{"id":3213,"date":"2024-06-13T18:03:55","date_gmt":"2024-06-13T18:03:55","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/calculus1\/?post_type=chapter&#038;p=3213"},"modified":"2024-08-05T02:12:22","modified_gmt":"2024-08-05T02:12:22","slug":"linear-approximations-and-differentials-fresh-take","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/calculus1\/chapter\/linear-approximations-and-differentials-fresh-take\/","title":{"raw":"Linear Approximations and Differentials: Fresh Take","rendered":"Linear Approximations and Differentials: Fresh Take"},"content":{"raw":"<section class=\"textbox learningGoals\">\r\n<ul>\r\n\t<li>Explain and use linearization to approximate a function\u2019s value near a specific point<\/li>\r\n\t<li>Calculate and interpret differentials to estimate small changes in function values<\/li>\r\n\t<li>Measure the accuracy of approximations made with differentials by calculating relative and percentage errors<\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2>Linear Approximation of a Function at a Point<\/h2>\r\n<div class=\"textbox shaded\">\r\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\r\n<ul>\r\n\t<li class=\"whitespace-normal break-words\">Linear approximation (or tangent line approximation) of [latex]f[\/latex] at [latex]x=a[\/latex] is given by:\r\n\r\n\r\n<ul>\r\n\t<li class=\"whitespace-normal break-words\">[latex]L(x) = f(a) + f'(a)(x-a)[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Interpretation:\r\n\r\n\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n\t<li class=\"whitespace-normal break-words\">Uses the tangent line at a point to estimate function values nearby<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Accurate for [latex]x[\/latex] close to [latex]a[\/latex]<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Based on the idea that smooth functions look linear when zoomed in sufficiently<\/li>\r\n<\/ul>\r\n<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Key Formula:\r\n\r\n\r\n<ul>\r\n\t<li class=\"whitespace-normal break-words\">[latex]f(x) \\approx f(a) + f'(a)(x-a)[\/latex] for [latex]x[\/latex] near [latex]a[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Applications:\r\n\r\n\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n\t<li class=\"whitespace-normal break-words\">Estimating function values<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Root approximation<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Simplifying complex calculations<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Basis for Newton's method<\/li>\r\n<\/ul>\r\n<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Limitations:\r\n\r\n\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n\t<li class=\"whitespace-normal break-words\">Accuracy decreases as [latex]x[\/latex] moves away from [latex]a[\/latex]<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Not suitable for functions with sharp turns or discontinuities at [latex]a[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/div>\r\n<section class=\"textbox example\">\r\n<p id=\"fs-id1165042444934\">Find the local linear approximation to [latex]f(x)=\\sqrt[3]{x}[\/latex] at [latex]x=8[\/latex]. Use it to approximate [latex]\\sqrt[3]{8.1}[\/latex] to five decimal places.<\/p>\r\n<p>[reveal-answer q=\"29087744\"]Hint[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"29087744\"]<\/p>\r\n<p id=\"fs-id1165043321518\">[latex]L(x)=f(a)+f^{\\prime}(a)(x-a)[\/latex]<\/p>\r\n<p>[\/hidden-answer]<\/p>\r\n<p>[reveal-answer q=\"fs-id1165042980470\"]Show Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"fs-id1165042980470\"]<\/p>\r\n<p id=\"fs-id1165042980470\">[latex]L(x)=2+\\frac{1}{12}(x-8)[\/latex]; 2.00833<\/p>\r\n<p>[\/hidden-answer]<\/p>\r\n<\/section>\r\n<section class=\"textbox example\">\r\n<p id=\"fs-id1165043060547\">Find the linear approximation for [latex]f(x)= \\cos x[\/latex] at [latex]x=\\dfrac{\\pi }{2}[\/latex].<\/p>\r\n<p>[reveal-answer q=\"8834672\"]Hint[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"8834672\"]<\/p>\r\n<p>[latex]L(x)=f(a)+f^{\\prime}(a)(x-a)[\/latex]<\/p>\r\n<p>[\/hidden-answer]<\/p>\r\n<p>[reveal-answer q=\"902505\"]Show Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"902505\"]<\/p>\r\n<p>[latex]L(x)=\u2212x+\\frac{\\pi}{2}[\/latex]<\/p>\r\n<p>[\/hidden-answer]<\/p>\r\n<\/section>\r\n<section class=\"textbox example\">\r\n<p id=\"fs-id1165043394826\">Find the linear approximation of [latex]f(x)=(1+x)^4[\/latex] at [latex]x=0[\/latex] without using the result from the preceding example.