Subsections

• Purpose
• Getting Started
• Background
  • Partial derivatives
  
  
• Exercises

Partial Derivatives and their Geometric Interpretation

Purpose

The purpose of this lab is to acquaint you with using Maple to compute partial derivatives.

Getting Started

To assist you, there is a worksheet associated with this lab that contains examples. You can copy that worksheet to your home directory with the following command, which must be run in a terminal window, not in Maple.

cp /math/calclab/MA1024/Partials_start.mws ~/My_Documents

You can copy the worksheet now, but you should read through the lab before you load it into Maple. Once you have read to the exercises, start up Maple, load the getting started worksheet and go through it carefully. Then you can start working on the exercises.

Background

For a function $f(x)$ of a single real variable, the derivative $f'(x)$ gives information on whether the graph of $f$ is increasing or decreasing. Finding where the derivative is zero was important in finding extreme values. For a function $F(x,y)$ of two (or more) variables, the situation is more complicated.

Partial derivatives

A differentiable function, $F(x,y)$, of two variables has two partial derivatives: $\partial F /\partial x$ and $\partial F /\partial y$. As you have learned in class, computing partial derivatives is very much like computing regular derivatives. The main difference is that when you are computing $\partial F /\partial x$, you must treat the variable $y$ as if it was a constant and vice-versa when computing $\partial F /\partial y$.

The Maple commands for computing partial derivatives are D and diff. The Getting Started worksheet has examples of how to use these commands to compute partial derivatives.

Exercises

1. Compute the three distinct second order partial derivatives of
   
   $$f(x,y)=\frac{\sin(x+y)}{(1+y^2)}$$
   
   
   at the point $(-1,1)$ using the diff command and then again using the D command.

2. Given the function
   
   $$k(x,y)=4x^3y-5xy^3+x-x^2y^2-x^2y$$
   
   

   a)

   Plot the function and the plane $y=-4$ on the same graph. Use intervals $-5 \leq x \leq 5$, $-5 \leq y \leq 5$, $-700 \leq z \leq 700$ and don't use scaling=constrained.

   b)

   Find the derivative of $k$ in the $y=-4$ plane.

   c)

   Graph the two-dimensional intersection of the plane $y=-4$ and $k$ on the same graph.

   d)

   Does your two-dimensional graph look like the intersection from your three-dimensional graph? Be sure to use the same ranges to properly compare and rotate the 3-D graph.

3. Find two points at which the plane tangent to the surface given by the function $$z=\frac{\sin(x)}{y^2+1}$$ is horizontal. Use Maple's solve command to find one of them (it will only yield one solution even though there are infinitely many) and use Maple's fsolve command to find another critical point in the ranges $2 \leq x \leq 6$ and $0 \leq y \leq 5$. Find both tangent planes and plot them along with the function on the same graph over the intervals $0 \leq x \leq 2 \pi$ and $-1 \leq y \leq 1$. Rotate the graph so that you can see that the tangent planes are horizontal.