secant_tangent_lines_by_hand.png<\/a> is required.<\/p>\n\n\n\n\\documentclass{article} % Basic LaTeX document setup\n\n\n% ========== Packages for extra features ==========\n\\usepackage{amsmath,amsthm} % For better math typesetting\n\\usepackage[colorlinks]{hyperref} % For clickable links and PDF bookmarks\n\\usepackage{unicode-math} % Maps LaTeX commands for mathematical characters to the Unicode Mathematical Alphanumeric Symbols block.\n\\usepackage{graphicx} % For including images\n\\pagestyle{empty} % (Optional) Removes page numbers (nicer for examples)\n\n\\usepackage{authblk}\t\t% Helps with author names and affiliations\n\n\\newtheorem{theorem}{Theorem}[section] % Numbering resets when a section changes\n\\newtheorem{corollary}{Corollary}[theorem] % Numbering resets when a new theorem happens\n\\newtheorem{lemma}[theorem]{Lemma} % Uses the same counter as Theorem\n\\newtheorem{definition}{Definition} % Uses the same counter as Theorem\n\\newtheorem{example}{Example} % Uses the same counter as Theorem\n\n\\usepackage{tikz}\n\\usetikzlibrary{positioning}\n\n\\usepackage[executable=python.exe]{pyluatex}\n\n% ========== Main document begins ==========\n\\begin{document}\n\t\n\\title{Limits}\n\\date{19 July 2026}\n\\author{Keith E Emmert}\n\\affil{Tarleton State University Mathematics Department}\n\n\\pagestyle{headings}\n\n\\maketitle\n\n\\section{Best Practices for Maximum Accessibility}\n\n\\begin{definition}\n\tThe \\textbf{secant line} of the function $f(x)$ through the points $(a,f(a))$ and $(x,f(x))$ is the line passing through these two points whose slope is given by $m_{sec} = \\dfrac{f(x)-f(a)}{x-a}$ provided that $x \\not= a$.\n\\end{definition}\n\n\\begin{definition}\n\tThe \\textbf{slope of a tangent line} is found by letting $m_{tan} = \\lim_{x\\to a}\u2061 \\dfrac{f(x)-f(a)}{x-a}$.\n\\end{definition}\n\n\\begin{example}\n\tConsider the function $f(x) = -x^3 + 4 x^2 - x - 6$ and suppose that $a=\\frac{3}{2}$. Consider three points $x=-\\frac{3}{2}, -\\frac{1}{2}, \\frac{1}{2}$ where we wish to draw secant lines. A figure illustrating these ideas where the three secant lines drawn in red, green and brown, and the tangent line drawn in black are shown on the graph of $f(x)$ drawn in blue. See Figures~\\ref{fig-sec-tan} or~\\ref{fig-sec-tan-2}.\n\\end{example}\n\n% --- Figure with embedded alternative text (accessibility for images) ---\n\\begin{figure}[h]\n\t\\centering\n\t% Use the 'alt' key directly within \\includegraphics\n\t\\includegraphics[width=\\linewidth]{secant_tangent_lines_by_hand.png}\n\t\\caption{The graph of $f(x)$ with its tangent line and several secant lines.