Skip to main content

Calculus Small Changes, Big Ideas

Bill Shillito

Section 12.4 Taylor approximations in higher dimensions

Introduction goes here.
🔗
  • Begin by recalling the single-variable Taylor expansion:
    \begin{equation*} f(x) \approx f(a) + f'(a)(x-a) + \frac{1}{2} f''(a)(x-a)^{2}\text{.} \end{equation*}
    🔗
    The tangent line is just the first-order approximation, and adding higher-order terms improves accuracy.
    🔗
    🔗
  • Connect to the previous section: The tangent plane
    \begin{equation*} z \approx z_{0} + \pdv{f}{x}(x-x_{0}) + \pdv{f}{y}(y-y_{0}) \end{equation*}
    is the first-order Taylor polynomial for \(f(x,y)\text{.}\)
    🔗
    🔗
  • Introduce second-order terms using higher partial derivatives:
    \begin{equation*} f(x,y) \approx f(x_{0},y_{0}) + \pdv{f}{x}(x-x_{0}) + \pdv{f}{y}(y-y_{0}) \end{equation*}
    \begin{equation*} \quad + \frac{1}{2} \pdv[2]{f}{x}(x-x_{0})^{2} + \pdv[2]{f}{x}{y}(x-x_{0})(y-y_{0}) + \frac{1}{2} \pdv[2]{f}{y}(y-y_{0})^{2}. \end{equation*}
    🔗
    🔗
  • Interpret each term:
    • First-order terms describe tilt.
      🔗
      🔗
    • Pure second derivatives describe curvature in coordinate directions.
      🔗
      🔗
    • Mixed partials describe interaction between variables.
      🔗
      🔗
    🔗
    🔗
  • Emphasize symmetry of mixed partials when conditions allow (Clairaut’s theorem).
    🔗
    🔗
  • Provide a worked example:
    • Compute first and second partial derivatives at a point.
      🔗
      🔗
    • Write the quadratic Taylor approximation.
      🔗
      🔗
    • Use it to estimate a nearby value.
      🔗
      🔗
    • Compare to the actual value.
      🔗
      🔗
    🔗
    🔗
  • Geometric meaning:
    • First-order: tangent plane.
      🔗
      🔗
    • Second-order: quadratic surface approximating curvature.
      🔗
      🔗
    • The quadratic form determines local shape.
      🔗
      🔗
    🔗
    🔗
  • General multivariable Taylor expansion:
    \begin{equation*} f(x,y) = \sum_{k=0}^{n}\text{(all $k$th-order partial terms in $(x-x_{0}),(y-y_{0})$)}+ R_{n},\text{,} \end{equation*}
    where each term involves partial derivatives of total order \(k\) evaluated at \((x_{0},y_{0})\text{.}\)
    🔗
    Each degree \(k\) term consists of all products
    \begin{equation*} (x-x_{0})^{i} (y-y_{0})^{j} \end{equation*}
    with \(i+j=k\text{,}\) multiplied by the corresponding partial derivative
    \begin{equation*} \frac{\partial^{k}f}{\partial x^{i}\partial y^{j}}(x_{0},y_{0}). \end{equation*}
    🔗
    The approximation improves as higher-order terms are included.
    🔗
    🔗
🔗
PrevTopNext