Calculus Small Changes, Big Ideas

Section 12.2 Tangent plane approximations

• Start with total differential:

  \begin{equation*} \dd z=\pdv{f}{x}\dd{x}+\pdv{f}{y}\dd{y} \end{equation*}

  Show how this can be used to approximate changes in \(z\text{:}\)

  \begin{equation*} \Delta\,z \approx\pdv{f}{x}\,\Delta x+\pdv{f}{y}\,\Delta y \end{equation*}

  🔗
  🔗
• If \(\Delta x=x-x_0\) etc, we get:

  \begin{equation*} z-z_0=\pdv{f}{x} \qty(x-x_0)+\pdv{f}{y} \qty(y-y_0) \end{equation*}

  This is an equation of the tangent plane to \(z=f(x,y)\) at the point \((x_{0},y_{0},z_{0})\text{.}\)

  🔗
  🔗
• Interpret geometrically:

  • The surface is locally approximated by a plane.

    🔗
    🔗
  • The coefficients \(\pdv{f}{x}\) and \(\pdv{f}{y}\) represent slopes in the \(x\)- and \(y\)-trace directions.

    🔗
    🔗
  • The tangent plane matches the surface exactly at the base point and closely nearby.

    🔗
    🔗

  🔗
  🔗
• Emphasize the connection to single-variable calculus: The tangent plane is the two-variable analogue of the tangent line

  \begin{equation*} y=f(a)+f'(a)(x-a). \end{equation*}

  🔗
  🔗
• Include a worked example:

  • Compute partial derivatives at a point.

    🔗
    🔗
  • Write the tangent plane equation.

    🔗
    🔗
  • Use it to approximate a nearby function value.

    🔗
    🔗
  • Compare to the actual value to assess accuracy.

    🔗
    🔗

  🔗
  🔗
• Briefly discuss limitations:

  • Accuracy improves as \((x,y)\) approaches \((x_{0},y_{0})\text{.}\)

    🔗
    🔗
  • Non-differentiable points do not admit a tangent plane.

    🔗
    🔗
  • Curvature effects motivate higher-order approximations.

    🔗
    🔗

  🔗
  🔗