<\/p>\r\n<p>[reveal-answer q=\"477002\"]Hint[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"477002\"]<\/p>\r\n<p id=\"fs-id1165042964925\">[latex]f^{\\prime}(x)=4(1+x)^3[\/latex]<\/p>\r\n<p>[\/hidden-answer]<\/p>\r\n<p>[reveal-answer q=\"fs-id1165043253559\"]Show Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"fs-id1165043253559\"]<\/p>\r\n<p id=\"fs-id1165043253559\">[latex]L(x)=1+4x[\/latex]<\/p>\r\n<p>[\/hidden-answer]<\/p>\r\n<\/section>\r\n<h2>Differentials and Amount of Error<\/h2>\r\n<div class=\"textbox shaded\">\r\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\r\n<ul>\r\n\t<li class=\"whitespace-normal break-words\">For a function [latex]y = f(x)[\/latex], the differential [latex]dy[\/latex] is defined as: [latex]dy = f'(x) , dx[\/latex]\r\n\r\n\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n\t<li class=\"whitespace-normal break-words\">[latex]dx[\/latex] is an independent variable that can be any nonzero real number<\/li>\r\n<\/ul>\r\n<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Relationship to Linear Approximation:\r\n\r\n\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n\t<li class=\"whitespace-normal break-words\">[latex]\\Delta y \\approx dy[\/latex] for small changes in [latex]x[\/latex]<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Based on the linear approximation: [latex]f(a + dx) \\approx f(a) + f'(a) , dx[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Error Estimation:\r\n\r\n\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n\t<li class=\"whitespace-normal break-words\">Propagated Error: [latex]\\Delta y = f(a + dx) - f(a)[\/latex]<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Estimated Error: [latex]\\Delta y \\approx dy \\approx f'(a + dx) , dx[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Relative and Percentage Error:\r\n\r\n\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n\t<li class=\"whitespace-normal break-words\">Relative Error: [latex]\\frac{\\Delta q}{q}[\/latex], where [latex]q[\/latex] is the actual value<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Percentage Error: Relative error expressed as a percentage<\/li>\r\n<\/ul>\r\n<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Applications:\r\n\r\n\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n\t<li class=\"whitespace-normal break-words\">Approximating function value changes<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Estimating measurement errors in calculations<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Analyzing accuracy in scientific measurements<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/div>\r\n<section class=\"textbox example\">\r\n<p id=\"fs-id1165042367595\">For [latex]y=e^{x^2}[\/latex], find [latex]dy[\/latex].<\/p>\r\n<p>[reveal-answer q=\"9076558\"]Hint[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"9076558\"]<\/p>\r\n<p id=\"fs-id1165043256948\">[latex]dy=f^{\\prime}(x)dx[\/latex]<\/p>\r\n<p>[\/hidden-answer]<\/p>\r\n<p>[reveal-answer q=\"fs-id1165043326695\"]Show Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"fs-id1165043326695\"]<\/p>\r\n<p id=\"fs-id1165043326695\">[latex]dy=2xe^{x^2} \\, dx[\/latex]<\/p>\r\n<p>[\/hidden-answer]<\/p>\r\n<\/section>\r\n<section class=\"textbox example\">\r\n<p id=\"fs-id1165042637207\">For [latex]y=x^2+2x[\/latex], find [latex]\\Delta y[\/latex] and [latex]dy[\/latex] at [latex]x=3[\/latex] if [latex]dx=0.2[\/latex].<\/p>\r\n<p>[reveal-answer q=\"33228801\"]Hint[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"33228801\"]<\/p>\r\n<p id=\"fs-id1165042925900\">[latex]dy=f^{\\prime}(3) \\, dx[\/latex], [latex]\\Delta y=f(3.2)-f(3)[\/latex]<\/p>\r\n<p>[\/hidden-answer]<\/p>\r\n<p>[reveal-answer q=\"fs-id1165043033503\"]Show Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"fs-id1165043033503\"]<\/p>\r\n<p id=\"fs-id1165043033503\">[latex]dy=1.6[\/latex], [latex]\\Delta y=1.64[\/latex]<\/p>\r\n<p>[\/hidden-answer]<\/p>\r\n<\/section>\r\n<section class=\"textbox example\">\r\n<p id=\"fs-id1165043253780\">Estimate the error in the computed volume of a cube if the side length is measured to be 6 cm with an accuracy of 0.2 cm.