}\\label{fig-sec-tan}\n\\end{figure}\n\n\\begin{python}\nimport numpy as np \nimport matplotlib.pyplot as plt \n\nx = np.linspace(-4, 6, 1000) \nf = -x**3 + 4*x**2 - x - 6 \n\nl1 = -13*x\/4 + 3 \nl2 = 5*x\/4 - 15\/4 \nl3 = 15*x\/4 - 15\/2 \nl4 = 17*x\/4 - 33\/4 \n\nfig, ax = plt.subplots(figsize=(8, 3)) \n\nax.plot(x, f, color='blue', linewidth=2, label='f(x) = -x^3+4x^2-x-6') \nax.plot(x, l1, '--', color='red', linewidth=1.5, label='y = -13x\/4 + 3') \nax.plot(x, l2, '--', color='green', linewidth=1.5, label='y = 5x\/4 - 15\/4') \nax.plot(x, l3, '--', color='brown', linewidth=1.5, label='y = 15x\/4 - 15\/2') \nax.plot(x, l4, color='black', linewidth=1.5, label='y = 17x\/4 - 33\/4') \n\nx0, y0 = 3\/2, -15\/8 \nax.plot(x0, y0, 'ko', markersize=6) \nax.plot(-3\/2, 63\/8, 'ro') \nax.plot(-1\/2, -35\/8, 'go', markersize=6) \nax.plot(1\/2, -45\/8, 'o', color = 'brown', markersize=6) \n\nax.set_xlim(x.min(), x.max()) \nax.set_ylim(-10, 10) \nax.axhline(0, color='gray', lw=0.6) \nax.axvline(0, color='gray', lw=0.6) \nax.set_xlabel('x')\nax.set_ylabel('y') \nax.legend(loc='best', fontsize=9) \nax.grid(alpha=0.3)\n\nfig.savefig('secant-tangent-python.pdf')\nplt.close()\n\\end{python}\n\n% Display the figure.\n\\begin{figure}[ht]\n\t\\centering\n\t\\includegraphics[width=\\linewidth]{secant-tangent-python.pdf}\n\t\\caption{The graph of $f(x)$ with its tangent line and several secant lines. Generated by Python embedded in the \\LaTeX\\ document.}\\label{fig-sec-tan-2}\n\\end{figure}\n\n\\begin{example}\n\tBelow is a table illustrating how the secant line slopes change. See Table~\\ref{tab-sec-slope}.\n\t\n\t\\begin{table}[h]\n\t\t\\centering\n\t\t\\caption{Slopes of three different secant lines.}\\label{tab-sec-slope}\n\t\t\\begin{tabular}{c|c}\n\t\t\t\\hline\n\t\t\t$x$ & $m_{sec} = \\dfrac{f(x)-f(a)}{x-a}$ \\\\\\hline\n\t\t\t$-\\dfrac{3}{2}$ & $-\\dfrac{13}{4}$ \\\\\n\t\t\t$\\dfrac{1}{2}$ & $\\dfrac{5}{4}$ \\\\\n\t\t\t$\\dfrac{1}{2}$ & $\\dfrac{17}{4}$ \n\t\t\\end{tabular}\n\t\\end{table}\n\\end{example}\n\n\n\\section{Derivatives}\n\n\\begin{definition}\n\tThe \\textbf{derivative at the point $x = a$} is found by computing $f'(a) = \\lim_{x \\to a} \\dfrac{f(x) - f(a)}{x - a}$.\n\\end{definition}\n\n\\section{Using TikZ}\n\nOften people utilize TikZ to generate figures.\n\n\n\\begin{tikzpicture}[\n\troundnode\/.style={circle, draw=green!60, fill=green!5, very thick, minimum size=7mm},\n\tsquarednode\/.style={rectangle, draw=red!60, fill=red!5, very thick, minimum size=5mm},\n\t]{alt=TikZ figure.}\n\t%Nodes\n\t\\node[squarednode] (maintopic) {2};\n\t\\node[roundnode] (uppercircle) [above=of maintopic] {1};\n\t\\node[squarednode] (rightsquare) [right=of maintopic] {3};\n\t\\node[roundnode] (lowercircle) [below=of maintopic] {4};\n\t\n\t%Lines\n\t\\draw[->] (uppercircle.south) -- (maintopic.north);\n\t\\draw[->] (maintopic.east) -- (rightsquare.west);\n\t\\draw[->] (rightsquare.south) .. controls +(down:7mm) and +(right:7mm) .. (lowercircle.east);\n\\end{tikzpicture}\n\n\\end{document}<\/code><\/pre>\n","protected":false},"excerpt":{"rendered":"Below you will find LaTeX code which will generate a PDF document with a very low accessibility score according to Canvas’ Ally tool. Note that the file secant_tangent_lines_by_hand.png is required.