's Method is an example of the use of а tangent line to approximate the graph of а function. ln this section, you wi/1 study other situations in which the graph of а function can Ье approximated Ьу а straight line. То begin, consider а function f that is differentiaЬle at с. The equation for the tangent line at the point (с,/ (с)) is = f'(c)(x - с) У = f(c) + .f"'(c)(x - с) у - !(с) -с.. and is called the tangent line approximation (or linear approximation) of f at с. Because _с is а constant, у is а linear function of х. Moreover, Ьу restricting the values of х to those sufficiently close to с, the values of у сап Ье used as approximations (to апу desired degree of accuracy) of the values of the function f. ln other words, as х approaches с, the limit of у is f (с). Ample 1: Find th.e tange�t lin� approximation of f (х) = 1 + sin х at the point (О, 1). Then use а tаЫе to compare the Тvalues of the linear function with those of f (х) on an open interval containing х = О. f l'l. f '(j. - f(i> ::: 1 f'� Т (о, 1) ::: 14-:stt"""')x. 'о: (°[с + - + 'б - +, ��)( :?:о� [_С) t � 1� ::- СО5С): ( EJ � �t-1 с o..n AL.&J 4. � .Pr ls f'{_t:. -6 ) l ] (_6� ( -с.') �k)6'e.,tvj Tangcn1 linc т (о, 1) Lti.,� Бt:> к 4 -1 -0.5 f(x) = 1 + sinx y=l+x ... s2.1 -0.1 -0.01 о 0.01 0.1 0.5 " 2 Differentials Tangent line at (с, f(c)) When the tangent line to the graph of f at the point (с, f (с)) • У= f(с)+ f '(с)(х- с) is used as ап approximation of the graph of /, the quantity ._.is called the change in x,and is denoted Ьу Лх, as shown in Figure 3.54. When Лх is small, the change in у (denoted Ьу Лу) сап Ье approximated as shown. L Лу=/(с+Лх)-/(с) ] �t'(с)Лх ft.c+ЛJ.) -+-----'----"J....-_..,_ ,r r с+Лх Лх For such ап approximation, the quantity Лх is traditionally denoted Ьу dx, and is called the differential of х. When лх is small, ду = f(c + ЛХ)- f(c) is approximated Ьу f'(с)ЛХ. The expression f '(x)dx is denoted Ьу dy, and is called the differential of у. � �& Definition of Differentials ,1,.. ,6 Let у = f(x) rep1·e ent а function tbat is differe11tiaЫe оп ап o p n interval �= � Х e real num er. V ('-.. coп taiпing х. Тhе s ff 1· n i t' х (deno ted Ьу dx) is апу noпzero d e e t alo b e deп o t d i d e e t al o i e b /. J Th i ft' r_ _п i _ _ _f_y_( _ _ _ _ _ _y_d_y_) s � _ - _ _ dy_ _ =f'(x)dx. ---------' � <:: t, . U,.. � _ lО ln many types of applications, the differential of у сап Ье used as ап approximation of the change in у. That is, Лу�dу ог Лу �f'(x)dx. д.'r� 4а l� tLO Example 2: Let у= х2 . Find dy when х = 1 and dx= 0.01. Compare this value with Лу for х = and i == х?· ==-'> �' = � r/� J--л � = '2( \ � 2.х �(с.. ... ::: f � 1 ЛХ = 0.01. Ь.><. \+.01 (_\. О 1 - - • �(с) -((1) '2.. - (\ � �Dt);;; .DL Ь'Х_..Д6'J • C�Atv ()� I N ,-Н � �o...,c.::r (ё)N 1 \S +н ..д (W�IМА-г'ё" 0 LW\� '1'-' -rн� RJ�ctcotJ us,ЛJ� il-4€ '(Дf..)С,ё"-'"f L--t ..., � . Ь. О J� 1 СЬ -t"� )' Note: ду is the exact change in the function, where dy is the approximate change in the function using the tangent line. Тгу comparing other values of dy and Лу. You will see that the values become c\oser to each other as dx (or Лх) approaches О. Тhе change in у, лу, 1s approXimated Ьу the differential of у, dy. Figure 3.55 Error Propagation Physicists and engineers tend to make liberal use of the approximation of Лу Ьу dy. One way this occurs in practic� is in estimation of errors propagated Ьу physica\ measuring devices. For example, if you let х represent the measured е of а variaЬle and let х + ЛХ represent the exact value, then ЛХ is the error in measurement. Finally, if the measured value х is used to compute another value f (х), then the difference between f (х + ЛХ) and f (х) is the propagated error. Measuгen1e111 Propagated т СIТОГ еr ог ,л-._, ,л-._, .f(x + Лх) - .f(x) = Лу '-----v-----' Exact va!L1e '-у-' Mea.suгed value Example З: The measured radius of а Ьа\1 bearing is 0.7 inch, as shown in figure. The measurement is correct to within 0.01 inch. Estimate the propagated error in the ygipme V of the bal\ bearing. --О\ f��� ... O\ ( �(') ..... 0.7 8all Ьearing witl1 measured radius t11at is corтect to Witnln 0.01 ,ncn. 2.. V = 4,r1 8 dvxЛv � 'J � " 4-п- ( ::l,°)�(_z"o,) ?i:.. ;±. О(о VoY Would you say that the propagated error in Example 3 is large or small? The answer is best given in relative terms Ьу comparing dVwith V. The ratio is called the relative error. dV C."A.�G&:. I /.J V • l�\�.;. (' ::: .. '1 \ "'"' VoLVM б: AGTt,AL. vo t..t:>�'6 dv - = 4,r ( .. 1- 2.( ±.ьr) ½т (-1-J� 12Ь!-А тw� �Q.QDQ.. °" 'Г = ..\- ,О\ У\ -± . О-1- Z q ± ,;-:z.Cf 1;;:.\.,. '-1 Calculating Differentials Each of the differentiation rules that we have studied сап Ье written in differential form. For example, let и and v Ье differentiaЬle functions of х. Ву the definition of differentials, you have du=u'dx and dv = v'dx. So, you сап write the differential form of the Product Rule as shown below. • d cl[uvJ = )иv] dx = [uv' + vu '] dx = uv' dx + vu' dx = и dv + v du d. Differential Fornшlas Let II алd' v Ье iff'eientiaЫe func • ons ot· х. • •• f • Consrmt1 multiple: d[cu] ""= с +1, Sum 01· diffeгence: d[u ± 1'] = du ± dv Producr: d[uv] = и dv + ,, du Quorienf: v du - и dv !!_ d[ ] = 2 V V Example 4: Finding Differentials �=� .¼=2� а. у= х 2 ь. у= Differential Derivative Function Jx 'с. у= 2 inx d. у= хех 1 е. У=­ х �= L,..Gc::6� х. � ::: е С \-+ '1-- • �--:.!­ � - )<1- The notation in Example 4 is called the Leibniz notation for derivatives and differentials, named after the German mathematician Gottfried Wilhelm Leibniz (1646-1716). The beauty of this notation is that it provides an easy way to remember several important calculus formulas Ьу making it seem as though the formulas were derived from algebraic manipulations of differentials. Differentials сап Ье used to approximate function values. То do this for the function given Ьу у = f (х), use the formula f(x + дх) = .f(x) + dy = f(x) + f' (:х) dx which is derived from the approximation Лу = f (х + Лх) - f (х) � dy. The key to using this formula is to choose а value for х that makes the calculations easier. (This formula is equivalent t. the tangent line approximation.