<\/p>\r\n<p>[reveal-answer q=\"8011542\"]Hint[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"8011542\"]<\/p>\r\n<p id=\"fs-id1165042561334\">[latex]dV=3x^2 \\, dx[\/latex]<\/p>\r\n<p>[\/hidden-answer]<\/p>\r\n<p>[reveal-answer q=\"fs-id1165042946479\"]Show Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"fs-id1165042946479\"]<\/p>\r\n<p id=\"fs-id1165042946479\">The volume measurement is accurate to within [latex]21.6 \\, \\text{cm}^3[\/latex].<\/p>\r\n<p>[\/hidden-answer]<\/p>\r\n<\/section>\r\n<p>&nbsp;<\/p>","rendered":"<section class=\"textbox learningGoals\">\n<ul>\n<li>Explain and use linearization to approximate a function\u2019s value near a specific point<\/li>\n<li>Calculate and interpret differentials to estimate small changes in function values<\/li>\n<li>Measure the accuracy of approximations made with differentials by calculating relative and percentage errors<\/li>\n<\/ul>\n<\/section>\n<h2>Linear Approximation of a Function at a Point<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\n<ul>\n<li class=\"whitespace-normal break-words\">Linear approximation (or tangent line approximation) of [latex]f[\/latex] at [latex]x=a[\/latex] is given by:\n<ul>\n<li class=\"whitespace-normal break-words\">[latex]L(x) = f(a) + f'(a)(x-a)[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Interpretation:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Uses the tangent line at a point to estimate function values nearby<\/li>\n<li class=\"whitespace-normal break-words\">Accurate for [latex]x[\/latex] close to [latex]a[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Based on the idea that smooth functions look linear when zoomed in sufficiently<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Key Formula:\n<ul>\n<li class=\"whitespace-normal break-words\">[latex]f(x) \\approx f(a) + f'(a)(x-a)[\/latex] for [latex]x[\/latex] near [latex]a[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Applications:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Estimating function values<\/li>\n<li class=\"whitespace-normal break-words\">Root approximation<\/li>\n<li class=\"whitespace-normal break-words\">Simplifying complex calculations<\/li>\n<li class=\"whitespace-normal break-words\">Basis for Newton&#8217;s method<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Limitations:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Accuracy decreases as [latex]x[\/latex] moves away from [latex]a[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Not suitable for functions with sharp turns or discontinuities at [latex]a[\/latex]<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/div>\n<section class=\"textbox example\">\n<p id=\"fs-id1165042444934\">Find the local linear approximation to [latex]f(x)=\\sqrt[3]{x}[\/latex] at [latex]x=8[\/latex]. Use it to approximate [latex]\\sqrt[3]{8.1}[\/latex] to five decimal places.<\/p>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q29087744\">Hint<\/button><\/p>\n<div id=\"q29087744\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165043321518\">[latex]L(x)=f(a)+f^{\\prime}(a)(x-a)[\/latex]<\/p>\n<\/div>\n<\/div>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qfs-id1165042980470\">Show Solution<\/button><\/p>\n<div id=\"qfs-id1165042980470\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165042980470\">[latex]L(x)=2+\\frac{1}{12}(x-8)[\/latex]; 2.00833<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\">\n<p id=\"fs-id1165043060547\">Find the linear approximation for [latex]f(x)= \\cos x[\/latex] at [latex]x=\\dfrac{\\pi }{2}[\/latex].<\/p>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q8834672\">Hint<\/button><\/p>\n<div id=\"q8834672\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]L(x)=f(a)+f^{\\prime}(a)(x-a)[\/latex]<\/p>\n<\/div>\n<\/div>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q902505\">Show Solution<\/button><\/p>\n<div id=\"q902505\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]L(x)=\u2212x+\\frac{\\pi}{2}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\">\n<p id=\"fs-id1165043394826\">Find the linear approximation of [latex]f(x)=(1+x)^4[\/latex] at [latex]x=0[\/latex] without using the result from the preceding example.