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