Polar Derivatives on TI-Nspire CAS

The following question about how to compute derivatives of polar functions was posted on the College Board’s AP Calculus Community bulletin board today.

From what I can tell, there are no direct ways to get derivative values for polar functions.  There are two ways I imagined to get the polar derivative value, one graphically and the other CAS-powered.  The CAS approach is much  more accurate, especially in locations where the value of the derivative changes quickly, but I don’t think it’s necessarily more intuitive unless you’re comfortable using CAS commands.  For an example, I’ll use $r=2+3sin(\theta )$ and assume you want the derivative at $\theta = \frac{\pi }{6}$.

METHOD 1:  Graphical

Remember that a derivative at a point is the slope of the tangent line to the curve at that point.  So, finding an equation of a tangent line to the polar curve at the point of interest should find the desired result.

Create a graphing window and enter your polar equation (menu –> 3:Graph Entry –> 4:Polar).  Then drop a tangent line on the polar curve (menu –> 8:Geometry –> 1:Points&Lines –> 7:Tangent).  You would then click on the polar curve once to select the curve and a second time to place the tangent line.  Then press ESC to exit the Tangent Line command.

To get the current coordinates of the point and the equation of the tangent line, use the Coordinates & Equation tool (menu –> 1:Actions –> 8:Coordinates and Equations).  Click on the point and the line to get the current location’s information.  After each click, you’ll need to click again to tell the nSpire where you want the information displayed.

To get the tangent line at $\theta =\frac{\pi }{6}$, you could drag the point, but the graph settings seem to produce only Cartesian coordinates.  Converting $\theta =\frac{\pi }{6}$ on $r=2+3sin(\theta )$ to Cartesian gives

$\left( x,y \right) = \left( r \cdot cos(\theta ), r \cdot sin(\theta ) \right)=\left( \frac{7\sqrt{3}}{4},\frac{7}{4} \right)$ .

So the x-coordinate is $\frac{7\sqrt{3}}{4} \approx 3.031$.  Drag the point to find the approximate slope, $\frac{dy}{dx} \approx 8.37$.  Because the slope of the tangent line changes rapidly at this location on this polar curve, this value of 8.37 will be shown in the next method to be a bit off.

Unfortunately, I tried to double-click the x-coordinate to set it to exactly $\frac{7\sqrt{3}}{4}$, but that property is also disabled in polar mode.

METHOD 2:  CAS

Using the Chain Rule, $\displaystyle \frac{dy}{dx} = \frac{dy}{d\theta }\cdot \frac{d\theta }{dx} = \frac{\frac{dy}{d\theta }}{\frac{dx}{d\theta }}$.  I can use this and the nSpire’s ability to define user-created functions to create a $\displaystyle \frac{dy}{dx}$ polar differentiator for any polar function $r=a(\theta )$.  On a Calculator page, I use the Define function (menu –> 1:Actions –> 1:Define) to make the polar differentiator.  All you need to do is enter the expression for a as shown in line 2 below.

This can be evaluated exactly or approximately at $\theta=\frac{\pi }{6}$ to show $\displaystyle \frac{dy}{dx} = 5\sqrt{3}=\approx 8.660$.

Conclusion:

As with all technologies, getting the answers you want often boils down to learning what questions to ask and how to phrase them.