A2.6

determine, through investigation using technology, the exponential function $f(x)=a^x$ $(a>0,a\ne 1)$ for which $f^{\prime }(x)=f(x)$ (e.g., by using graphing technology to create a slider that varies the value of $a$ in order to determine the exponential function whose graph is the same as the graph of its derivative), identify the number $e$ to be the value of $a$ for which $f^{\prime }(x)=f(x)$ [i.e., given $f(x)=e^x$, $f^{\prime }(x)=e^x$], and recognize that for the exponential function $f(x)=e^x$ the slope of the tangent at any point on the function is equal to the value of the function at that point Sample problem: Use graphing technology to determine an approximate value of $e$ by graphing $f(x)=a^x$ $(a>0,a\ne 1)$ for various values of $a$, comparing the slope of the tangent at a point with the value of the function at that point, and identifying the value of $a$ for which they are equal.