{"id":3771,"date":"2021-05-12T20:24:06","date_gmt":"2021-05-12T20:24:06","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/calculus1\/chapter\/review-for-section-2-1-a-preview-of-calculus\/"},"modified":"2021-05-12T20:24:06","modified_gmt":"2021-05-12T20:24:06","slug":"review-for-section-2-1-a-preview-of-calculus","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/calculus1\/chapter\/review-for-section-2-1-a-preview-of-calculus\/","title":{"raw":"Skills Review for A Preview of Calculus","rendered":"Skills Review for A Preview of Calculus"},"content":{"raw":"\n<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n \t<li>Use the slope formula to find the slope of a line between two points<\/li>\n<\/ul>\n<\/div>\nA Preview of Calculus discusses some of the various applications of calculus including how calculus is used to calculate the instantaneous rate of change or slope of a tangent line. To help build the idea of calculating the instantaneous rate of change, how to calculate the slope of a line using two points is reviewed here.\n<h2>Calculate the Slope of a Line<\/h2>\nThe <strong>slope<\/strong> of a line refers to the ratio of the vertical change in <em>y<\/em> over the horizontal change in <em>x<\/em> between any two points on a line. It indicates the direction in which a line slants as well as its steepness. Slope is sometimes described as rise over run.\n<div style=\"text-align: center;\">[latex]m=\\frac{{y}_{2}-{y}_{1}}{{x}_{2}-{x}_{1}}[\/latex]<\/div>\nIf the slope is positive, the line slants upward to the right. If the slope is negative, the line slants downward to the right. As the slope increases, the line becomes steeper. Some examples are shown below. The lines indicate the following slopes: [latex]m=-3[\/latex], [latex]m=2[\/latex], and [latex]m=\\frac{1}{3}[\/latex].\n\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/11185922\/CNX_CAT_Figure_02_02_002.jpg\" alt=\"Coordinate plane with the x and y axes ranging from negative 10 to 10. Three linear functions are plotted: y = negative 3 times x minus 2; y = 2 times x plus 1; and y = x over 3 plus 2.\" width=\"487\" height=\"442\">\n<div class=\"textbox\">\n<h3>A General Note: The Slope of a Line<\/h3>\nThe slope of a line, <em>m<\/em>, represents the change in <em>y<\/em> over the change in <em>x.<\/em> Given two points, [latex]\\left({x}_{1},{y}_{1}\\right)[\/latex] and [latex]\\left({x}_{2},{y}_{2}\\right)[\/latex], the following formula determines the slope of a line containing these points:\n<div style=\"text-align: center;\">[latex]m=\\frac{{y}_{2}-{y}_{1}}{{x}_{2}-{x}_{1}}[\/latex]<\/div>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Finding the Slope of a Line Given Two Points<\/h3>\nFind the slope of a line that passes through the points [latex]\\left(2,-1\\right)[\/latex] and [latex]\\left(-5,3\\right)[\/latex].\n[reveal-answer q=\"688301\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"688301\"]\n\nWe substitute the <em>y-<\/em>values and the <em>x-<\/em>values into the formula.\n<div style=\"text-align: center;\">[latex]\\begin{array}{l}m\\hfill&amp;=\\frac{3-\\left(-1\\right)}{-5 - 2}\\hfill \\\\ \\hfill&amp;=\\frac{4}{-7}\\hfill \\\\ \\hfill&amp;=-\\frac{4}{7}\\hfill \\end{array}[\/latex]<\/div>\nThe slope is [latex]-\\frac{4}{7}[\/latex].\n<h4>Analysis of the Solution<\/h4>\nIt does not matter which point is called [latex]\\left({x}_{1},{y}_{1}\\right)[\/latex] or [latex]\\left({x}_{2},{y}_{2}\\right)[\/latex]. As long as we are consistent with the order of the <em>y<\/em> terms and the order of the <em>x<\/em> terms in the numerator and denominator, the calculation will yield the same result.\n[\/hidden-answer]\n\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\nFind the slope of the line that passes through the points [latex]\\left(-2,6\\right)[\/latex] and [latex]\\left(1,4\\right)[\/latex].\n\n[reveal-answer q=\"196055\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"196055\"]\n\nslope[latex]=m=\\dfrac{-2}{3}=-\\dfrac{2}{3}[\/latex][\/hidden-answer]\n\n[ohm_question]1719[\/ohm_question]\n\n<\/div>\n","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Use the slope formula to find the slope of a line between two points<\/li>\n<\/ul>\n<\/div>\n<p>A Preview of Calculus discusses some of the various applications of calculus including how calculus is used to calculate the instantaneous rate of change or slope of a tangent line. To help build the idea of calculating the instantaneous rate of change, how to calculate the slope of a line using two points is reviewed here.