In Problem 1, students investigate the how the value of b affects the shape of the graph f(x) = b^x.

In Problem 2, students use a tangent line to the curve at a point to explore the relationship between the slope of the tangent line at that point and the value of the function at the same point.

In the third problem, students combine the ideas of the graphs from the previous two problems. They observe the changing values to form conclusions about how the slope of the tangent line and the value of the function are related.

At the end of this activity, students will discover that there is exactly one value of b, Euler’s number, for which the slope of the tangent and value of the function are equal.

Students will also explore and model applications of exponential growth in given situations.