Tangent Lines - Math & Matter

> [!summary] A tangent line is the limit of a secant line as the distance between two points A and B reduces to zero. > **Key equation:** > Slope of a tangent line: > $\displaystyle \lim_{ h\to 0 } \frac{f (a+h) - f (a)}{h}$ or $\displaystyle \lim_{ x\to a } \frac{f (x) - f (a)}{x-a}$ > Equation of a Tangent Line: $f(x) = f(a) + f'(a)(x-a)$ >[!info]+ Read Time **⏱ 3 mis** # Definition The slope of a tangent line is the [[Limits|limit]] of a [[Secant Lines|secant line]] as $P$ approaches $A$. This slope is sometimes called the [[Instantaneous|instantaneous]] [[Rate of Change|rate of change]] at a point $A$ (shown below) > [!note]+ Diagram of the secant line to the tangent line > > | ![[tl_2.png]] | ![[tl_2.png]] | > | :-----------: | :-----------: | > [^1] > Secant line with point P and point A. As the distance between point P and A decreases (as P approaches A) it creates a tangent line > [!warning] Assumption From the image above, assume the horizontal distance between $P$ and $A$ to be called $h$ To align with our distance from the equation in [[Secant Lines|secant line]] ($\frac{f(a+h) - f(a)}{h}$) $ \begin{array}{c} \text{If our tangent line is the limit as our distance from P and A go to zero, then:} \\ \displaystyle \lim_{ h\to 0 } \frac{f(a+h) - f(a)}{h} \\ \\ \text{Another way is saying as you move one function closer to the other}\\ \displaystyle \lim_{ x\to a } \frac{f(x) - f(a)}{x-a} \end{array} $ > [!bug] Common Misconception A common misconception is that the definition of a tangent line is a line that touches a curve at exactly 1 point. This is technically wrong; refer to the resources below. ## Equation of a Tangent Line Notice from the image above that our tangent line is straight. So we could use [[Point-Slope Form Equation of a Line|point slope form]] for lines to find the equation of a tangent line. > [!warning] Assumptions To make our equation of a tangent line easier to look at let's assume this notion: > - Use the notion that $y_{2} = f(x), y_{1} = f(a),x_{2} = x, x_{1} = a$ to simplify notion > - Slope $=m=f'(a) = \displaystyle \lim_{ h\to 0 } \frac{f (a+h) - f (a)}{h}$ > - $f(x)-f(a) = m(x -a )$ from [[Point-Slope Form Equation of a Line|point slope form]] Notice that $f'(a)$ is our slope so just using point slope form we can just make this equation. $ \begin{array}{c} f(x) - f(a) = f'(a)(x-a) \\ \\ \text{We often like it in this notion below to make math later easier} \\ f(x) = f(a) + f'(a)(x-a) \end{array} $ # Resources <iframe width="560" height="315" src="https://www.youtube.com/embed/O_cwTAfjgAQ?si=MUYidx69DcA3UkpH" title="YouTube video player" frameborder="0" allow="accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share" referrerpolicy="strict-origin-when-cross-origin" allowfullscreen></iframe> ---  <div class="mm-form-light"> <iframe src="https://updates.cyberleadhub.com/widget/form/Y0kpQVpjJQuxEfX59m17" id="inline-Y0kpQVpjJQuxEfX59m17" title="Join Math & Matter Newsletter (Light)" data-height="900" scrolling="no" allowtransparency="true" loading="lazy" style="width:100%;height:350px;border:none;border-radius:10px;background:transparent;overflow:hidden" ></iframe> </div>  <div class="mm-form-dark"> <iframe src="https://updates.cyberleadhub.com/widget/form/lbeDLm24VjuaFxhjccA1" id="inline-lbeDLm24VjuaFxhjccA1" title="Join Math & Matter Newsletter (Dark)" data-height="900" scrolling="no" allowtransparency="true" loading="lazy" style="width:100%;height:350px;border:none;border-radius:10px;background:transparent;overflow:hidden" ></iframe> </div>  <script src="https://updates.cyberleadhub.com/js/form_embed.js"></script> [^1]: Adapted from https://www.integral-domain.org/lwilliams/Resources/TikzImg/secant.tex