Derivatives - Rolle Theorem (Continued)

</section>
<section style="font-size: 50px";>
<h2 id="application-of-the-rolle-theorem-1">Application of the Rolle Theorem</h2>
<ul>
<li>The Rolle Theorem is often used in calculus to prove other theorems and results.</li>
</ul>
</section>
<section style="font-size: 50px";>
<h2 id="example-2">Example 2</h2>
<p>Prove that the function f(x) = sin(x) - x has at most one root.
<em>Solution:</em></p>
<ul>
<li>First, we need to show that f(x) is continuous on the closed interval [0, π] and differentiable on the open interval (0, π).</li>
<li>Second, we need to show that f(0) = f(π) = 0.
<ul>
<li>f(0) = sin(0) - 0 = 0</li>
<li>f(π) = sin(π) - π = 0</li>
<li>Therefore, f(0) = f(π) = 0</li>
</ul>
</li>
<li>As all the conditions are satisfied, we can conclude that there exists at least one solution c in (0, π) such that f’(c) = 0.</li>
<li>Since f(x) is a smooth function and the derivative f’(x) = cos(x) - 1 never equals zero on the interval (0, π), there can be at most one root for the equation f(x) = 0.

</section>
<section style="font-size: 50px";>
<h2 id="example-12">Example 12</h2>
<p>Find the derivative of the function f(x) = sin(3x^2 - 2x).
<em>Solution:</em></p>
<ul>
<li>Using the chain rule, the derivative is given by:
<ul>
<li>d/dx(sin(3x^2 - 2x)) = cos(3x^2 - 2x) * d/dx(3x^2 - 2x)</li>
<li>Applying the power rule and constant rule, we have:</li>
<li>d/dx(sin(3x^2 - 2x)) = cos(3x^2 - 2</li>
</ul>
</li>
</ul>
</section>

</div>
				</div>

function transformCallouts() {
      var slides = document.querySelectorAll('.reveal .slides section');
      slides.forEach(function (slide) {
        var data = document.getElementsByTagName("Success");
        for (let i = 0; i < data.length; i++) {
          data[i].classList.add("callout");
          data[i].classList.add("callout-success");
        }

var data = document.getElementsByTagName("Default");
        for (let i = 0; i < data.length; i++) {
          data[i].classList.add("callout");
          data[i].classList.add("callout-default");
        }

var data = document.getElementsByTagName("Warning");
        for (let i = 0; i < data.length; i++) {
          data[i].classList.add("callout");
          data[i].classList.add("callout-warning");
        }

var data = document.getElementsByTagName("Primary");
        for (let i = 0; i < data.length; i++) {
          data[i].classList.add("callout");
          data[i].classList.add("callout-primary");
        }

var data = document.getElementsByTagName("Danger");
        for (let i = 0; i < data.length; i++) {
          data[i].classList.add("callout");
          data[i].classList.add("callout-danger");
        }
      });
    };

function addrfit() {
      var slides = document.querySelectorAll('.reveal');
      slides.forEach(function (slide) {
        var temp = document.getElementsByTagName("h2");

if (temp.length > 0) {
          for (let i = 0; i < temp.length; i++) {
            temp[i].classList.add("r-fit-text");
          }
        }

var temp3 = document.getElementsByTagName("h1");

if (temp3.length > 0) {
          for (let i = 0; i < temp3.length; i++) {
            temp3[i].classList.add("r-fit-text");
          }
        }

var temp6 = document.getElementsByClassName("text-container");

if (temp6.length > 0) {
          for (let i = 0; i < temp6.length; i++) {
            var charcount = 0;
            var ptags = temp6[i].getElementsByTagName("p");
            try {
              for (let j = 0; j < ptags.length; j++) {
                temp = ptags[j].outerText.replaceAll(" ", "");
                temp = temp.replaceAll("\n", "");
                charcount += temp.length;
              }
            } catch (err) {
              console.log(err);
            }

var font = 35;

if (charcount > 50 && charcount < 100) {
              font = 35;
            } else if (charcount > 100 && charcount < 120) {
              font = 33;
            } else if (charcount > 120 && charcount < 150) {
              font = 33;
            } else if (charcount > 150 && charcount < 170) {
              font = 31;
            } else if (charcount > 170 && charcount < 200) {
              font = 30;
            } else if (charcount > 200 && charcount < 230) {
              font = 29;
            } else if (charcount > 230 && charcount < 250) {
              font = 28;
            } else if (charcount > 250 && charcount < 270) {
              font = 27;
            } else if (charcount > 270 && charcount < 300) {
              font = 25;
            } else if (charcount > 300 && charcount < 330) {
              font = 23;
            } else if (charcount > 330 && charcount < 360) {
              font = 24;
            } else if (charcount > 360 && charcount < 390) {
              font = 22;
            } else if (charcount > 390 && charcount < 420) {
              font = 21;
            } else if (charcount > 420) {
              font = 20;
            }

temp6[i].style.fontSize = `${font}px`;
          }
        }
      })
    }

Reveal.addEventListener('ready', addrfit);
    Reveal.addEventListener('ready', transformCallouts);
  </script>