[Analysis] More examples

semester3/analysis-ii/cheat-sheet-rb/parts/05_diff.tex

@@ -291,6 +291,19 @@ $$
         & T_2f(y;x_0) = f(x_0) + \nabla f(x_0) \cdot y + \frac{1}{2} \Bigl( y^\top \cdot \textbf{H}_f(x_0) \cdot y\Bigr)
 \end{align*}
 
+\begin{footnotesize}
+    \textbf{Example:} Approximate $f(x,y) = x\sqrt{y}$ at $f(1.1, 4.4)$ using $(1, 4)$.
+    \begin{align*}
+        T_1f(y;(1, 4)) &= f(1,4) + \nabla f(1, 4) \cdot (y_1-1, y_2-4) \\
+        &= 2 + \begin{bmatrix}
+            2 \\
+            \frac{1}{4}
+        \end{bmatrix} \cdot (y_1-1, y_2-4) \\
+        &= 2 + 2(y_1-1) + \frac{1}{4}(y_2-4)
+    \end{align*}
+    Thus $T_1f\bigl((1.1, 4.4); (1, 4)\bigr) = 2 + 2(1.1-1) + \frac{1}{4}(4.4-4) = 2.3$
+\end{footnotesize}
+
 \method Calculating $T_kf(y;x_0)$ also yields $\textbf{H}_f$ for $k \geq 2$.
 \begin{align*}
     & T_2f((x_0,y_0);(x,y)) = \ldots + ax^2 + by^2 + cxy \\