Calculus > Calculus of Several Variables - Partial Derivatives

Brooks/Cole TI-89 Tools Guide Screen | Keying in the HP Functions | Finding Slope on Surface | Getting Additional Information

Brooks/Cole TI-89 Tools Guide Screen

Here is Brooks/Cole Math Tools Guide TI-89 screen from Chapter 12 Calculus of Several Variables .The description of the first partial derivative of f with respect to y at (a,b) is not shown.

bc_derivatives_smFigure 1 - Chapter 12 Calculus of Several Variables - (Interactive Video Skillbuilder CD-ROM)

Next, we look at using the HP Prime for this page. The function and partial derivatives can be entered directly in the CAS view. This works for HP Prime Lite / Free as well.

Keying in HP Functions f, fx, and fy - Using the Functions

We will use four screen collage approach to explain our HP Prime solutions. The four screen collage allows us to have multiple problems shown in a small area using Textbook display. Press the CAS key to change to CAS view. Use the Clear key to erase any previous work.

Key in the function, f(x,y):=, as shown in the command line of upper left screen, figure 2. Key in the partial derivatives as shown in the upper right screen, figure 2. To key in the line 1’s  partial derivative function, type fx(x,y):=. Use the partial derivate template from the Math palette and key in partial derivative as shown. To key in the line 2’s  partial derivative function, highlight the first line's content and use soft key Copy to move to the command line. Edit, so that we have  fy(x,y):=. with partial derivative shown as being with respect to y..

In the lower left screen, figure 2, we see that line 3 and 5 give us the partial derivative results. Key in  fx(x,y) and fy(x,y)  respecively. In he lower right screen, figure 2, we substitute in 1 and 2 into fx and fy to get -2 and 8.

Note: Unlike pivmat on the Finite Math, Simplex section, Begin return End were unnecessary and the HP Connectivity Kit did not crashed when Kit tried to display Pro/Virtual CAS Vars f, fx, and fy.

partial_derivativeFigure 2 - HP Prime partial derivaatives

Finding the Slope on a Surface in the x and y Directions

We will use four screen collage approach to explain this. The four screen collage allows us to have multiple problems shown in a small area using Textbook display. Press the Apps key to bring up the Application Library screen. Select the Graph 3D app.

Note: The Graph 3D app is not available to the Lite / Free HP Prime application. As shown above in the lower right screen, figure 2, we can calculate the slopes value for fx and fv with the Lite / Free HP Prime app. However, without the Graph 3D app, we are unable to picture the geometric interpretation of the tangent lines on the surface in the x and y directions.

Key in the functions, FZ1(X,Y), FZ2(X,Y), and FZ3(X,Y) as shown in the upper left screen, figure 3. The second and third functions allow us to draw very close approximations of the y plane y=2 and x plane x=1. The approximate planes are shown in the lower left screen and lower right screen, figure 3. Before leaving the Graph 3D Symbolic view, we need to temporarily disable entry three, FZ3(X,Y) by unchecking its selection box.

Press the Plot Setup key and enter values as own in the upper right screen, figure 3. Press the Plot key to bring up the lower left screen, figure 3. Rotate the 3D graph until the axis are as shown by the miniature x,y,z icon. Press soft keys Menu, and GoTo. Enter 1 for X and 2 for Y on the Go To Screen (not show).Press the soft key OK. Press the soft key Menu again. Notice the cross hair at FZ1(1,2)=5. its location is labeled at the bottom. The labeled point, that is the cross hair, is on the intersection of FZ1 graph and Y=2 plane. Picture a tangent line to the surface with slope of -2 at this point and on the Y=2 plane. The -2 is our value for fx(1,2) in lower right screen, figure 2.

Uncheck FZ2(X,Y) and check FZ3(X,Y) in upper left screen, figure 3. Press the Plot key to bring up the lower right screen, figure 3. Notice the cross hair at FZ1(1,2)=5. With the colors of the surface and the plane, the cross hair is hard to see. Its location is labeled at bottom. The labeled point, that is the cross hair, is on the intersection of FZ1 graph and X=1 plane. Picture a tangent line to the surface with slope of 8 at this point and on the X=1 plane. The 8 is our value for fy(1,2) in lower right screen, figure 2.

graph_3d_part_derFigure 3 - HP Prime Graph 3D partial derivaatives

Getting Additional Information

At the present time only the HP Prime Guide Algebra Fundamentals is available in eBook and Print. As demonstrated by this example, the Finite Math, and Trigonometry examples, the HP Prime Free / Lite, Pro, virtual and handheld are able to perform anything the TI-89 could perform. With programming and Apps added, the HP Prime options are unlimited.

I am working on material for additional volumes. For now you can Google HP Prime and a topic for additional information. You can also see threads such as my thread on Xcas Youtube videos that has 52 screens shots of math examples with multiple commands per screen shot, that can provide you with some assistance with all sorts of math topics.

 
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