This worksheet is designed to strengthen the conceptual
understanding of the chain rule for differentiation of functions of a
single variable. The worksheet uses an applet that can be found
by following the link provided:
Follow the link and click on the "Chain Rule" button to launch the applet.
1) The first issue to work through is to connect the graphs of
the functions f, g, and h=f(g). Spread the windows out and drag
your mouse back and forth in the "3D View" window to see various
views and to verify that the green curve is the intersection of
the red y=g(x) surface and the blue z=f(y) surface. Each surface
has a corresponding lighter colored tangent plane, but we will ignore
them for now.
In the control panel, uncheck the "f(y)" box. The blue curves and surfaces should disappear. Rotate the "3D View" window so that you are looking directly up the z-axis. (This can be done by dragging, or by selecting the obvious menu item fro the "View" menu. From that viewpoint, describe the relationship of the red and green curves. What is the relationship of these curves with the pink tangent plane? How does what you see in the "3D View" compare with what you see in the "2D View"?
Now unclick the g(x) box, recheck the f(x) box and repeat the questions above with the natural color changes. In this case we want to look up the x-axis.
Now turn both f(x) and g(x) off, look up the y-axis, and compare the two views.
2) Zoom in by taking del down to 0.001. Notice that at this
resolution the surfaces are indistinguishable from their tangent planes
and the curves are indistinguishable form their tangent lines.
Now look at the tangent lines window. Give the heights where the three tangent lines intersect the line x=1.
What is the radius of the pink circle?
Using the readouts in the control panel, give the coordinates of the point where the blue line intersects the line x=g'(P_1). How does this compare to other values we have noted?
How does this compare with the value we expect from the chain rule? Use the scroll button in the control panel to change the value of t0. What happens to your pictures?
3) Now experiment with pairs of functions that you know are inverses. Start with g(x)= x^2 and f(x)=sqrt(x). (Technical detail - be sure to hit enter with the cursor in both the f(x) and the g(x) box to register the change.) Using the control panel, scroll t0 from -1 to 1. Do you get the expected values? Are the two functions inverses of each other? Repeat the process with f(x)=asin(x) and g(x)=sin(x).
Return to the SLU Calculus Applet page.
Last updated By Mike May, S.J., February 11, 2006.