Subsections

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

MA 1024, Partial derivatives

Purpose

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:

\\filer\calclab

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

Pardiff_start_C13.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.

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

2. Given the function
   
   $$k(x,y)=2xy^3-3x^3y+y-x^2y^2-xy^2$$
   
   
   do the following:

   a)

   Plot the function and the plane $y=-4$ on the same graph. Use plotting ranges $-5 \leq x \leq 5$, $-5 \leq y \leq 5$, $-500 \leq z \leq 500$.

   b)

   Graph the two-dimensional intersection of the plane $y=-4$. Be sure to use $x$ and $z$ ranges that are consistent with your ranges in part a.

   c)

   Does your two-dimensional graph look like the intersection from your tree-dimensional graph?

3. Given:
   
   $$j(x,y) = \frac{x+y}{\sqrt{1+9x^2+9y^2}}$$
   
   
   , find the equation of the plane tangent to the surface at
   
   $$(-\frac{1}{2},-\frac{1}{2})$$
   
   
   . Plot the surface and the tangent plane on the same graph over the intervals $-3 \leq x \leq 3$, $-3 \leq y \leq 3$.

4. Given:
   
   $$j(x,y) = \frac{x+y}{(1+2x^2+2y^2)}$$
   
   

   a)

   Find all points on the graph where the tangent plane is horizontal.

   b)

   Find the equation of each horizontal plane and plot them on the same graph as the function $j(x,y)$. Use plotting ranges $-3 \leq x \leq 3$, $-3 \leq y \leq 3$.