Small Group Activity: 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

Instructor's Guide

Main Ideas

  • 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

Prerequisite Knowledge

Students should be able to:
  • Use the measurement tool to approximate a derivative from a plastic surface

Props/Equipment

  • Tabletop Whiteboard with markers
  • Plastic surface with contour map and inclinometer
  • A handout for each student

Student Conversations

  • Some students may believe that vectors are tied to the coordinate axes, and not the contour maps; that moving the coordinate axes moves the vector along with them so that no matter the direction of the coordinate axes, the vector is always pointing in the same direction relative to those axes.
  • Some students may believe that the gradient will always point towards the top of the hill.
  • Some groups may make choose arbitrary vectors that trivialize the calculations in this activity.
  • Some students may believe that the gradient is a vector's property and not a point's property; that every vector at a given point has its own, but not necessarily unique, gradient. These students may justify their reasoning by saying that the steepest direction for any given vector must be along that vector.
  • Some students may believe that longer vectors always have a greater slope / rate of change than shorter vectors, or that vector length is what denotes magnitude of slope / rate of change; that a vector's rate of change and its magnitude represent the same value.
  • Some students may believe that every vector pointing in the same direction as the gradient is identical to the gradient vector.
  • Some students may believe that a short vector on the contour map will always correspond to a short vector on the surface.
  • Some students may believe that taking the dot product of two vectors produces a vector.
Keywords
Directional derivatives
Learning Outcomes