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 and even solutions to some of the exercises. You can copy that worksheet to your home directory by going to your computer's Start menu and choose run. In the run field type:

\\storage\academics\math\calclab

when you hit enter, you can then choose MA1024 and then choose the worksheet

Pardiff_start_B14.mw

Remember to immediately save it in your own home directory. Once you've copied and saved the worksheet, read through the background on the internet and the background of the worksheet before starting 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
   
   $$f(x,y)=\cos(2y) + \sin(x^2-y^2)$$
   
   
   at the point $$(\frac{\pi}{6},-\frac{\pi}{6})$$ using the diff command and then again using the D command.

2. Given the same function
   
   $$f(x,y)=\cos(2y) + \sin(x^2-y^2)$$
   
   

   a)

   Plot the function and the plane $$x=\frac{\pi}{2}$$ on the same graph. Use intervals $-3 \leq x \leq 3$, $-3 \leq y \leq 3$, $-3 \leq z \leq 3$.

   b)

   Find the derivative of $f$ in the $$x=\frac{\pi}{2}$$ plane. Evaluate this derivative at $y=2$ and then find the equation of the line tangent to the two-dimensional intersection of the plane $$x=\frac{\pi}{2}$$ and $f$ at the point $$(\frac{\pi}{2},2)$$.

   c)

   Plot the tangent line and the two-dimensional intersection of the plane $$x=\frac{\pi}{2}$$ and $f$ on the same graph. Be sure to use $y$ and $z$ ranges that are consistent with your ranges in part a.

   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 the equation of the plane tangent to
   
   $$f(x,y)=\frac{x \sin(x+y)}{x^2 + y^2 +4}$$
   
   
   at the point $(-1,2,z)$. Plot the function $f(x,y)$ and the tangent plane over the intervals $-3 \leq x \leq 1$ and $0 \leq y \leq 4$ and rotate the plot so that you can see that the plane is tangent.