Unit Plan: Introduction to Calculus
To understand the underlying ideas about derivatives conceptually through graphs, geometry, and algebra.
Review some geometric properties of circles and tangents of circles and introduce the utilities of graphing calculator, zooming and scrolling along som discrete values of n such as to change the size or location of a circle, while exploring some of the ideas about circles in the process.
Day 2-3:(depending on class length and how far you get)
Begin the introduction to tangents of curves involving the fairly simple curve of y=2x^.5 at the particular point (1,2) by using the properties of tangents of circles and algebra. HERE!
The graphing calculator file is already set up here for after students have explored some ideas on their own as directed in the lesson plan. (*For some reason the gcf file won't come up so I set it up as a webpage so you could at least see the equations and the starting picture to get an idea of what it looks like).
Students work together to develop a general equation for determining the tangent points using the Descartes' Method from the previous activity depending upon the variable x.
The class should graph this curve and examine it's properties.
If the class progresses well, then the students should explore other types of curves beginning with transformations of the square root curve.
The class discusses the effects of transformation done to the curve on the tangent curve equation and its graph.
They also further explore if they didn't have much time on day 4 other types of curves and discuss which types of curves seem best suited for this method and which are difficult to determine.
The class then is introduced to the typical methods behind finding tangents to curves. On day 6, they review the idea of slope in general and how it is found for a line and then begin to use this idea for a simple curve such as y=x^2 at particular points and then generalize. Here (*) is a graphing calculator file that can be used similarly to the one above for this curve at the point (1,1). It graphs a line through that point and another point on the curve depending on the slider value and calculates the slope so that students can see what the slope is when the line looks fairly close to tangent. The line is in slope-intercept form so it can be adjusted for different values along the curve for which you may want to find the tangent.
On day 7, they begin to try this method on other curves and discuss some general findings they discover about the relationship between the curve and the tangent curve.
The students are introduced to the formal definition of the derivative involving limits (possibly with a review of limits). They determine the derivatives of some of the curves they explored earlier and prove some of the discoveries relating to patterns they found and make further generalizations.