<\/p>\n","protected":false},"author":1002,"featured_media":580,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"template-fullwidth.php","meta":{"_acf_changed":false,"inline_featured_image":false,"advgb_blocks_editor_width":"","advgb_blocks_columns_visual_guide":"","footnotes":""},"class_list":["post-2594","page","type-page","status-publish","has-post-thumbnail","hentry"],"acf":[],"coauthors":[],"author_meta":{"author_link":"https:\/\/www.tarleton.edu\/math\/author\/emmerttarleton-edu\/","display_name":"Emmert, Dr. Keith"},"relative_dates":{"created":"Posted 2 days ago","modified":"Updated 2 days ago"},"absolute_dates":{"created":"Posted on July 29, 2026","modified":"Updated on July 29, 2026"},"absolute_dates_time":{"created":"Posted on July 29, 2026 1:57 pm","modified":"Updated on July 29, 2026 4:53 pm"},"featured_img_caption":"","featured_img":false,"series_order":"","_links":{"self":[{"href":"https:\/\/www.tarleton.edu\/math\/wp-json\/wp\/v2\/pages\/2594","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.tarleton.edu\/math\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/www.tarleton.edu\/math\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/www.tarleton.edu\/math\/wp-json\/wp\/v2\/users\/1002"}],"replies":[{"embeddable":true,"href":"https:\/\/www.tarleton.edu\/math\/wp-json\/wp\/v2\/comments?post=2594"}],"version-history":[{"count":2,"href":"https:\/\/www.tarleton.edu\/math\/wp-json\/wp\/v2\/pages\/2594\/revisions"}],"predecessor-version":[{"id":2608,"href":"https:\/\/www.tarleton.edu\/math\/wp-json\/wp\/v2\/pages\/2594\/revisions\/2608"}],"wp:attachment":[{"href":"https:\/\/www.tarleton.edu\/math\/wp-json\/wp\/v2\/media?parent=2594"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}
![secant_tangent_lines_by_hand.png<\/a> is required.<\/p>\n\n\n\n\\documentclass{article} % Basic LaTeX document setup\n\n\n% ========== Packages for extra features ==========\n\\usepackage{amsmath,amsthm} % For better math typesetting\n\\usepackage[colorlinks]{hyperref} % For clickable links and PDF bookmarks\n\\usepackage{unicode-math} % Maps LaTeX commands for mathematical characters to the Unicode Mathematical Alphanumeric Symbols block.\n\\usepackage{graphicx} % For including images\n\\pagestyle{empty} % (Optional) Removes page numbers (nicer for examples)\n\n\\usepackage{authblk}\t\t% Helps with author names and affiliations\n\n\\newtheorem{theorem}{Theorem}[section] % Numbering resets when a section changes\n\\newtheorem{corollary}{Corollary}[theorem] % Numbering resets when a new theorem happens\n\\newtheorem{lemma}[theorem]{Lemma} % Uses the same counter as Theorem\n\\newtheorem{definition}{Definition} % Uses the same counter as Theorem\n\\newtheorem{example}{Example} % Uses the same counter as Theorem\n\n\\usepackage{tikz}\n\\usetikzlibrary{positioning}\n\n\\usepackage[executable=python.exe]{pyluatex}\n\n% ========== Main document begins ==========\n\\begin{document}\n\t\n\\title{Limits}\n\\date{19 July 2026}\n\\author{Keith E Emmert}\n\\affil{Tarleton State University Mathematics Department}\n\n\\pagestyle{headings}\n\n\\maketitle\n\n\\section{Best Practices for Maximum Accessibility}\n\n\\begin{definition}\n\tThe \\textbf{secant line} of the function $f(x)$ through the points $(a,f(a))$ and $(x,f(x))$ is the line passing through these two points whose slope is given by $m_{sec} = \\dfrac{f(x)-f(a)}{x-a}$ provided that $x \\not= a$.