Exploration: Exponential Functions and Their Derivatives

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This exploration uses a small sample of the applets in the Mathlets package together to explore exponential functions and their derivatives in depth.
Part 1 of this exploration uses the Exponential Functions applet to show graphs of the general exponential equation y=Cekx, using the positions of two points on the graph to determine the values of the constants C and k.
Part 2 uses the Limits applet, applied to the limit of a difference quotient for the function f(x)=ex, to compute the derivative of this exponential function, f '(x)=ex.
Part 3 uses the Tangent Lines applet to show how the derivative formula from Part 2 can be used to compute the slopes of tangent lines for the exponential function.
Part 4 uses the Tangent Lines applet again to show that the derivative of an exponential function is not computed using the Power Rule for derivatives, by showing that the resulting slopes do not correspond to tangent lines.
Part 5 uses the Differential Equations applet, applied to the differential equation y'=y (suggested by the derivative formula from Part 2) to show from the graphs that solutions of this differential equation are, in fact, exponential functions.

This exploration, and the applets in the Mathlets package, were written by:
Tom Leathrum
Assistant Professor of Mathematics
Department of Mathematics, Computing, and Information Sciences
Jacksonville State University
Jacksonville, AL
[For copyright information, see the Mathlets main page.]