Explore the shape of the gradient function for various exponential functions [math]y = a^x[/math]. The graph of [math]y = a^x[/math] is shown, a point P on the curve is shown with the tangent at that point. The gradient of the tangent is measured, and the value of the gradient is plotted at Q.
Choose a value for a. Press the "trace Q" button. Use the slider to adjust P. 1. What do you notice about the shape of the gradient function of [math]y = a^x[/math]? Use the slider to set P to be 0. Press "don't trace Q". Use the slider to adjust a. 2. There is an important value of [math]a[/math] which leads to a special result. Write down the important value of [math]a[/math] as accurately as you can. Keep a on this important value. Press "Trace Q". Use the slider to adjust P. 3. What do you notice about the gradient function of [math]y = a^x[/math] for this important value of [math]a[/math]?