Secant to Tangent

The secant line runs through $(a, f(a))$ and the other end of an interval of width $w$. Press Right, Left, or Both to set where the interval sits around $a$. Shrink divides the width by 10. The worked column computes the average rate of change over the interval, as in the Lesson 25 Notes.

The shrinking-intervals table gives the average rate of change from each side at widths 1 down to 0.00000001. The table marks the first width where both sides round to the same value at three decimals and both sides at the next smaller width round to that value too. These two widths meet the stopping test of the Lesson 25 Notes. The rounded average rate of change at the marked width is the derivative estimate. Show tangent line draws the tangent line at $a$ and writes its equation.

A typed function is read with R’s rules. A makeFun line works too.

Under More, Show earlier secants keeps a faint copy of the secant line for each wider interval in the table.

Show R gives the R code that builds the average rate of change as a function f_change of the two ends and evaluates it on the interval.

Keys: press [ or ] to pick the point at $a$ or the other end. Press the left or right arrow key to move it one step. Hold Shift with an arrow key to move it five steps. Press Esc to let go. Press + or − to zoom.