\n\\end{definition}\n\n\\begin{definition}\n\tThe \\textbf{slope of a tangent line} is found by letting $m_{tan} = \\lim_{x\\to a}\u2061 \\dfrac{f(x)-f(a)}{x-a}$.\n\\end{definition}\n\n\\begin{example}\n\tConsider the function $f(x) = -x^3 + 4 x^2 - x - 6$ and suppose that $a=\\frac{3}{2}$. Consider three points $x=-\\frac{3}{2}, -\\frac{1}{2}, \\frac{1}{2}$ where we wish to draw secant lines. A figure illustrating these ideas where the three secant lines drawn in red, green and brown, and the tangent line drawn in black are shown on the graph of $f(x)$ drawn in blue. See Figures~\\ref{fig-sec-tan} or~\\ref{fig-sec-tan-2}.\n\\end{example}\n\n% --- Figure with embedded alternative text (accessibility for images) ---\n\\begin{figure}[h]\n\t\\centering\n\t% Use the 'alt' key directly within \\includegraphics\n\t\\includegraphics[width=\\linewidth]{secant_tangent_lines_by_hand.png}\n\t\\caption{The graph of $f(x)$ with its tangent line and several secant lines.}\\label{fig-sec-tan}\n\\end{figure}\n\n\\begin{python}\nimport numpy as np \nimport matplotlib.pyplot as plt \n\nx = np.linspace(-4, 6, 1000) \nf = -x**3 + 4*x**2 - x - 6 \n\nl1 = -13*x\/4 + 3 \nl2 = 5*x\/4 - 15\/4 \nl3 = 15*x\/4 - 15\/2 \nl4 = 17*x\/4 - 33\/4 \n\nfig, ax = plt.subplots(figsize=(8, 3)) \n\nax.plot(x, f, color='blue', linewidth=2, label='f(x) = -x^3+4x^2-x-6') \nax.plot(x, l1, '--', color='red', linewidth=1.5, label='y = -13x\/4 + 3') \nax.plot(x, l2, '--', color='green', linewidth=1.5, label='y = 5x\/4 - 15\/4') \nax.plot(x, l3, '--', color='brown', linewidth=1.5, label='y = 15x\/4 - 15\/2') \nax.plot(x, l4, color='black', linewidth=1.5, label='y = 17x\/4 - 33\/4') \n\nx0, y0 = 3\/2, -15\/8 \nax.plot(x0, y0, 'ko', markersize=6) \nax.plot(-3\/2, 63\/8, 'ro') \nax.plot(-1\/2, -35\/8, 'go', markersize=6) \nax.plot(1\/2, -45\/8, 'o', color = 'brown', markersize=6) \n\nax.set_xlim(x.min(), x.max()) \nax.set_ylim(-10, 10) \nax.axhline(0, color='gray', lw=0.6) \nax.axvline(0, color='gray', lw=0.6) \nax.set_xlabel('x')\nax.set_ylabel('y') \nax.legend(loc='best', fontsize=9) \nax.grid(alpha=0.3)\n\nfig.savefig('secant-tangent-python.pdf')\nplt.close()\n\\end{python}\n\n% Display the figure.\n\\begin{figure}[ht]\n\t\\centering\n\t\\includegraphics[width=\\linewidth]{secant-tangent-python.pdf}\n\t\\caption{The graph of $f(x)$ with its tangent line and several secant lines. Generated by Python embedded in the \\LaTeX\\ document.}\\label{fig-sec-tan-2}\n\\end{figure}\n\n\\begin{example}\n\tBelow is a table illustrating how the secant line slopes change. See Table~\\ref{tab-sec-slope}.\n\t\n\t\\begin{table}[h]\n\t\t\\centering\n\t\t\\caption{Slopes of three different secant lines.}\\label{tab-sec-slope}\n\t\t\\begin{tabular}{c|c}\n\t\t\t\\hline\n\t\t\t$x$ & $m_{sec} = \\dfrac{f(x)-f(a)}{x-a}$ \\\\\\hline\n\t\t\t$-\\dfrac{3}{2}$ & $-\\dfrac{13}{4}$ \\\\\n\t\t\t$\\dfrac{1}{2}$ & $\\dfrac{5}{4}$ \\\\\n\t\t\t$\\dfrac{1}{2}$ & $\\dfrac{17}{4}$ \n\t\t\\end{tabular}\n\t\\end{table}\n\\end{example}\n\n\n\\section{Derivatives}\n\n\\begin{definition}\n\tThe \\textbf{derivative at the point $x = a$} is found by computing $f'(a) = \\lim_{x \\to a} \\dfrac{f(x) - f(a)}{x - a}$.