$\newcommand{\dis}{\displaystyle} \newcommand{\m}{\hspace{1em}} \newcommand{\mm}{\hspace{2em}} \newcommand{\x}{\vspace*{1ex}} \newcommand{\xx}{\vspace*{2ex}} \let\limm\lim \renewcommand{\lim}{\dis\limm} \let\fracc\frac \renewcommand{\frac}{\dis\fracc} \let\summ\sum \renewcommand{\sum}{\dis\summ} \let\intt\int \renewcommand{\int}{\dis\intt} \newcommand{\sech}{\text{sech}} \newcommand{\csch}{\text{csch}} \newcommand{\Ln}{\text{Ln}} \newcommand{\p}{\partial} $

Lecture 15, 9/26/2022. This page is for Section 1 only.
ACMS 20550: Applied Mathematics Method I
Instructor: Bei Hu, b1hu@nd.edu, Hurley 174A

$\hspace{.6em}$ Went over Mathlab lesson #3.

  1. More chain rules: putting everything together.
      Review partial derivatves when another variable is fixed.
    1. Example: Finding $\Big(\frac{\p z}{\p x}\Big)_r$ for $ z = e^{2x} + \sin y^2$.
      (a) Reduce the expression to one $\fcolorbox{white}{yellow}{containing $x$ and $r$ only}$. (b) Differentiate.
      Sol. (a) $ z = e^{2x} + \sin y^2 = e^{2x} + \sin (y^2+x^2 -x^2) = e^{2x} +\sin(r^2 -x^2)$.
      $\m$ (b) $\Big(\frac{\p z}{\p x}\Big)_r = 2 e^{2x} - 2x \sin(r^2 -x^2)$.
    2. Example: Find the tangent line of $xy^3-yx^3 =6$ at $(1,2)$. - working with implicit differentiations:
      Sol. Diff on both sides with respect to $x$,
      $ \Big(y^3+ x \cdot 3y^2 \frac{dy}{dx}\Big)-\Big(\frac{dy}{dx} x^3 + y \cdot 3x^2 \Big) =0$.
      Plug in $x=1$, $y=2$: $\mm 8 + 12 \frac{dy}{dx} - \frac{dy}{dx} - 6 = 0,\m$ i.e., $\mm\frac{dy}{dx} = - \frac2{11}$.
      Tangent line: $ y-2 = - \frac{2}{11}(x-1)$.
    3. Example: If $z=x^2y^3$ and $x=\sin(s+t)$, $y = e^{s-t}$. Find $\frac{\partial z}{\partial s}$. Use differentials.
    4. Example: Went over the example 3 on page 205. Use differentials.