Calc III - Graphing a Function of Multiple Variables by hand

In summary, the conversation discusses how to graph equations by hand, specifically focusing on the example of z = sin(xy). The speaker suggests setting z equal to a value and solving for y in terms of x, and also mentions the use of a calculator for this method. They also mention the concept of contour maps and how to plot curves on the x, y plane.
  • #1
Cloudless
15
0
I come across questions where I have to match the equation with its contour map and graph.

Examples:

z = sin(xy)

z = sin(x-y)


Right now I'm using Wolfram Alpha for all of these, but supposing these appear on an exam... how do I graph it by hand? :confused: For example, for z = sin(xy) I'm just plugging in random values for x and y... but doesn't that mean the entire xy graph should be filled then?
 
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  • #2
A contour of (for example) sin(xy) is a curve on the x, y plane where this function is constant. This in turn means that the product xy must be constant. So you could write

c = xy
y = c/x

And for each c you get a different contour. Plotting the curves once you have them in that form is probably a little more familiar to you.
 
  • #3
Cloudless said:
I come across questions where I have to match the equation with its contour map and graph.

Examples:

z = sin(xy)

z = sin(x-y)


Right now I'm using Wolfram Alpha for all of these, but supposing these appear on an exam... how do I graph it by hand? :confused: For example, for z = sin(xy) I'm just plugging in random values for x and y... but doesn't that mean the entire xy graph should be filled then?
Set z equal to some value (-1 ≤ z ≤ 1, why?), then solve for y in terms of x.
 
  • #4
If you're doing it by hand I think it's easier to do what I wrote, for a few simple c values and afterwards evaluate what z is on each curve. Of course your way works too.
 
  • #5
A contour of (for example) sin(xy) is a curve on the x, y plane where this function is constant. This in turn means that the product xy must be constant. So you could write

c = xy
y = c/x

And for each c you get a different contour. Plotting the curves once you have them in that form is probably a little more familiar to you.

Wow, this is brilliant @_@ Thank you

Set z equal to some value (-1 ≤ z ≤ 1, why?), then solve for y in terms of x.
You would need a calculator for this method though, right? Inverse sin of (-1 to 1) = xy.
 

1. How do I graph a function of multiple variables by hand?

To graph a function of multiple variables by hand, you will need to create a three-dimensional coordinate system with x, y, and z axes. Then, plug in different values for each variable and plot the corresponding point on the graph. Once you have several points, you can connect them to create a 3D surface that represents the function.

2. What is the purpose of graphing a function of multiple variables by hand?

Graphing a function of multiple variables by hand allows you to visualize the relationship between the variables and the output of the function. It can also help you identify any patterns or trends in the data.

3. What is the difference between graphing a function of one variable and multiple variables?

Graphing a function of one variable results in a 2D graph, while graphing a function of multiple variables results in a 3D graph. This is because multiple variables add an extra dimension to the graph.

4. Are there any shortcuts for graphing a function of multiple variables by hand?

While there are no shortcuts for graphing a function of multiple variables by hand, there are some techniques that can make the process easier. For example, you can use a table of values to plot points instead of calculating them manually, or you can use technology like a graphing calculator to help you visualize the function.

5. What are some common mistakes to avoid when graphing a function of multiple variables by hand?

Some common mistakes to avoid when graphing a function of multiple variables by hand include not labeling the axes, not using a consistent scale for each axis, and not plotting enough points to accurately represent the function. It's also important to double-check your calculations and make sure you are using the correct equations for the function.

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