# Partial Derivatives: Geometric Visualization

#### Summary

This write-pair-share activity presents Calculus III students with a worksheet containing several exercises that require them to find partial derivatives of functions of two variables. Afterwards, a series of Web-based animations are used to illustrate the surface of each function, the path of the indicated partial derivative for a specified value of the variable, and the value of the derivative at each point along the path.

Students find it difficult if not impossible to visualize most three-dimensional surfaces without assistance; the Web-based animation gallery provides an excellent source of visual illustrations that allow students to connect their abstract mathematical computations with geometric representations.

## Learning Goals

To enable students to:

- exercise the skill of finding partial derivatives of functions of two variables
- develop their understanding of the geometric nature of partial derivatives
- deepen their understanding of derivative as the slope of a tangent line to a curve

## Context for Use

This activity can be carried out in a small class or in a large lecture setting anytime during or after students have been introduced to the concept of partial derivatives. This activity takes approximately 30 minutes to complete, not including the final phase involving additional practice.

## Description and Teaching Materials

Activity description

- Students are given a write-pair-share activity worksheet (Acrobat (PDF) 26kB Aug3 06) and allowed time to work together in pairs. For several functions of two variables, students are directed to find indicated partial derivates. (~5-10 minutes)
- Afterwards, the instructor reviews the correct answers with the students in order to correct any misunderstandings concerning the process of finding partial derivatives. (~5 minutes)
- In addition, the instructor displays Web-based animations of each function and a geometric view of its indicated partial derivative by using the gallery of animations listed below. This allows students to clearly and easily see the geometric nature of a partial derivative. (~20 minutes)
- Partial Derivative Geometrically Gallery

A gallery of colorful and effective animations of partial derivatives for a small collection of functions of two variables

- Partial Derivative Geometrically Gallery

- MERLOT description of this resource

http://www.merlot.org/artifact/ArtifactDetail.po?oid=1010000000000324300 - Lastly, students are directed to practice additional exercises involving partial derivatives through an interactive Java applet. This practice can be assigned for homework or can take place in a computer lab or in a wireless classroom with laptops.
- Calculus on the Web (COW)

A collection of interactive calculus exercises

https://math.temple.edu/~reich/cowdemo/cowhome

Access path: Calculus Book III/Functions/Derivatives/Partial Derivatives - MERLOT description of this resource

https://www.merlot.org/merlot/viewMaterial.htm?id=83836&hitlist=keywords%3DCalculus%2520on%2520the%2520Web&fromUnified=true

- Calculus on the Web (COW)

## Teaching Notes and Tips

While students can generally perform the computations necessary to find partial derivatives, their geometric understanding is very limited. The animation gallery is an excellent resource for developing their conceptual and geometric understanding.

Each animation consists of three parts: a text description, a slicing of the surface, and the tangent line; each part takes approximately 20 seconds and each animation lasts for about a minute.

In the third part of an animation, it is instructive to have students note the values of the derivative in relation to the values of the pertinent independent variable. This reinforces the sometimes overlooked fact that partial derivatives are themselves functions of a variable and that their values vary along with those of the associated independent variable. It is helpful to replay an animation and focus student attention on these values.

It is also helpful to point out to students that the slice through the surface results in a subset of points on the surface of the original function; however, this curve on the surface is not the graph of the partial derivative. This can be clearly demonstrated via the first function in the gallery: f(x,y)= -x^2+y in which the partial derivative with respect to y is f(y)=1. The graph of the partial derivative is a horizontal line while the slice through the surface reveals a graph of a line having slope of 1.

## Assessment

This activity is intended to develop geometric understanding and is ungraded; students can be awarded credit for participation if desired.

## References and Resources

MIT OpenCourseWare-Calculus by Gilbert Strang

This online textbook provides explanatory material on partial derivatives.

MERLOT description of MIT OpenCourseWare-Calculus

Direct link to MIT OpenCourseWare-Calculus (Access path: Chapter 13/Sections 13.1-13.4/13.2 Partial Derivatives, p.277-281)