## Understanding the Derivative as the Limit of the Slope of a Line Segment (in R!)

Sometimes we want to know about the instantaneous rate of change at a point on a function. Most (but not all) of the time, this is because we want to know the maximum or minimum (or maxima or minima) of a function, which occur when the instantaneous rate of change is zero. This interactive R code walks through how we get from looking at the slope of the line segment that connects two points around the point of interest to the definition of the derivative. (It follows Section 5.3 in Gill 2006 very closely, with a bit of borrowing from Section 5.3 in Moore and Siegel 2013.)

Directions: (1) Scroll to the bottom and click "Activate Console". This will allow you to copy and paste the code, line by line, into the console. Make sure you run each line in the order the appear, as lines later in the code rely upon lines earlier in the code.

(2) You can enlarge the console space by dragging the horizontal line upward.

(3) When you finish running the code as is, feel free to change the function, the estimation point, and the bounds. This will allow you to explore the idea of the derivative a little further and try to get a better feel for it. When you reload the website, it should reset to the original code.

Directions: (1) Scroll to the bottom and click "Activate Console". This will allow you to copy and paste the code, line by line, into the console. Make sure you run each line in the order the appear, as lines later in the code rely upon lines earlier in the code.

(2) You can enlarge the console space by dragging the horizontal line upward.

(3) When you finish running the code as is, feel free to change the function, the estimation point, and the bounds. This will allow you to explore the idea of the derivative a little further and try to get a better feel for it. When you reload the website, it should reset to the original code.