<\/p>\n<h2>Calculate the Slope of a Line<\/h2>\n<p>The <strong>slope<\/strong> of a line refers to the ratio of the vertical change in <em>y<\/em> over the horizontal change in <em>x<\/em> between any two points on a line. It indicates the direction in which a line slants as well as its steepness. Slope is sometimes described as rise over run.<\/p>\n<div style=\"text-align: center;\">[latex]m=\\frac{{y}_{2}-{y}_{1}}{{x}_{2}-{x}_{1}}[\/latex]<\/div>\n<p>If the slope is positive, the line slants upward to the right. If the slope is negative, the line slants downward to the right. As the slope increases, the line becomes steeper. Some examples are shown below. The lines indicate the following slopes: [latex]m=-3[\/latex], [latex]m=2[\/latex], and [latex]m=\\frac{1}{3}[\/latex].<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/11185922\/CNX_CAT_Figure_02_02_002.jpg\" alt=\"Coordinate plane with the x and y axes ranging from negative 10 to 10. Three linear functions are plotted: y = negative 3 times x minus 2; y = 2 times x plus 1; and y = x over 3 plus 2.\" width=\"487\" height=\"442\" \/><\/p>\n<div class=\"textbox\">\n<h3>A General Note: The Slope of a Line<\/h3>\n<p>The slope of a line, <em>m<\/em>, represents the change in <em>y<\/em> over the change in <em>x.<\/em> Given two points, [latex]\\left({x}_{1},{y}_{1}\\right)[\/latex] and [latex]\\left({x}_{2},{y}_{2}\\right)[\/latex], the following formula determines the slope of a line containing these points:<\/p>\n<div style=\"text-align: center;\">[latex]m=\\frac{{y}_{2}-{y}_{1}}{{x}_{2}-{x}_{1}}[\/latex]<\/div>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Finding the Slope of a Line Given Two Points<\/h3>\n<p>Find the slope of a line that passes through the points [latex]\\left(2,-1\\right)[\/latex] and [latex]\\left(-5,3\\right)[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q688301\">Show Solution<\/span><\/p>\n<div id=\"q688301\" class=\"hidden-answer\" style=\"display: none\">\n<p>We substitute the <em>y-<\/em>values and the <em>x-<\/em>values into the formula.<\/p>\n<div style=\"text-align: center;\">[latex]\\begin{array}{l}m\\hfill&=\\frac{3-\\left(-1\\right)}{-5 - 2}\\hfill \\\\ \\hfill&=\\frac{4}{-7}\\hfill \\\\ \\hfill&=-\\frac{4}{7}\\hfill \\end{array}[\/latex]<\/div>\n<p>The slope is [latex]-\\frac{4}{7}[\/latex].<\/p>\n<h4>Analysis of the Solution<\/h4>\n<p>It does not matter which point is called [latex]\\left({x}_{1},{y}_{1}\\right)[\/latex] or [latex]\\left({x}_{2},{y}_{2}\\right)[\/latex]. As long as we are consistent with the order of the <em>y<\/em> terms and the order of the <em>x<\/em> terms in the numerator and denominator, the calculation will yield the same result.\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Find the slope of the line that passes through the points [latex]\\left(-2,6\\right)[\/latex] and [latex]\\left(1,4\\right)[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q196055\">Show Solution<\/span><\/p>\n<div id=\"q196055\" class=\"hidden-answer\" style=\"display: none\">\n<p>slope[latex]=m=\\dfrac{-2}{3}=-\\dfrac{2}{3}[\/latex]<\/p><\/div>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"ohm1719\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=1719&theme=oea&iframe_resize_id=ohm1719&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n\n\t\t\t <section class=\"citations-section\" role=\"contentinfo\">\n\t\t\t <h3>Candela Citations<\/h3>\n\t\t\t\t\t <div>\n\t\t\t\t\t\t <div id=\"citation-list-3771\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Original<\/div><ul class=\"citation-list\"><li>Modification and Revision. <strong>Provided by<\/strong>: Lumen Learning. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>College Algebra Corequisite. <strong>Provided by<\/strong>: Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/courses.lumenlearning.com\/waymakercollegealgebracorequisite\/\">https:\/\/courses.lumenlearning.com\/waymakercollegealgebracorequisite\/<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Precalculus. <strong>Provided by<\/strong>: Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/courses.lumenlearning.com\/precalculus\/\">https:\/\/courses.lumenlearning.com\/precalculus\/<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section>","protected":false},"author":17533,"menu_order":1,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"College Algebra Corequisite\",\"author\":\"\",\"organization\":\"Lumen 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