# Exploring the properties of 'e' and the graph of f(x) = e^x

- Author:
- Andy Greasley

Investigating the gradient properties of f(x) = e^x

1. As you change the base (b) describe what happens to the graph of f(x)
2. What is the y-intercept for f(x), is this true for any value of b? Why/why not?
3. The value of e is defined such that if f(x) = e^x, then f'(x) = e^x. In other words, with graph of e^x, the gradient at any point on the curve is equivalent to the y-value. What would the gradient of of f(x)=e^x be at x=0?
4. Based on Q3, move the slider, and look at the gradient of the tangent line, can you estimate the value of 'e'.
5. Once you have estimated the value of e, push the button to set the graph to exactly f(x) = e^x. Was your estimate close? Move the point along the curve and compare the gradient at that point to the y-value of that point.