<\/p>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q477002\">Hint<\/button><\/p>\n<div id=\"q477002\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165042964925\">[latex]f^{\\prime}(x)=4(1+x)^3[\/latex]<\/p>\n<\/div>\n<\/div>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qfs-id1165043253559\">Show Solution<\/button><\/p>\n<div id=\"qfs-id1165043253559\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165043253559\">[latex]L(x)=1+4x[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/section>\n<h2>Differentials and Amount of Error<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\n<ul>\n<li class=\"whitespace-normal break-words\">For a function [latex]y = f(x)[\/latex], the differential [latex]dy[\/latex] is defined as: [latex]dy = f'(x) , dx[\/latex]\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">[latex]dx[\/latex] is an independent variable that can be any nonzero real number<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Relationship to Linear Approximation:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">[latex]\\Delta y \\approx dy[\/latex] for small changes in [latex]x[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Based on the linear approximation: [latex]f(a + dx) \\approx f(a) + f'(a) , dx[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Error Estimation:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Propagated Error: [latex]\\Delta y = f(a + dx) - f(a)[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Estimated Error: [latex]\\Delta y \\approx dy \\approx f'(a + dx) , dx[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Relative and Percentage Error:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Relative Error: [latex]\\frac{\\Delta q}{q}[\/latex], where [latex]q[\/latex] is the actual value<\/li>\n<li class=\"whitespace-normal break-words\">Percentage Error: Relative error expressed as a percentage<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Applications:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Approximating function value changes<\/li>\n<li class=\"whitespace-normal break-words\">Estimating measurement errors in calculations<\/li>\n<li class=\"whitespace-normal break-words\">Analyzing accuracy in scientific measurements<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/div>\n<section class=\"textbox example\">\n<p id=\"fs-id1165042367595\">For [latex]y=e^{x^2}[\/latex], find [latex]dy[\/latex].<\/p>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q9076558\">Hint<\/button><\/p>\n<div id=\"q9076558\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165043256948\">[latex]dy=f^{\\prime}(x)dx[\/latex]<\/p>\n<\/div>\n<\/div>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qfs-id1165043326695\">Show Solution<\/button><\/p>\n<div id=\"qfs-id1165043326695\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165043326695\">[latex]dy=2xe^{x^2} \\, dx[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\">\n<p id=\"fs-id1165042637207\">For [latex]y=x^2+2x[\/latex], find [latex]\\Delta y[\/latex] and [latex]dy[\/latex] at [latex]x=3[\/latex] if [latex]dx=0.2[\/latex].<\/p>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q33228801\">Hint<\/button><\/p>\n<div id=\"q33228801\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165042925900\">[latex]dy=f^{\\prime}(3) \\, dx[\/latex], [latex]\\Delta y=f(3.2)-f(3)[\/latex]<\/p>\n<\/div>\n<\/div>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qfs-id1165043033503\">Show Solution<\/button><\/p>\n<div id=\"qfs-id1165043033503\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165043033503\">[latex]dy=1.6[\/latex], [latex]\\Delta y=1.64[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\">\n<p id=\"fs-id1165043253780\">Estimate the error in the computed volume of a cube if the side length is measured to be 6 cm with an accuracy of 0.2 cm.<\/p>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q8011542\">Hint<\/button><\/p>\n<div id=\"q8011542\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165042561334\">[latex]dV=3x^2 \\, dx[\/latex]<\/p>\n<\/div>\n<\/div>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qfs-id1165042946479\">Show Solution<\/button><\/p>\n<div id=\"qfs-id1165042946479\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165042946479\">The volume measurement is accurate to within [latex]21.6 \\, 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