diff --git a/blog/2024/taylor-series/content.md b/blog/2024/taylor-series/content.md index 04541bf..a1b5f4d 100644 --- a/blog/2024/taylor-series/content.md +++ b/blog/2024/taylor-series/content.md @@ -14,25 +14,35 @@ need a more general method. We can approximate some non-polynomial functions by constructing a polynomial with the _same derivatives_ as the function. This is called a _Taylor Polynomial_. -> [!NOTE] In general, if $c \neq 0$, it's called a Taylor Polynomial. If $c = 0$, then it's a + +> [!NOTE] +> In general, if $c \neq 0$, it's called a Taylor Polynomial. If $c = 0$, then it's a > Maclaurin Polynomial. -> [!CAUTION] test lorem ipsum dolor sit amet consectetur adipiscing elit sed do eiusmod tempor + +> [!CAUTION] +> test lorem ipsum dolor sit amet consectetur adipiscing elit sed do eiusmod tempor > incididunt ut labore et dolore magna aliqua. Ut enim ad minim veniam quis nostrud exercitation > ullamco laboris nisi ut aliquip ex ea commodo consequat. Duis aute irure dolor in reprehenderit in > voluptate velit esse cillum dolore eu fugiat nulla pariatur. -> [!WARNING] test lorem ipsum dolor sit amet consectetur adipiscing elit sed do eiusmod tempor + +> [!WARNING] +> test lorem ipsum dolor sit amet consectetur adipiscing elit sed do eiusmod tempor > incididunt ut labore et dolore magna aliqua. Ut enim ad minim veniam quis nostrud exercitation > ullamco laboris nisi ut aliquip ex ea commodo consequat. Duis aute irure dolor in reprehenderit in > voluptate velit esse cillum dolore eu fugiat nulla pariatur. -> [!TIP] test lorem ipsum dolor sit amet consectetur adipiscing elit sed do eiusmod tempor + +> [!TIP] +> test lorem ipsum dolor sit amet consectetur adipiscing elit sed do eiusmod tempor > incididunt ut labore et dolore magna aliqua. Ut enim ad minim veniam quis nostrud exercitation > ullamco laboris nisi ut aliquip ex ea commodo consequat. Duis aute irure dolor in reprehenderit in > voluptate velit esse cillum dolore eu fugiat nulla pariatur. -> [!IMPORTANT] test lorem ipsum dolor sit amet consectetur adipiscing elit sed do eiusmod tempor + +> [!IMPORTANT] +> test lorem ipsum dolor sit amet consectetur adipiscing elit sed do eiusmod tempor > incididunt ut labore et dolore magna aliqua. Ut enim ad minim veniam quis nostrud exercitation > ullamco laboris nisi ut aliquip ex ea commodo consequat. Duis aute irure dolor in reprehenderit in > voluptate velit esse cillum dolore eu fugiat nulla pariatur. @@ -77,7 +87,9 @@ You can confirm that this polynomial has the same first, second, and third deriv Taking repeated derivatives like this leads to a common pattern in all Taylor polynomials. -> [!NOTE] We use the notation $P_n(x)$ to denote the $n^{th}$ Taylor polynomial + +> [!NOTE] +> We use the notation $P_n(x)$ to denote the $n^{th}$ Taylor polynomial Taylor polynomials take the form: @@ -132,13 +144,17 @@ $$ z \in [c,\,x] $$ -> [!TIP] This is a fancy way of saying that $z$ is between $c$ and $x$. + +> [!TIP] +> This is a fancy way of saying that $z$ is between $c$ and $x$. $$ \text{Error} = \left|R_n(x)\right| $$ -> [!NOTE] > $R_n$ would be the "next term" in $P_n(x)$ except we put $z$ instead of $c$. + +> [!NOTE] +> $R_n$ would be the "next term" in $P_n(x)$ except we put $z$ instead of $c$. #### Applying to $e^x$