\n\\end{definition}\n\n\\section{Using TikZ}\n\nOften people utilize TikZ to generate figures.\n\n\n\\begin{tikzpicture}[\n\troundnode\/.style={circle, draw=green!60, fill=green!5, very thick, minimum size=7mm},\n\tsquarednode\/.style={rectangle, draw=red!60, fill=red!5, very thick, minimum size=5mm},\n\t]{alt=TikZ figure.}\n\t%Nodes\n\t\\node[squarednode] (maintopic) {2};\n\t\\node[roundnode] (uppercircle) [above=of maintopic] {1};\n\t\\node[squarednode] (rightsquare) [right=of maintopic] {3};\n\t\\node[roundnode] (lowercircle) [below=of maintopic] {4};\n\t\n\t%Lines\n\t\\draw[->] (uppercircle.south) -- (maintopic.north);\n\t\\draw[->] (maintopic.east) -- (rightsquare.west);\n\t\\draw[->] (rightsquare.south) .. controls +(down:7mm) and +(right:7mm) .. (lowercircle.east);\n\\end{tikzpicture}\n\n\\end{document}<\/code><\/pre>\n","protected":false},"excerpt":{"rendered":"Below you will find LaTeX code which will generate a PDF document with a very low accessibility score according to Canvas’ Ally tool. Note that the file secant_tangent_lines_by_hand.png is required.<\/p>\n","protected":false},"author":1002,"featured_media":580,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"template-fullwidth.php","meta":{"_acf_changed":false,"inline_featured_image":false,"advgb_blocks_editor_width":"","advgb_blocks_columns_visual_guide":"","footnotes":""},"class_list":["post-2594","page","type-page","status-publish","has-post-thumbnail","hentry"],"acf":[],"coauthors":[],"author_meta":{"author_link":"https:\/\/www.tarleton.edu\/math\/author\/emmerttarleton-edu\/","display_name":"Emmert, Dr. Keith"},"relative_dates":{"created":"Posted 2 days ago","modified":"Updated 2 days ago"},"absolute_dates":{"created":"Posted on July 29, 2026","modified":"Updated on July 29, 2026"},"absolute_dates_time":{"created":"Posted on July 29, 2026 1:57 pm","modified":"Updated on July 29, 2026 4:53 pm"},"featured_img_caption":"","featured_img":false,"series_order":"","_links":{"self":[{"href":"https:\/\/www.tarleton.edu\/math\/wp-json\/wp\/v2\/pages\/2594","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.tarleton.edu\/math\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/www.tarleton.edu\/math\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/www.tarleton.edu\/math\/wp-json\/wp\/v2\/users\/1002"}],"replies":[{"embeddable":true,"href":"https:\/\/www.tarleton.edu\/math\/wp-json\/wp\/v2\/comments?post=2594"}],"version-history":[{"count":2,"href":"https:\/\/www.tarleton.edu\/math\/wp-json\/wp\/v2\/pages\/2594\/revisions"}],"predecessor-version":[{"id":2608,"href":"https:\/\/www.tarleton.edu\/math\/wp-json\/wp\/v2\/pages\/2594\/revisions\/2608"}],"wp:attachment":[{"href":"https:\/\/www.tarleton.edu\/math\/wp-json\/wp\/v2\/media?parent=2594"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}](\"https:\/\/www.tarleton.edu\/math\/wp-content\/uploads\/sites\/130\/2026\/07\/secant_tangent_lines_by_hand.png\"){kind=link}