Section 1.1 - Limits, a Heuristic Definition

The graph below is the graph of the function $f(x) = .25x^4$. I'd like to know the slope of the tangent line to the graph at $x = 0.5$. Unfortunately, we have no way to find that since our slope formula requires two points. What we can do is get an approximation of the value by looking at the slopes of secant lines that are really close to the tangent line.

The slope formula isn't a difficult one to work with but calculating multiple values can get tedious. Fortunately, computers are really good at doing tedious things. If you enter an x value into the textbook below and click the "Add Secant" button, the page will calculate the slope of the secant line through $(0.5, f(0.5)$, the point that we're interested in, and the point with the x value you entered. Try picking several x values each one a little closer to $x=0.5$ and see if you can spot a trend in the slope's. We'll talk about what's going on here in the first lecture on the next page.