changed a sentence

59275d31 · Isabel Slingerland · 8baa8370 · 59275d31
Commit 59275d31 authored 6 months ago by Isabel Slingerland
--- a/book/numerical_methods/2-derivative.ipynb
+++ b/book/numerical_methods/2-derivative.ipynb
@@ -49,7 +49,7 @@
    "\n",
    "\n",
    "\n",
-    "**What does the derivative represents?** A rate of change! In this case, how fast $f(x)$ changes with respect to $x$ (or in the direction of $x$). The derivative can also be with respect to time or another independent variable (e.g., $y,z,t,etc$). Look at the figure to the right. The first derivative evaluated at $x_0$ is illustrated as a simple slope. You can also see that the rate of change at $x_0$ is larger than at $x_1$.   "
+    "**What does the derivative represents?** A rate of change! In this case, how fast $f(x)$ changes with respect to $x$ (or in the direction of $x$). The derivative can also be with respect to time or another independent variable (e.g., $y,z,t,etc$). Look at figure 1. The first derivative evaluated at $x_0$ is illustrated as a simple slope. You can also see that the rate of change at $x_0$ is larger than at $x_1$.   "
   ]
  },
  {

 %% Cell type:markdown id: tags:

 # The First Derivative

 %% Cell type:markdown id: tags:

 ```{note}
 **Important things to retain from this block:**
 * Understand what the derivative represents
 * Recognize that the derivative can be approximated in different ways
 ```

 %% Cell type:markdown id: tags:

 :::{card} **Definition**:

 The derivative of a function $f(x)$ evaluated at the point $x_0$ is

 $$
 \frac{df}{dx}\bigg\rvert_{x_0}=f'(x_0)=\lim_{x \to x_0}\frac{f(x)-f(x_0)}{x-x_0}.
 $$
 :::

 :::{card} **Numerically**:

 $x - x_0$ **cannot tend to 0!** Thus:

 $$
 f'(x_0) \approx \frac{f(x)-f(x_0)}{\Delta x}, \hspace{3mm} \text{where } \Delta x=x-x_0.
 $$

 :::

 ```{figure} figs/derivative_new2.png
 :name: derivative_new2

 derivatives of a function $f(x)$ at a specific points $x_0$ and $x_1$
 ```



-**What does the derivative represents?** A rate of change! In this case, how fast $f(x)$ changes with respect to $x$ (or in the direction of $x$). The derivative can also be with respect to time or another independent variable (e.g., $y,z,t,etc$). Look at the figure to the right. The first derivative evaluated at $x_0$ is illustrated as a simple slope. You can also see that the rate of change at $x_0$ is larger than at $x_1$.
+**What does the derivative represents?** A rate of change! In this case, how fast $f(x)$ changes with respect to $x$ (or in the direction of $x$). The derivative can also be with respect to time or another independent variable (e.g., $y,z,t,etc$). Look at figure 1. The first derivative evaluated at $x_0$ is illustrated as a simple slope. You can also see that the rate of change at $x_0$ is larger than at $x_1$.

 %% Cell type:markdown id: tags:

 ## Numerical Freedom to compute derivatives


 In the Figure above, the derivative approximation was illustrated arbitarly using two points: the one at which the derivate was evaluated and another point in front of it. However, there are more possibilities. Instead of using points at $x_{-1,0,1}$ a more general notation is used: $x_{i-1,i,i+1}$. The simplest ways to approximate the derivative evaluated at the pont $x_i$ use two points:

 $$
 \text{forward: }\hspace{3mm} \frac{df}{dx}\bigg\rvert_{x_i}\approx\frac{f(x_{i+1})-f(x_{i})}{x_{i+1}-x_i}  \hspace{5mm} \text{backward: } \hspace{3mm} \frac{df}{dx}\bigg\rvert_{x_i}\approx\frac{f(x_{i})-f(x_{i-1})}{x_{i}-x_{i-1}}   \hspace{5mm} \text{central } \hspace{3mm} \frac{df}{dx}\bigg\rvert_{x_i}\approx\frac{f(x_{i+1})-f(x_{i-1})}{x_{i+1}-x_{i-1}}
 $$





 ```{figure} figs/derivative_ways.png
 :name: derivative_ways

 graphs illustrating three different ways to approximate derivatives.
 ```

 %% Cell type:markdown id: tags: