Lesson Plan: Geometric Approach to Derivatives

Brian Swanagan

**Student Audience:**

This lesson is primarily for students in a calculus course who are soon to learn about derivatives and have a grasp on some geometric properties relating circles and tangents.

**Objectives:**

1. To explore finding tangents for curves using DescartesŐ approach involving circles.

2. To demonstrate how different topics in mathematics can be related and helpful.

**Materials:**

1. Projector for computer

2. Graphing Calculator program

**Lesson:**

Teacher reviews the relationship between tangents on a circle and the radius of the circle by having the students recall and explain what they remember. Students then can discuss whether they think curves in general may have tangents. The teacher can also graph a circle on Graphing Calculator and then zoom in to demonstrate that curves often are fairly linear locally.

Then
the teacher should graph the equation of a particular curve, y=2x^{.5},
and ask the students to consider the point (1,2) on the curve in order to find
a tangent at that point. He/she
will also graph the circle equation (x-n)^{2}
+ y^{2} = (1-n)^{2} + 4 but not show the equation to the
students. Then, tell the students
that the circle they see passes through the point (1,2) but has a center along
the x-axis. You can hit the play
button at the bottom to make the values for n change and so students can see
the circle move back and forth along the x-axis. Have the students attempt to determine an equation for the
circle. Have students present some
attempts and possibilities.

When scrolling through different values for n, the students can see that the circle seems to fit better along the curve then at other times. With a zoomed in view on the point where the curve is fairly linear, they can observe the circle (which will also appear to be fairly linear in the zoomed view) running along the curve only for an instant and then clearly crossing with a different slope at other times. Have the students note some observations and draw their focus to when the circle appears to only touch once locally where the circle also appears to fit more closely to the curve. We can then find the tangent of the circle at this point for the closest fitting circle in order to determine the tangent of the curve.

First,
they will have to find a circle that only has one intersection with the
curve. So, they will have to solve
for n using the two equations.
Substitute 2x^{.5} for y and create a quadratic equation with
constants in terms of n. Ask the
students to think about when quadratics only have one
solution. Someone will likely
realize that the determinant equals zero so you can then solve b^{2}
– 2ac = 0 to find the value of n which will give you the location of the
center of the circle.

Ask the students to think about how we can then find the equation of the tangent line using this information. You may need to scaffold by reminding them that you could find the tangent at a point of a circle using methods mentioned earlier in class (they can find the radius through the point and the center and find the equation of the perpendicular line through that point). In fact, you may wish to graph the equations of the line through the point and center and its perpendicular through the point afterwards generally in terms of n. Then, let n change and see whether the tangent line fits the curve well enough at that point when the n hits the determined value and not so well elsewhere.

The students may wish to try the method they have discovered for other points to see if it works. Allow them to explore on Graphing Calculator if you have enough computers. Also, encourage the students to find a generic equation for the tangent lines depending on x.

**Resources:**

Eves, H. (1990). *An introduction to the
history of mathematics.* United States: Thomson Learning, Inc.,
348-352.

Toeplitz, O.
(1967). *The calculus: a genetic
approach*. Chicago: The University of Chicago Press, 77.