Student handout: Directional Derivatives

Vector Calculus I 2022
This small group activity using surfaces relates the geometric definition of directional derivatives to the components of the gradient vector. Students work in small groups to measure a directional derivative directly, then compare its components with measured partial derivatives in rectangular coordinates. The whole class wrap-up discussion emphasizes the relationship between the geometric gradient vector and directional derivatives.
What students learn Measuring slope along various directions at a point on the surface. Experimentally determining the gradient at a point on the surface. Using the Master Formula.

  1. Measurement
    1. Find the rate of change in the surface in the \(x\)-direction at the blue dot on your surface. Include units. \[ \frac{\partial{f}}{\partial{x}} = \underline{\hspace{2in}} \]
    2. Find the rate of change in the surface in the \(y\)-direction at the blue dot on your surface. Include units. \[ \frac{\partial{f}}{\partial{y}} = \underline{\hspace{2in}} \]
    3. Draw an arbitrary vector \(\boldsymbol{\vec u}\) at the blue dot on the contour mat. What are its components? \[ \boldsymbol{\vec u} = \underline{\hspace{2in}} \]
    4. Find the rate of change in the surface in the \(\boldsymbol{\vec u}\)-direction. Include units. \[ \frac{df}{ds} = \underline{\hspace{2in}} \]
  2. Computation
    1. Determine the gradient of \(f\) at the blue dot. \[ \boldsymbol{\vec{\nabla}} f = \underline{\hspace{2in}} \]
    2. Use the Master Formula to express \(\frac{df}{ds}\) in terms of \(\boldsymbol{\vec\nabla} f\), and compute the result. \[ \frac{df}{ds} =\underline{\hspace{2in}} \]
  3. Comparison
    • Compare your answers.


Copyright 2014 by The Raising Calculus Group

Keywords
Directional derivatives
Learning Outcomes