In this lab we will explore implicit functions (of two variables), including their graphs, derivatives, and tangent lines.
An example of an implicit function is given by the equation . This equation provides an implicit relation between and . Compare this to the equation , which gives explicitly in terms of .
Graphing an implicit function is fairly simple in Sage using the implicit_plot command. This command requires three arguments: an equation (using double equal sign), a plot range for the first variable, and a plot range for the second variable. I will add the optional "axes=true" and "frame=false" so that axes will be plotted instead of a frame.
Graph (circle of radius 5 centered at the origin).
Now that we can graph these functions, we want to compute the derivative of with respect to . This assumes that is a function of , so we need to tell Sage to assume this as well:
Now we can take the derivative.
Find if .
First, we take the derivative of the whole equation, then we'll solve for .
The diff(y(x),x) is the derivative .
The curvy-looking "d" you get when you use show is the symbol for a partial derivative (you'll learn about those in Calc 3). Since this is Calc 1, you should just think of those as a regular "d."
Now we can solve for the derivative:
This tells us that . [Note: Sage is treating as a function of , so it uses function notation . We usually write just .]
Find when .
Now that we can find the derivative of an implicit function, we can also find tangent lines.
Recall that the line tangent to a function at the point has equation .
Find an equation for the line tangent to the circle given by at the point .
Above we found . So the slope of the tangent line at is .
Thus, an equation for the tangent line is .
Let's graph the implicit function and the tangent line.
Find an equation for the tangent line to the graph of at the point .
We found the derivative above: .
Now we need to substitute and .
I will copy and paste this derivative from the calculation above, and then I will replace x with and y(x) with .
Now that we have the slope, we can find an equation of the tangent line: .
Let's check our answer by graphing:
Here is one final example that puts all the pieces together.
Find the derivative, find the tangent line at , and graph the curve and tangent line.
First, we find the derivative.
Now we define "a" and "b," copy and paste the derivative, and replace x with a and y(x) with b.
Next we define the tangent line, using the answer above for the slope.
Finally, we plot the original function and the tangent line (remember to "reset" y using %var y before graphing).
Sage can also plot an implicit function of three variables. We won't need this for our assignment, but here are a few examples.
[Note: you can make it bigger or smaller with the mouse wheel; click and drag to rotate]
Sphere of radius 5:
This one is a little more interesting: